# Critical dynamics of long range models on Dynamical Lévy Lattices

Riccardo Aiudi  Dipartimento di Scienze Matematiche, Fisiche e Informatiche,
Università degli Studi di Parma, Parco Area delle Scienze, 7/A 43124 Parma,
Italy  INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, 43124
Parma, Italy  Raffaella Burioni  Dipartimento di Scienze Matematiche, Fisiche
e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze, 7/A
43124 Parma, Italy  INFN, Gruppo Collegato di Parma, Parco Area delle Scienze
7/A, 43124 Parma, Italy  Alessandro Vezzani  Istituto dei Materiali per
l’Elettronica ed il Magnetismo (IMEM-CNR), Parco Area delle Scienze,
37/A-43124 Parma, Italy  Dipartimento di Scienze Matematiche, Fisiche e
Informatiche, Università degli Studi di Parma, Parco Area delle Scienze, 7/A
43124 Parma, Italy  INFN, Gruppo Collegato di Parma, Parco Area delle Scienze
7/A, 43124 Parma, Italy

###### Abstract

We investigate critical equilibrium and out of equilibrium properties of a
ferromagnetic Ising model in one and two dimension in the presence of long
range interactions, $J_{ij}\propto r^{-(d+\sigma)}$. We implement a novel
local dynamics on a _dynamical_ Lévy lattice, that correctly reproduces the
static critical exponents known in the literature, as a function of the
interaction parameter $\sigma$. Due to its locality the algorithm can be
applied to investigate dynamical properties, of both discrete and continuous
long range models. We consider the relaxation time at the critical temperature
and we measure the dynamical exponent $z$ as a function of the decay parameter
$\sigma$, highlighting that the onset of short range regime for the dynamical
critical properties appears to occur at a value of $\sigma$ which differs from
the equilibrium one.

††preprint: APS/123-QED

##  I Introduction

Systems with non-local long range interactions are known to give rise to
interesting physics in classical [1] and quantum [2] many-body regimes, both
at equilibrium and out-of equilibrium. Long range interactions can induce
spontaneous symmetry breaking even at low dimensions [3] and, at continuous
phase transitions, a sufficiently slow decay can modify the universality
classes, resulting in critical exponents which depends on the interaction
decay [4]. Out of equilibrium, long range interactions modify the dynamical
exponents in coarsening phenomena, [5, 6, 7, 8], also giving rise to non
trivial metastable states that can affect the dynamics [9, 10].

A wide interest in the field has been devoted to the investigation of magnetic
models on lattices in the presence of long range interactions of the type
$J_{ij}\propto r^{-(d+\sigma)}$. The accepted description of the critical
properties in the ferromagnetic case was given by Sak [11]. For $\sigma<d/2$,
the critical behavior is mean field, while in the so called ”long range”
regime, $d/2<\sigma<2-\eta_{SR}$, the critical exponents depend non-trivially
on $\sigma$ ($\eta_{SR}$ is the exponent of the critical correlation function
in the short range model). For large enough $\sigma>2-\eta_{SR}$, the short
range behavior is recovered.

In this class of models, very often studies are based on numerical approaches.
In _physical_ long range models, interactions involve all degrees of freedoms
since all pairs of sites interact and thus they form a fully connected model
on a complete graph with weighted links. This implies that the number of
interactions scales as $O(N^{2})$, requiring large resources for simulations.
In recent years, several solutions to this problem have been proposed. Cluster
algorithms [12, 13] can simulate long range model with a computational cost of
$O(N\log{N})$ or $O(N)$ and reduce the relaxation time, so that they are not
influenced by critical slowing down, at the price of a non local dynamical
evolution. On the other hand the kinetic Monte Carlo [7, 8, 9, 10] provides an
effective tools for simulating a local dynamics only at very low temperatures.

Another interesting solution are Lévy lattices [14, 15, 16, 17]. These are
random diluted graphs with interactions between pairs which are constant and
occur with probability $\propto r^{-(d+\sigma)}$, with the total number of
interactions being $O(N)$. Lévy lattices drastically reduce the computational
cost while keeping a local dynamics, however an average over different
realizations is required. In $1$ dimension, Lévy lattices seem to fall in a
different universality class than their fully connected counterparts, due to
long range correlations induced by the disorder in the random lattice
realization [18], while in $2$ dimension consistent results have been obtained
in the XY model [17, 19].

In this paper, we introduce an alternative local dynamics for the long range
model based on a _dynamical_ Lévy lattice, that is a dynamical sampling at
each time step of the long range interacting model, in the spirit of the
q-Ising model [20, 21]. In a nutshell, each spin interacts with a constant
interaction J only with q neighbours, randomly drawn from the corresponding
long range probability distribution. Unlike Lévy lattices, the underlying
graph is not fixed before the dynamics take place, but evolves dynamically
with the system and it is built during the simulation. The system can be
thought of as living on a temporal Lévy graph [22, 23, 24], in which at each
time step the underlying structure evolves and it is reshuffled. As a result,
the long range correlations characterizing the single disordered realization
of static Lévy lattices are eliminated. Such correlations are known to deeply
affect the behavior of the system, at least in low dimension as shown in [18].
Moreover, in the dynamical Lévy lattice physical quantities are computed by
averaging over the full dynamic evolution of the model and thus feature the
same symmetries of the statistical model on the fully connected graph. This is
expected to provide a significant numerical advantage with respect to
simulations on the static Lévy lattice in particular at small $\sigma$, which
are affected by large fluctuations in different quenched realizations at
finite size [19].

On the other hand, in the dynamical Lévy lattices detailed balance is not
guaranteed, so we need to compare our results with that of static fully
connected lattices to show that they belong to the same universality class.
Therefore, we first test the validity of the new algorithm by studying
numerically the critical behaviour of $1$ and $2$-dimensional Ising model in
the presence of long range interactions, comparing the results with the Sak’s
scenario and with state-of-the-art simulations [25, 26, 27, 28, 29]. We then
present a novel numerical measure of the dynamical exponent $z$ at the
critical temperature, which governs the behaviour of the relaxation time at
criticality. This is a critical dynamical quantity that cannot be measured
with the non-local cluster algorithm [12, 13] nor with the kinetic Monte Carlo
[7, 8]. We consider the full interesting range of $\sigma$. In the mean field
regime $\sigma<d/2$, we show that $z=d/2$, consistently with the relaxation
properties of a Curie Weiss model. At $\sigma=d/2$, the super-diffusive
behavior $z=\sigma$, characterising a free random walk with long range motion,
is recovered. At larger $\sigma$ interaction starts to play a role, giving
rise to perturbative contributions and, interestingly, the transition to the
short range behavior seems to occur at a value of $\sigma$ larger than that
for the short range regime at the equilibrium.

The paper is organized as follows: in section II we briefly describe the long
range model and we summarize the existing algorithms with their features and
limitations. In section III we present our dynamics and in section IV we
numerically test its validity comparing the results with the Sak’s prediction.
In section V we present our results for the dynamical critical exponent $z$.
Finally, in section VI we summarize and discuss future perspectives of our
work.

##  II The long range Ising Model

The ferromagnetic Ising model with long range interaction on a hypercubic
lattice of dimension $d$ is described by the Hamiltonian:

$H=-\frac{1}{2}\sum_{i,j=1}^{N}J_{ij}\sigma_{i}\sigma_{j},$ (1)

where $\sigma_{i}$ are the usual Ising spins and $J_{ij}\propto
r_{ij}^{-(d+\sigma)}$, with $r_{ij}$ the euclidean distance between $i$ and
$j$, $\sigma$ being the parameter for the decay of the long range interaction.
The interactions have here infinite range and connect all the spin, i.e. the
system is fully connected. According to Renormalization Group calculations [4,
11, 30], the model exhibits three different behaviours depending on the value
of $\sigma$:

  * •

$\sigma\in(0,\frac{d}{2})$: the system is in a mean field (MF) regime;

  * •

$\sigma\in(\frac{d}{2},2-\eta_{SR})$, where $\eta_{SR}$ is the critical
exponent $\eta$ characterizing the decay of the correlation function at
criticality in the short range model in $d$ dimension: the system belongs to a
universality class different from the MF one and the critical exponents depend
on the value of $\sigma$. This region will be referred as the long range (LR)
region;

  * •

$\sigma>2-\eta_{SR}$: the interaction decays so fast that the system becomes
equivalent to the corresponding short range (SR) model;

The boundary $\sigma=2-\eta_{SR}$ between LR and SR regions is indeed the main
point of Sak’s prediction. In $1$ dimension the LR phase is observed for
$1/2<\sigma<1$, while for $\sigma>1$ the system does not present a critical
transition at finite temperature, and at $\sigma=1$ a Kosterlitz-Thouless
transition occurs [31] with non trivial features [32, 33].

### Numerical approaches to long range models

As mentioned above, the simulation of the fully connected model is
computationally costly, and several algorithms and techniques have been
introduced to study long range models numerically. We now briefly discuss the
approaches introduced so far.

Cluster Algorithms. Following the idea of Swendsen-Wang and Wolff [34, 35],
cluster algorithms have been designed to simulate long range Ising models with
a computational cost of $O(N)$ [12, 13]. The basic step of these algorithms is
to flip arbitrary large clusters of spins by preserving the detailed balance
and reducing the relaxation time. Indeed large clusters are updated in a
single step and uncorrelated configurations are rapidly obtained even at the
critical point. This avoids the critical slowing down, which is the typical
bottleneck in terms of computational cost. This algorithm has been used to
test the Sak’s scenario and to confirm that it gives the correct description
of the critical behavior in the Ising case [25, 26, 27, 13, 29]. However, as
there is no local dynamics, the algorithm cannot be used to study the temporal
evolution. Moreover, identification of clusters is straightforward for the
Ising model while it cannot be implemented in models with continuous
variables. Recently, a cluster algorithm has also been introduced to study
long-range percolation [36].

The Kinetic Monte Carlo Algorithm. Recently an implementation of the Kinetic
Metropolis algorithm has been introduced for the long range Ising model [7, 8,
9, 10]. In this local algorithm, the time intervals in which no spin flip
occurs are directly estimated after each move and the rejection rate vanishes.
The implementation turns out to be very efficient at low temperatures, when
spin flips are very unlikely, while at higher temperatures (e.g. at
criticality) the efficiency is comparable to standard Metropolis. The Kinetic
Monte Carlo Algorithm has been applied to study non-equilibrium properties in
coarsening dynamics, that is the evolution of the system quenched from
$T=\infty$ to a temperature lower than the critical one [5, 6].

Lévy lattices. The Lévy lattice is a diluted graph with adjacency matrix
$A_{i,j}=0,1$, built to display the same properties of the long range fully
connected model [14, 15, 16, 17]. In particular, the long range ferromagnetic
model on a $d$-dimensional lattice is approximated by a graph in which two
sites are connected with a probability proportional to the long range
interaction, i.e. $A_{ij}=1$ with probability $P_{ij}\propto
r_{ij}^{-(d+\sigma)}$. On the diluted Lévy lattice, the usual Monte Carlo
simulations are implemented with saving of computational cost, since each node
is connected to a finite number of edges. There are no analytical proofs that
models on Lévy lattices are equivalent to their long range fully connected
counterparts. In particular, for $1$-dimensional lattices, numerical evidences
suggest that static long range correlations can modify the critical properties
in the free model, changing the spectral dimension [18]. On the other hand, in
the $2$-dimensional case, the spectral dimension of the Lévy lattice seems to
coincide with that of the long range, suggesting that the difference is
limited to low dimensions [17, 37]. At difference with the cluster algorithm,
the Lévy diluted lattice can be used to simulate continuous symmetry models.
Consistent simulations have been performed for the $2$-dimensional $XY$ model
with long range interaction [17, 19], in the presence of a Kosterlitz Thouless
phase transition [38]. On finite systems, simulations on Lévy lattices
strongly depend on the graph realization and averages over a large number of
samples are necessary to obtain stable results. In this context the problem of
self averaging is still an open issue; moreover the averaging procedure can be
very demanding from a computational point of view [19].

##  III The Dynamical Lévy lattice

The starting point of our approach is the q-Ising model [20, 21]. In this
algorithm, at each time step a spin $\sigma_{i}$ is chosen randomly and it
interacts with the field produced by $q$ neighbors, which are also randomly
drawn uniformly among the remaining $N-1$ spins. The spin $\sigma_{i}$ flips
according to a Metropolis [39] or Glauber [40] prescription. As shown in [41]
this model has two fluctuating variables, the spins and the links. The former
are in contact with the heat bath, while the links are randomly rewired during
the dynamics, without any acceptance-rejection procedure. The links can be
considered as being in thermal contact with a heat bath of temperature
$T=\infty$, which implies rewiring with probability $1$. With two different
heat baths governing the dynamics, detailed balance is not satisfied and
thermal equilibrium is not obvious.

Our idea is to simulate a long range $q$-Ising model, which we expect to
belong to the same universality class as the long range Ising model. In the
Dynamical Lévy lattice (DLL), first we select the spin $\sigma_{i}$ and
thereafter its $q$ neighbors are randomly chosen from the nodes of the
lattice, based on a power-law probability distribution $P_{ij}\propto
r_{ij}^{-(d+\sigma)}$. Then $\sigma_{i}$ flips, according to a constant
interaction $J$ with the $q$ randomly selected spins. Note that in this way
the decay of the interaction as a power law is recovered in a statistical
sense. In particular, the model can be regarded as living on a _dynamical_
Lévy lattice, because the graph has an adjacency matrix which evolves with
time, i.e. $J_{ij}\rightarrow J_{ij}(t)=J\times A(i,j,t)$, where $A(i,j,t)=1$
if at time $t$ the spin $\sigma_{j}$ is drawn as a neighbour of $\sigma_{i}$
and $0$ otherwise.

We observe that, by definition, random walks on a DLL exhibit the same
behavior as random walks on the fully connected long-range graph: in this
case, in fact, the walker jumps at each step from the starting site $i$ to any
nodes of the network $j$ with probability $P_{ij}\propto
r_{ij}^{-(d+\sigma)}$. This is exactly the same probability which is used on a
DLL at each step to choose the $q$-neighbors of the walker, and then the jumps
occur uniformly among these $q$ nodes. So the two dynamics coincide and,
hence, the spectral dimension, as measured from the return probability of the
random walker [42, 43], are the same on the fully connected graph and on DLL.
In contrast, on a Lévy lattice the walker evolves on a random static network,
thus in the presence of correlations. For example, when the walker crosses a
link connecting two long-distance sites, it has a significant probability of
going back along the same link that remains active on the static graph, while
it fails to reach in a few steps the lattice sites that are not connected by
long-distance links in that specific quenched realization. This induces static
long-range correlations that are not present in either the fully connected
lattice or the DLL. In particular, such correlations in $1$ dimension [18] are
able to modify the spectral dimension with respect to the fully connected long
range model.

Similarly, in the DLL the correlation functions
$\langle\sigma_{i}\sigma_{j}\rangle$ are computed by averaging over the
dynamic evolution of the model and thus naturally possess the symmetries of
the statistical model on the fully connected graph. In contrast, on a Lévy
lattice, the $\langle\sigma_{i}\sigma_{j}\rangle$ depend on the quenched
realization of the random structure and disorder breaks the original
translation invariance. Therefore, the original spatial symmetries are
recovered only after averaging over several realizations of the random
lattice. We observe that this averaging procedure can also be very numerically
demanding [19].

The method we propose appears very flexible and can be applied to several
statistical models both with discrete and continuous symmetry. Finally, the
algorithm displays the same efficiency at any temperature even if, due to its
local nature, we expect to observe the typical slowing down at criticality. An
important point is that since on the DLL detailed balance is not valid and the
equilibrium distribution is not known, the equivalence in terms of critical
exponents with the fully connected long range system has to be checked.

##  IV Equilibrium critical properties

Accordingly to Renormalization Group calculations, ferromagnetic long range
models have critical exponents which depends on the parameter $\sigma$ [4, 11,
30]. The three different regimes (MF, LR and SR) can be characterized by their
critical exponents, for example the one governing the scaling of the magnetic
susceptibility $\chi$, which can be defined as the magnetization $M$
fluctuations, $\chi\propto\langle M^{2}\rangle-\langle|M|\rangle^{2}$. We call
this exponent $y$.

For $\sigma<\frac{d}{2}$ the susceptibility exponent is equal to the MF one,
$y=\frac{d}{2}$ [26]; for $\frac{d}{2}<\sigma<2-\eta_{SR}$ the system exhibits
long range behaviour and the exponent is related to that of the spatial
correlation $2-\eta$ and depends on $\sigma$: $y=2-\eta=\sigma$; finally for
$\sigma>2-\eta_{SR}$ the system is equivalent to the corresponding short range
model, and so the exponent: $y=2-\eta=2-\eta_{SR}$. The Sak’s prediction for
the boundary between LR and SR regions is widely considered valid and many
numerical studies corroborate it [25, 26, 27, 29]. Thus, one way to test the
validity of the Dynamical Lévy lattice is to measure this exponent and to
compare it with the expected one. It is known that strong finite size effects
are present in long range models [28, 29]. One way to proceed is to include
higher order terms in the scaling form of the thermodynamic functions [29].

Close to criticality the scaling form of susceptibility is

$\chi\sim L^{y}\cdot\tilde{\chi}(tL^{\frac{1}{\nu}}),$ (2)

where $L$ is the linear size of the system, $t=\frac{T-T_{c}}{T_{c}}$ the
reduced temperature ($T_{c}$ is the critical temperature) and $\tilde{\chi}$
is a scaling function. In infinite systems $\chi$ diverges at $T_{c}$, but in
finite ones there is a size-dependent temperature $T_{c}(L)$ where it has a
maximum. According to Eq. (2), this maximum follows a power law behaviour
$\chi_{max}(L)\sim L^{y}$ which can be efficiently used to extrapolate the
exponent $y$ (see Appendix B).

###  IV.1 $1$-Dimensional chain

We first focus on the $1$-dimensional spin chain (see Appendix A for details).
We consider sizes from $L=2^{8}$ to $L=2^{15}$ and five values of the $\sigma$
parameter: $0.35$ for the MF region, $0.6$, $0.8$ and $0.9$ for the LR one and
finally $1.2$ for the SR region, where the $1$-dimensional classical Ising
model behaviour is recovered and the system does not undergo a phase
transition. We check this behaviour by studying the absolute value of the
magnetization per spin $m$, for different sizes. We find that $m$ goes to zero
for each temperature as the size is increased, as it is clearly shown in
Figure 1. For the other values of $\sigma$ we find good agreement with
theoretical expectations and we collect the results in Table 1 and Figure 2.
An interesting regimes in the $1$-dimensional case is observed when $\sigma$
approaches one and a Kosterlitz-Thouless transition occurs [31]. However, in
this limit we observe that simulations are quite demanding (see the large
error on the exponents in Figure 2) and a detailed study requires further
investigations.

Figure 1: $|m|$ behaviour with the temperature as we increase the system size
for $\sigma=1.2$ ($1$-dimensional case). Figure 2: $2-y$ critical exponent
extrapolated with our method, for the $1$-dimensional case. The black dashed
line is the theoretical expectation. Numerical data are provided in Table 1 in
Appendix D.

###  IV.2 $2$-Dimensional lattice

In the $2$-dimensional case we consider a wide range of $\sigma$’s in the MF,
LR and SR regimes. The results for the critical exponent $\eta$ are summarized
in Figure 3 and show a deviation from the theoretical prediction for value of
$\sigma$ at the LR-SR boundary, where it is well known that there are large
finite size effects as pointed out in [28, 29] and the finite system is
sensitive to boundary conditions (see Appendix A for discussion on boundary
conditions). Indeed, we analyzed the same range of $\sigma$ and systems of the
same size in the fully connected long-range model, using the traditional
cluster algorithm of Luijten and Böthe, and we verified that in the measure of
$\eta$ comparable finite size effects are present. Angelini et al. [29] indeed
argue that near $\sigma=1.75$ it is necessary to consider higher-order terms
in the correction to scaling. In particular, they consider a scaling form of
the type

$\chi\sim L^{2-\eta}(a+bL^{-\delta}),$ (3)

where $a,b,\delta$ are parameters which in principle could depend on $\sigma$.
They found that at $\sigma=1.75$ a value of $\delta\approx.42$ accounts for
the correct scaling correction and the analytical predictions of Sak´s
renormalization group are verified. In Appendix B we show indeed that a
correction to the scaling with $\delta=.42$ is consistent with our simulation
even for the other values of $\sigma$. In particular, Figure 3 shows that
taking into account this correction to the scaling we obtain a very good
agreement with Sak’s predictions in the whole range of $\sigma$.
Interestingly, in Appendix B we also show that the ratio $a/b$ between the
coefficients defined by Eq. (3) strongly depends on the exponent $\sigma$ and
it displays a sharp minimum at $\sigma=1.75$. The presence of such a minimum
is an evidence that finite size effects are particularly relevant at the
transition between the SR and the LR regimes. Our results are summarized in
Tables 2, 3 and Figure 3. More details about the numerical analysis and error
estimate can be found in the Appendix B.

We observe that the DLL, due to its locality, at criticality cannot be as
efficient as the cluster algorithm in [12] which involves non-local Montecarlo
moves. Here, however, our goal was to show that the algorithm provides the
expected equilibrium critical properties. In the next section, we use our
local algorithm to study the dynamical critical properties that cannot be
approached by non-local algorithms.

Figure 3: $2-y$ critical exponent extrapolated with our method, for the
$2$-dimensional case. The black dashed line is the theoretical expectation.
The brown circles are the results of our algorithm (DLL) and the blue star is
the value found when using next to leading order corrections $(\delta=0.42)$.
Numerical data are reported in Tables 2 and 3 in Appendix D

##  V  The dynamical critical exponent

###  V.1 Scaling analysis

We now turn our attention to long range dynamical properties at criticality, a
regime inaccessible to both cluster algorithms and kinetic Monte Carlo
approaches. We concentrate on the dynamical critical exponent $z$, which
governs the relaxation time. Indeed, near the critical point the
autocorrelation time $\tau$, which is the Monte Carlo time needed to get two
statistically independent samples, diverges as a power law $\tau\sim\xi^{z}$,
where $\xi$ is the correlation length. When dealing with a critical finite
size system, the only characteristic length scale is its linear size $L$,
which implies

$\tau\sim L^{z}.$ (4)

In this perspective we compute $\tau$ from the autocorrelation time of the
absolute magnetization per spin.

We expect $z$ to show different values in the three regimes considered in
equilibrium, as outlined in the Sak’s scenario. In the MF region, the system
behaves as a Curie-Weiss model. The underlying geometry disappears and the
only remaining information is the total number of spins, i.e. the volume $V$.
A simple argument shows that for the Curie-Weiss model, the relaxation time
goes as $\tau\sim V^{1/2}$. Let us consider the Fokker-Planck equation for the
magnetization:

$\displaystyle\frac{\partial P(m,\tau)}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
m}\Big{[}\,\Big{(}A(T-T_{c})m+Bm^{3}\Big{)}P(m,\tau)\Big{]}\,$ (5)
$\displaystyle+\frac{\tilde{D}}{V}\frac{\partial^{2}}{\partial
m^{2}}P(m,\tau),$

where $A$ and $B$ are arbitrary constant and the diffusion coefficient depends
on the system size as $\frac{\tilde{D}}{V}$. Then, at criticality ($T=T_{c}$),
the equilibrium distribution has the form
$P^{(eq)}(m)=Be^{-\frac{C}{V}m^{4}}$, which implies $\langle m^{2}\rangle\sim
V^{-1/2}$. Since at criticality the magnetization remains small and the
dynamics is expected to be purely diffusive, if we initialize the system at
$m=0$ we get that $\langle m^{2}\rangle\approx\frac{D}{V}t$. Thus we can
conclude that the time needed to reach the equilibrium is $t\sim V^{1/2}$.
Consequently, in the MF region we expect $\tau\sim L^{d/2}$. At the crossing
point $\sigma=d/2$, Renormalization Group calculations show that the model is
Gaussian and the system is in a free super-diffusive regime in which the
dynamical exponent $z$ is expected to be

$z=\frac{2d}{d_{s}},$ (6)

where $d_{s}$ is the spectral dimension, as defined in [42]. In long range
systems the spectral dimension is exactly $d_{s}^{(LR)}=2d/\sigma$ [44], which
means $z=\sigma$. As soon as $\sigma>d/2$ we expect the free random walk
behavior $z=\sigma$ to be perturbed by the presence of interactions, leading
to a slightly larger value of $z$. This is consistent with the behavior of
short range models, in which a small perturbation to the free diffusion is
observed. In $d=1$ such a perturbed diffusive behavior should characterize the
critical dynamics up to $\sigma=1$ where critical dynamics disappears. In
$d=2,3$ we know that for the equilibrium critical exponents the short range
behavior is recovered at $\sigma=2-\eta_{\rm SR}$. On the other hand, the non
interacting dynamics of free random walks shows anomalous diffusion $z=\sigma$
up to $\sigma=2$ where normal diffusion ($z=2$) is recovered. In this
perspective, for the dynamical exponent $z$ is not clear if the short range
behavior is recovered at $\sigma=2-\eta_{\rm SR}$ or at $\sigma=2$. We show
that our simulations seem to support the second hypothesis.

###  V.2 The numerical measure of the autocorrelation time

Let us call $m_{i}$ ($i=1\dots N$) the time series of the magnetization at a
Monte Carlo step. The autocorrelation time $\tau$ of $m_{i}$ is related to the
error $\mathcal{E}$ on the mean $\overline{m}$ by the following relation [45]:

$\frac{\mathcal{E}^{2}}{\mathcal{E}_{1}^{2}}=\tau,$ (7)

where
$\mathcal{E}_{1}^{2}=\frac{\sum_{i=1}^{N}(m_{i}-\overline{m})^{2}}{N(N-1)}$ is
the estimate of the error on $\overline{m}$ as if the system was uncorrelated.
The error $\mathcal{E}$ can be estimated by using the the Jackknife resampling
method. We construct new samples aggregating bigger and bigger temporally
consecutive blocks of magnetization. The aggregation is performed by taking
the mean value of the block and then we calculate the error of the new sample,
i.e.

$\mathcal{E}(t_{BS})^{2}=\frac{\sum_{i=1}^{N/t_{BS}}(m^{BS}_{i}-\langle
m\rangle)^{2}}{N/t_{BS}(N/t_{BS}-1)},$ (8)

where $t_{BS}$ indicates the block size and $m^{BS}_{i}$ is the mean value of
the $i$-th block of aggregated magnetization. At large enough block sizes
$t_{BS}$, $\mathcal{E}(t_{BS})^{2}$ turns out to be independent of $t_{BS}$
and the limit value of $\mathcal{E}(t_{BS})$ is the best estimator of the
error on the mean of the time series. At criticality in a finite system we
expect the model to scale with the only intrinsic time in the dynamical
evolution i.e. $L^{z}$. Since the block size of the Jackknife procedure is a
time length, that we introduce to probe the system, we obtain:

$\mathcal{E}_{N}(t_{BS},L)\equiv\frac{\mathcal{E}(t_{BS},L)}{\mathcal{E}(1,L)}=L^{z/2}\tilde{\mathcal{E}}_{c}(\sqrt{t_{BS}L^{-z}}),$
(9)

where $\tilde{\mathcal{E}}_{c}$ is a scaling function. Notice that for
$t_{BS}\gg\tau$ we have $\mathcal{E}_{N}(t_{BS},L)=\tau^{1/2}\sim L^{z/2}$, so
that $\tilde{\mathcal{E}}_{c}(x)$ is constant in the limit of large $x$. On
the other hand, for $t_{BS}\ll\tau$ the variance of the resampled system in
blocks $t_{BS}$ should be equal to the variance of the original time series.
Therefore from Eq. (8,9) we obtain $\tilde{\mathcal{E}}_{c}(x)\sim{x}$ for
$x\sim 0$. Eq. (9) has been obtained exactly at $T=T_{c}$. Close to
criticality we expect a general scaling relation
$\mathcal{E}_{N}(t_{BS},t,L)=L^{z/2}\tilde{\mathcal{E}}(\sqrt{t_{BS}L^{-z}},tL^{1/\nu})$,
where $t$ is the reduced temperature. Since our simulations have been
performed for each size $L$ at a temperature corresponding to the peak of the
susceptibility $\chi$, from Eq. (2), $tL^{1/\nu}=x_{M}$ is constant in the
different simulations. So we obtain
$\mathcal{E}_{N}(t_{BS},L)=L^{z/2}\tilde{\mathcal{E}}(\sqrt{t_{BS}L^{-z}},x_{M})$
where the new scaling function
$\tilde{\mathcal{E}}(\sqrt{t_{BS}L^{-z}},x_{M})=\tilde{\mathcal{E}}(\sqrt{t_{BS}L^{-z}})$
displays the same asymptotic features of the critical scaling function
$\tilde{\mathcal{E}}_{c}(\sqrt{t_{BS}L^{-z}})=\tilde{\mathcal{E}}(\sqrt{t_{BS}L^{-z}},0)$.

The numerical data are analyzed using the exponent $z$ that gives the best
collapse at different sizes $L$ for the rescaled functions
$L^{-z/2}\mathcal{E}_{N}$ as a function of $\sqrt{t_{BS}L^{-z}}$, see the
Appendix C for details and error estimate.

In Figure 4 we show an example with $\sigma=1.75$, before and after rescaling.
In upper panel, we clearly see the effect of critical slowing down, by looking
to the fast increasing plateau with the size. The rescaling is performed with
the best value of $z$, showing a collapse which improves with increasing
system size.

The method we introduced turns out to be quite efficient. In fact, for systems
with strong finite size effects, a good estimate of $z$ requires including in
the scaling analysis even sizes where the simulation times are not much larger
than the decorrelation time. This can be observed in figure 4 for $L=512$,
where a plateau is not yet reached at large block sizes $t_{BS}$ and the
decorrelation time cannot be directly calculated by Eq. (7) by measuring a
stable asymptotic value. Nevertheless, with our method the system at $L=512$
can be included in finite-dimensional rescaling, providing an important
contribution to the estimate of $z$. An alternative way to measure the $z$
exponents using even shorter simulations is through a critical quench [46], in
which equilibration of the system is not even required. This method, however,
implies knowledge of the static equilibrium critical exponents.

Figure 4: Upper: $\mathcal{E}^{2}_{N}$ for $\sigma=1.75$. The value of the
autocorrelation time is the plateau. Lower: Rescaled $\mathcal{E}_{N}$ with
$z=1.75$, the collapse improves with increasing size. The differences at small
sizes are due to finite size effects.

###  V.3 Results in $1$ and $2$ dimension

We collect the results for both $1$ and $2$-dimensional case in Tables 4 and
5, and in Figures 5 and 6. We recall that, in the $1$-dimensional case, there
is no phase transition in the SR region. In the $2$-dimensional case for this
region we compare our results with $z=2.14\pm 0.02$, as taken from [47].

The values obtained for the $1$-dimensional case are consistent with our
analysis. For the more interesting $2$-dimensional case, in Figure 6 the
darker line represents the short range behavior according the Sak’s hypothesis
on the onset of LR and SR regimes. If $\sigma\in(1.75,2)$ corresponds to SR
region, we should observe a rapid deviation from the free super-diffusive
behavior $z=\sigma$ so that the short range behavior should be recovered at
$\sigma=1.75$. Instead we found that it seems to be recovered smoothly at
$\sigma=2$. In this perspective, in order to obtain a clear picture on the
transition in the dynamical exponent from the LR to the SR regime, more
extensive simulations are required to clarify the picture in the region
$\sigma\in(1.75,2)$, together with some analytic Renormalization Group
argument which is beyond the scope of our work.

Figure 5: $z$ The dynamical critical exponent as obtained from DLL (see
Appendix C) for the $1$-dimensional case. The black dashed line represents the
theoretical prediction at leading order, as discussed in the text. Numerical
data are illustrated in Table (4) in Appendix D Figure 6: The dynamical
critical exponent as obtained from DLL (see Appendix C), for the
$2$-dimensional case. Again, the black dashed line represents the leading
order, following the Sak’s static prediction. The lighter dashed line
represents the leading order in the case where the SR behaviour is recovered
at $\sigma=2$. The dark horizontal line at higher values of $\sigma$ is the
value of $z$ with its error as found in [47], $z=2.14(2)$. Numerical data are
illustrated in Table (5) in Appendix D.

##  VI Conclusion and perspectives

In this work we have presented an analysis of the equilibrium critical
exponents and of the dynamical exponent $z$ at criticality for $1$ and
$2$-dimensional Ising models with long range interactions. The numerical
values of the exponents have been obtained using a new dynamical algorithm,
the Dynamical Levy Lattice, designed for the study of models with long range
interactions. The algorithm corresponds, in practice, to a Monte Carlo
analysis on a diluted temporal graph. The dynamics has relatively small
computational cost, avoiding the typical scalability problems of long range
interactions, preserves locality, and can be adapted to a wide range of
situations beyond Ising interactions. However, since detailed balance is not
satisfied and the stationary distribution is not known, the equivalence
between our model and the long range Ising equilibrium distribution has been
checked. Our results are in line with the present literature. Exploiting the
local nature of our algorithm, we then obtain a first measure of the dynamical
critical exponent $z$, which shows a peculiar behaviour near the LR-SR
crossing point. Interestingly, the SR regime seems to be reached at larger
values of $\sigma$ than in the equilibrium case. Extensive simulations and new
analytical arguments are needed to clarify this scenario. The effect of a
different choice of $q$ on the dynamics is still to be carefully tested. Our
choice $q=3$ is driven by the results of the fully connected model, in which
analytical calculations can be performed exactly. [20, 21] Our choice gives a
quite effective dynamics, however more efficient choices could be possible.
The locality of the algorithm makes it widely applicable to study out-of-
equilibrium properties, for example to investigate aging in critical quenches
or the effect of long range interaction in coarsening phenomena [8], in which
topology is expected to have non trivial effects such as pinning [9, 48, 49,
50]. Finally, the numerical approach can be easily extended to different
models. In particular, the XY model in $d=2$ is expected to show non trivial
features in the presence long range interactions, due to the three different
phases, i.e. paramagnetic, ferromagnetic and Kosterlitz Thouless [38].

###### Acknowledgements.

We warmly thank Maria Chiara Angelini and Federico Corberi for very useful
discussions and suggestions. We thank Matteo Ghizzi for providing part of the
numerical data for the $1$-dimensional equilibrium case. This research
benefits from the HPC (High Performance Computing) facility of the University
of Parma.

##  Appendix A Algorithm implementation

In the algorithm of the DLL, we start by choosing uniformly random a spin
$\sigma_{i}$. Next we choose a list $S_{q}$ of $q$ spins, which will be
neighbours of $i$ during this temporal step. Such $q$ spins $\sigma_{j}$ are
drawn from the distribution $P(r_{ij})\propto r_{ij}^{-(d+\sigma)}$, with
$r_{ij}$ being the distance between nodes $i$ and $j$ and $\sigma$ the decay
parameter. A precise definition of $P(r_{ij})$ requires the introduction of
boundary conditions, that will be discussed later. Chosen $i$ and its $q$
neighbors, we apply the standard dynamics, for example Metropolis or Glauber,
using as flipping energy $\Delta E=2\cdot J\cdot\sigma_{i}\cdot\sum_{j\in
S_{q}}\sigma_{j}$. Applying $N$ times this procedure defines a Monte Carlo
step. The procedure is summarized in Algorithm 1.

Algorithm 1 DLL (single Monte Carlo step)

1:$q\in\mathbb{N}$ and $\sigma\in\mathbb{R}^{+}$

2:Choose a random spin $\sigma_{i}$

3:Draw a node $j\not=i$ from the probability distribution $P(r_{ij})\sim
r_{i,j}^{-(d+\sigma)}$ where $r_{ij}$ is the distance between nodes $i$ and
$j$. Repeat the extraction $q$ times so that you get $q$ random nodes, (allow
for the extraction of the same node more than one time in the list $S_{q}$).

4:Calculate the interaction energy $\Delta E=2\cdot
J\cdot\sigma_{i}\cdot\sum_{j\in S_{q}}\sigma_{j}$

5:Flip $\sigma_{i}$ following a dynamics, Metropolis or Glauber, using $\Delta
E$ as flipping energy

6:Repeat steps 1-5 $O(N)$ times

In the fully connected standard q-Ising model i.e. $P(r_{ij})=constant$,
analytical calculations show that for $q<2$ the system does not present a
phase transition at finite temperature i.e. $T_{c}=0$; with Metropolis
dynamics, for $q=3$ there is a continuous phase transition of the mean field
universality class while for $q>2$ a first order transition is observed [20,
21]. For Glauber dynamics instead, a continuous transition is always observed
for $q>2$. For this reason we choose the Glauber dynamics and set $q=3$ so
that we expect a second order phase transition, as we indeed observe in our
simulations. In particular, we verify that both lowering and increasing
temperature the system never presents hysteresis. A small value of $q$ allows
for an efficient implementation of the algorithm while large values are
typically more demanding since the algorithm requires the extraction of $q$
random numbers in a microscopic step. In general the study of the dependence
on $q$ of the dynamical evolution is an interesting open issue. Let us briefly
discuss the choice of the boundary conditions. As it is well known for the
fully connected long range model the results at small $\sigma$ are strongly
affected by finite size effects and by the choice of boundary conditions. The
most natural choice is to extract as a distance $r_{ij}$ a $d$-dimensional
integer vector, from the probability distribution
$p(\vec{r}_{ij})\sim|\vec{r}_{ij}|^{-(d+\sigma)}$ with
$\sqrt{d}/2<|\vec{r}_{ij}|<L/2$ and then impose periodic boundary conditions.
In [29] it is shown that a more efficient choice of the boundary condition is
to use copies (images) of the original configuration; in our DLL this means to
drawn the integer vector $\vec{r}_{ij}$ from the probability distribution
$p(\vec{r}_{ij})\sim|\vec{r}_{ij}|^{-(d+\sigma)}$ with
$\sqrt{d}/2<|\vec{r}_{ij}|$ (no upper limit to the distribution). The site $j$
is then obtained with periodic boundary condition where now $\vec{r}_{ij}$ can
wind around the torus an arbitrary number of times. We verify indeed that also
for our model this choice is more efficient in simulations. A further choice
of the boundary conditions turns out to be even slightly better. In
particular, we draw again the vector $\vec{r}_{ij}$ from the integer
probability distribution $p(\vec{r}_{ij})\sim|\vec{r}_{ij}|^{-(d+\sigma)}$
with $\sqrt{d}/2<|\vec{r}_{ij}|$ (no upper limit); then we fix $\sigma_{j}=0$
if $|\vec{r}_{ij}|>L/2$; while for $|\vec{r}|<L/2$ the node $j$ is obtained
imposing periodic boundary conditions; in this way when faraway coordinates
are considered we take into account that interaction occurs with nodes of zero
magnetization on average, while spin-spin correlations are relevant only at
distances smaller than $L$. In $d=2$ finite size effects are large and we
adopt this last prescription for the boundary conditions, which better
reproduces Sak predictions on the exponent $\eta$. On the other hand, in $d=1$
the results seem independent of the choice since we deal with very large $L$.

##  Appendix B Simulation details - the susceptibility

During the Monte Carlo simulation we save the value of the magnetization
$m=L^{-d}\sum_{i=1}^{N}\sigma_{i}$, for a total of $O(10^{6})$ realization for
each temperature. Then we compute the susceptibility as

$\chi=k_{b}TL^{-d}(\langle m^{2}\rangle-\langle|m|\rangle^{2}$ (10)

where the constant $k_{b}$ is set as equal to $1$. The error for each $\chi$
is calculated taking the standard deviation of different Markov chain
realizations. These errors are propagated in every successive fit we make,
always checking that the reduced $\chi^{2}$ is compatible with the unity, for
each one of those. To find the maximum value of $\chi$, we concentrate our
simulations near the peak and then we extract the maximum with a quadratic
fit, propagating the errors as mentioned before. A sketch of this procedure
can be seen at Figure 7.

Figure 7: Parable extrapolation of the maximum of $\chi$ for $d=2$,
$\sigma=1.2$ and $L=16$, which result is drawn in the same color of the
points. The star is the extrapolated maximum of the curve, with its respective
error bar.

Therefore, we perform a linear fit with the logarithm of the maximum, in
particular from equation (2) we get:

$\log{(\chi_{max}(L))}=y\cdot\log{L}+C,$ (11)

with $\chi_{max}$ the maximum of $\chi$ and C a constant.

Typically Eq. (11) is affected by strong finite size effects and the linear
behavior in log-log plot is not observed. Thus our strategy is to measure the
variation of the slope $y$ as a function of the size $L$ as illustrated in
Figure 8. For such size dependent slopes $y(L)$ we extrapolate the value at
$L=\infty$ by means of a linear fit against $L^{-1}$, i.e. we assume the
intercept of this fit as an estimate of the exponent. Again, at this step we
propagate the errors coming from the quadratic fits. The result of this simple
extrapolation procedure is shown in the circle of Figure 3 evidencing that
still important finite size effects are present in the estimate. Indeed, in
[29] it has been shown that the correction to the scaling at $\sigma=1.75$
vanishes more slowly than linearly according to Eq. (3). In this perspective
we extrapolate the value of $y$ by fitting linearly the size dependent
estimates $y(L)$ as a function of $(1/L)^{\delta}$ and we obtain the data
shown with blue stars in Figure 3 which confirm Sak’s prediction on the
exponent $\eta$. In this second approach errors are larger (see Figure 3)
since it assumes larger scaling corrections.

Figure 8: Finite size extrapolation example for $d=1$ and $\sigma=0.35$. Each
line is the result of a linear fit of two consecutive sizes and it is clear
how the slope changes with increasing sizes.

Figure 9 shows that corrections to scaling according to Eq. (3) are indeed
consistent with our numerical simulations. In particular, the quantity $\chi
L^{-(2-\eta)}$ displays a linear dependence respect to $L^{-\delta}$ with
$\delta=0.42$. Then we study the ratio $\frac{a}{b}$ where $a$ and $b$ are the
coefficient in Eq. (3) obtained from the linear fit in Figure 9. In Figure 10
we find that $\frac{a}{b}$ has a sharp minimum for $\sigma=1.75$ and it grows
when moving away from the SR-LR crossing point. This confirms that the
corrections to scaling are bigger in this transition regime.

Figure 9: Linear dependence of $\chi L^{-(2-\eta)}$ for $\sigma$ values near
the SR-LR crossing point.Straight line are the result of the linear fit
respect to $L^{-\delta}$

.

Figure 10: $\frac{a}{b}$ ratio for different values of $\sigma$. The
logarithmic scale on the $y$ axis, shows how fast this ratio grows as we move
away from the SR-LR crossing point.

##  Appendix C Simulation details - $z$ extrapolation

In this section we briefly explain the method we used to calculate the $z$
dynamical critical exponent, since there is no standard way to measure it. We
define a function which depends on $z$, whose minimum indicates the best
estimate of the exponent. The function is defined as follows. First, we take a
value of $z$ and we rescale $\mathcal{E}_{N}$, as mentioned previously in the
text, i.e we look at the curves $L^{-z/2}\mathcal{E}_{N}(\tilde{t},L)$ as
function of $\tilde{t}\equiv(t_{BS}L^{-z})^{1/2}$, at different sizes $L$.
Then, for each value of $\tilde{t}$, we compute the difference between
consecutive curves in size, defining the function:

$\Delta\mathcal{E}_{N}(\tilde{t},L)\equiv\mathcal{E}_{N}(\tilde{t},L)L^{-z/2}-\mathcal{E}_{N}(\tilde{t},L/2)(L/2)^{-z/2}$
(12)

To take into account finite size effects, we extrapolate the thermodynamic
limit of $\Delta\mathcal{E}_{N}(\tilde{t},L)$, by assuming a linear dependence
of $L^{-1}$:

$\Delta\mathcal{E}_{N}(\tilde{t},L)=AL^{-1}+\Delta\mathcal{E}_{N}(\tilde{t},\infty).$
(13)

In this case, a next to leading correction to the scaling e.g. as
$L^{-\delta}$ with $\delta\not=1$, seems to have a small effect on the final
value of $z$, at least for the size and errors we are considering in our
simulation. Finally, we take the square sum of
$\Delta\mathcal{E}_{N}(\tilde{t},\infty)$ for all the values of $\tilde{t}$,
to penalize the discrepancies from $0$. Indeed, the best value of $z$ is the
one for which, in the thermodynamic limit, the curves at different size
collapse. Thus, for each value of $z$ we have defined the function:

$\Delta\mathcal{E}_{N}(z)\equiv\sum_{\\{\tilde{t}\\}}\Delta\mathcal{E}_{N}^{2}(\tilde{t},\infty),$
(14)

where the sum is extended to all value of $\tilde{t}$. Our best estimate of
$z$ is the one for which this function has a minimum. For the error, we look
at the values of $z_{e}$ for which
$\Delta\mathcal{E}_{N}(z_{e})=5\times\Delta\mathcal{E}_{N}(z_{min})$.

##  Appendix D Numerical data

$\sigma$ | Experiment | $\Delta y$
---|---|---
$0.35$ | $0.507\pm 0.015$ | $0.007\pm 0.015$
$0.6$ | $0.612\pm 0.016$ | $0.012\pm 0.016$
$0.8$ | $0.761\pm 0.032$ | $0.039\pm 0.032$
$0.9$ | $0.861\pm 0.041$ | $0.039\pm 0.041$
Table 1: $y$ exponent extrapolation for $1$-dimensional spin chain. The last column indicates the discrepancy with respect to the expected theoretical exponent. $\sigma$ | Experiment | $\Delta y$
---|---|---
$0.5$ | $0.95\pm 0.03$ | $0.05\pm 0.03$
$0.9$ | $1.019\pm 0.025$ | $0.019\pm 0.025$
$1.2$ | $1.168\pm 0.033$ | $0.032\pm 0.033$
$1.5$ | $1.424\pm 0.061$ | $0.076\pm 0.061$
$1.75$ | $1.608\pm 0.032$ | $0.142\pm 0.032$
$1.9$ | $1.679\pm 0.046$ | $0.071\pm 0.046$
$2.0$ | $1.685\pm 0.048$ | $0.065\pm 0.048$
$2.15$ | $1.705\pm 0.038$ | $0.045\pm 0.038$
$2.5$ | $1.754\pm 0.026$ | $0.004\pm 0.026$
Table 2: $y$ exponent extrapolation for $2$-dimensional lattice. The last column indicates the discrepancy with respect to the expected theoretical exponent. $\sigma$ | Experiment | $\Delta y$
---|---|---
$0.5$ | $1.022\pm 0.059$ | $0.022\pm 0.059$
$0.9$ | $0.995\pm 0.051$ | $0.005\pm 0.051$
$1.2$ | $1.169\pm 0.064$ | $0.031\pm 0.064$
$1.5$ | $1.516\pm 0.062$ | $0.016\pm 0.062$
$1.75$ | $1.766\pm 0.068$ | $0.016\pm 0.068$
$1.9$ | $1.766\pm 0.069$ | $0.016\pm 0.069$
$2.0$ | $1.761\pm 0.07$ | $0.011\pm 0.07$
$2.15$ | $1.793\pm 0.076$ | $0.043\pm 0.076$
$2.5$ | $1.789\pm 0.075$ | $0.039\pm 0.075$
Table 3: $y$ exponent extrapolation for $2$-dimensional lattice when using second order correction $\delta=0.42$, as discussed in the text. $\sigma$ | $z$
---|---
$0.35$ | $0.52\pm 0.05$
$0.6$ | $0.63\pm 0.04$
$0.8$ | $0.85\pm 0.04$
$0.9$ | $0.97\pm 0.03$
Table 4: $z$ exponent extrapolation for the $1$-dimensional case. $\sigma$ | $z$
---|---
$0.5$ | $1.01\pm 0.04$
$0.9$ | $1.01\pm 0.04$
$1.2$ | $1.2\pm 0.035$
$1.5$ | $1.55\pm 0.04$
$1.75$ | $1.76\pm 0.02$
$1.9$ | $1.91\pm 0.03$
$2.0$ | $2.016\pm 0.035$
$2.15$ | $2.07\pm 0.06$
$2.5$ | $2.12\pm 0.04$
Table 5: $z$ exponent extrapolation for the $2$-dimensional case.

## References

  * Campa  _et al._ [2014] A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo,  _Physics of Long-Range Interacting Systems_, 1st ed. (Oxford University PressOxford, 2014).
  * Defenu  _et al._ [2021] N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo, and A. Trombettoni, Long-range interacting quantum systems (2021), arXiv.2109.01063.
  * Dyson [1969] F. Dyson, Existence of a phase-transition in a one-dimensional ising ferromagnet., Commun. Math. Phys. 12, 91 (1969).
  * Fisher  _et al._ [1972] M. E. Fisher, S.-k. Ma, and B. G. Nickel, Critical Exponents for Long-Range Interactions, Physical Review Letters 29, 917 (1972).
  * Bray [1993] A. J. Bray, Domain-growth scaling in systems with long-range interactions, Phys. Rev. E 47, 3191 (1993).
  * Bray and Rutenberg [1994] A. J. Bray and A. D. Rutenberg, Growth laws for phase ordering, Phys. Rev. E 49, R27 (1994).
  * Christiansen  _et al._ [2019] H. Christiansen, S. Majumder, and W. Janke, Phase ordering kinetics of the long-range Ising model, Physical Review E 99, 011301 (2019).
  * Christiansen  _et al._ [2020] H. Christiansen, S. Majumder, M. Henkel, and W. Janke, Aging in the Long-Range Ising Model, Physical Review Letters 125, 180601 (2020).
  * Agrawal  _et al._ [2021] R. Agrawal, F. Corberi, E. Lippiello, P. Politi, and S. Puri, Kinetics of the two-dimensional long-range ising model at low temperatures, Phys. Rev. E 103, 012108 (2021).
  * Agrawal  _et al._ [2022] R. Agrawal, F. Corberi, F. Insalata, and S. Puri, Asymptotic states of ising ferromagnets with long-range interactions, Phys. Rev. E 105, 034131 (2022).
  * Sak [1973] J. Sak, Recursion Relations and Fixed Points for Ferromagnets with Long-Range Interactions, Physical Review B 8, 281 (1973).
  * Luijten and Blöte [1995] E. Luijten and H. W. Blöte, Monte carlo method for spin models with long-range interactions, International Journal of Modern Physics C 06, 359 (1995).
  * Fukui and Todo [2009] K. Fukui and S. Todo, Order-N cluster Monte Carlo method for spin systems with long-range interactions, Journal of Computational Physics 228, 2629 (2009).
  * Leuzzi  _et al._ [2008] L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz-Lorenzo, Dilute One-Dimensional Spin Glasses with Power Law Decaying Interactions, Physical Review Letters 101, 107203 (2008).
  * Leuzzi  _et al._ [2009] L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz-Lorenzo, Ising spin-glass transition in a magnetic field outside the limit of validity of mean-field theory, Phys. Rev. Lett. 103, 267201 (2009).
  * Katzgraber  _et al._ [2009] H. G. Katzgraber, D. Larson, and A. P. Young, Study of the de almeida–thouless line using power-law diluted one-dimensional ising spin glasses, Phys. Rev. Lett. 102, 177205 (2009).
  * Berganza and Leuzzi [2013] M. I. Berganza and L. Leuzzi, Critical behavior of the X Y model in complex topologies, Physical Review B 88, 144104 (2013).
  * Millán  _et al._ [2021] A. P. Millán, G. Gori, F. Battiston, T. Enss, and N. Defenu, Complex networks with tuneable spectral dimension as a universality playground, Phys. Rev. Res. 3, 023015 (2021).
  * Cescatti  _et al._ [2019] F. Cescatti, M. Ibáñez Berganza, A. Vezzani, and R. Burioni, Analysis of the low-temperature phase in the two-dimensional long-range diluted xy model, Phys. Rev. B 100, 054203 (2019).
  * Jȩdrzejewski  _et al._ [2015] A. Jȩdrzejewski, A. Chmiel, and K. Sznajd-Weron, Oscillating hysteresis in the $q$-neighbor ising model, Phys. Rev. E 92, 052105 (2015).
  * Chmiel  _et al._ [2018] A. Chmiel, T. Gradowski, and A. Krawiecki, q-Neighbor Ising model on random networks, International Journal of Modern Physics C 29, 1850041 (2018).
  * Holme and Saramäki [2012] P. Holme and J. Saramäki, Temporal networks, Physics Reports 519, 97 (2012).
  * Ubaldi  _et al._ [2016] E. Ubaldi, N. Perra, M. Karsai, A. Vezzani, R. Burioni, and A. Vespignani, Asymptotic theory of time-varying social networks with heterogeneous activity and tie allocation, Scientific Reports 6, 35724 (2016).
  * Ubaldi  _et al._ [2017] E. Ubaldi, A. Vezzani, M. Karsai, N. Perra, and R. Burioni, Burstiness and tie activation strategies in time-varying social networks, Scientific Reports 7, 46225 (2017).
  * Luijten and Blöte [1996] E. Luijten and H. W. J. Blöte, Finite-size scaling and universality above the upper critical dimensionality, Phys. Rev. Lett. 76, 1557 (1996).
  * Luijten and Blöte [1997] E. Luijten and H. W. J. Blöte, Classical critical behavior of spin models with long-range interactions, Physical Review B 56, 8945 (1997).
  * Luijten and Blöte [2002] E. Luijten and H. W. J. Blöte, Boundary between Long-Range and Short-Range Critical Behavior in Systems with Algebraic Interactions, Physical Review Letters 89, 025703 (2002).
  * Picco [2012] M. Picco, Critical behavior of the Ising model with long range interactions (2012), arXiv:1207.1018 [cond-mat, physics:hep-th].
  * Angelini  _et al._ [2014] M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, Relations between short-range and long-range Ising models, Physical Review E 89, 062120 (2014).
  * Defenu  _et al._ [2015] N. Defenu, A. Trombettoni, and A. Codello, Fixed-point structure and effective fractional dimensionality for o($n$) models with long-range interactions, Phys. Rev. E 92, 052113 (2015).
  * Cardy [1981] J. Cardy, One-dimensional models with 1/r2 interactions, J. Phys. A: Math. Gen. 14, 1407 (1981).
  * Bar and Mukamel [2014] A. Bar and D. Mukamel, Mixed-order phase transition in a one-dimensional model, Phys. Rev. Lett. 112, 015701 (2014).
  * Barma  _et al._ [2019] M. Barma, S. N. Majumdar, and D. Mukamel, Fluctuation-dominated phase ordering at a mixed order transition, Journal of Physics A: Mathematical and Theoretical 52, 254001 (2019).
  * Swendsen and Wang [1987] R. H. Swendsen and J.-S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Physical Review Letters 58, 86 (1987).
  * Wolff [1989] U. Wolff, Collective Monte Carlo Updating for Spin Systems, Physical Review Letters 62, 361 (1989).
  * Gori  _et al._ [2017] G. Gori, M. Michelangeli, N. Defenu, and A. Trombettoni, One-dimensional long-range percolation: A numerical study, Phys. Rev. E 96, 012108 (2017).
  * Bighin  _et al._ [2022] G. Bighin, T. Enss, and N. Defenu, Universal scaling in fractional dimension (2022), arXiv:2211.13302 [cond-mat].
  * Giachetti  _et al._ [2021] G. Giachetti, N. Defenu, S. Ruffo, and A. Trombettoni, Berezinskii-Kosterlitz-Thouless Phase Transitions with Long-Range Couplings, Physical Review Letters 127, 156801 (2021).
  * Metropolis and Ulam [1949] N. Metropolis and S. Ulam, The Monte Carlo Method, Journal of the American Statistical Association 44, 335 (1949).
  * Glauber [1963] R. J. Glauber, Time‐Dependent Statistics of the Ising Model, Journal of Mathematical Physics 4, 294 (1963).
  * Park and Noh [2017] J.-M. Park and J. D. Noh, Tricritical behavior of nonequilibrium Ising spins in fluctuating environments, Physical Review E 95, 042106 (2017).
  * Alexander and Orbach [1982] S. Alexander and R. Orbach, Density of states on fractals: fractons, J. Physique Lett. 43, 625 (1982).
  * Burioni and Cassi [2005] R. Burioni and D. Cassi, Random walks on graphs: ideas, techniques and results, Journal of Physics A: Mathematical and General 38, R45 (2005).
  * Burioni and Cassi [1997] R. Burioni and D. Cassi, Geometrical Universality in Vibrational Dynamics, Modern Physics Letters B 11, 1095 (1997).
  * Sokal [1997] A. Sokal, Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms, in  _Functional Integration_ , NATO ASI Series, Vol. 361 (Springer US, 1997) pp. 131–192.
  * Hasenbusch [2020] M. Hasenbusch, Dynamic critical exponent $z$ of the three-dimensional ising universality class: Monte carlo simulations of the improved blume-capel model, Phys. Rev. E 101, 022126 (2020).
  * Adzhemyan  _et al._ [2022] L. Adzhemyan, D. Evdokimov, M. Hnatič, E. Ivanova, M. Kompaniets, A. Kudlis, and D. Zakharov, The dynamic critical exponent z for 2d and 3d ising models from five-loop $\epsilon$ expansion, Physics Letters A 425, 127870 (2022).
  * Burioni  _et al._ [2007] R. Burioni, D. Cassi, F. Corberi, and A. Vezzani, Phase-ordering kinetics on graphs, Phys. Rev. E 75, 011113 (2007).
  * Burioni  _et al._ [2013] R. Burioni, F. Corberi, and A. Vezzani, Topological regulation of activation barriers on fractal substrates, Phys. Rev. E 87, 032160 (2013).
  * Corberi  _et al._ [2015] F. Corberi, M. Zannetti, E. Lippiello, R. Burioni, and A. Vezzani, Phase ordering in disordered and inhomogeneous systems, Phys. Rev. E 91, 062122 (2015).


$p\tilde{c}_{ij}\,\tilde{c}_{i}(\mathbf{x}^{*}_{i})^{p-1}=r^{*}_{j}+\kappa^{*}_{ij}\text{
for all }i\in\mathcal{N},j\in\mathcal{M}.$ (15)

Since $\tilde{c}_{ij}>0$, then Equation 15 implies that
$p\,\tilde{c}_{i}(\mathbf{x}^{*}_{i})^{p-1}=\frac{r^{*}_{j}+\kappa^{*}_{ij}}{\tilde{c}_{ij}}$.
Additionally, if $x^{*}_{ij}>0$, Equation 13 implies that $\kappa^{*}_{ij}=0$,
and therefore,
$p\,\tilde{c}_{i}(\mathbf{x}^{*}_{i})^{p-1}=\frac{r^{*}_{j}}{\tilde{c}_{ij}}$.
Hence, if $x^{*}_{ij}>0$, for any $\ell\in\mathcal{M}$ we have

$\frac{r^{*}_{j}}{\tilde{c}_{ij}}=p\,\tilde{c}_{i}(\mathbf{x}^{*}_{i})^{p-1}=\frac{r^{*}_{\ell}+\kappa^{*}_{i\ell}}{\tilde{c}_{i\ell}}\geq\frac{r^{*}_{\ell}}{\tilde{c}_{i\ell}}.$
(16)

#### Market Interpretation.

Let $e^{*}_{i}\coloneqq p\,\tilde{c}_{i}(\mathbf{x}_{i}^{*})^{p}$. We will
show that $(\mathbf{x}^{*},\mathbf{r}^{*},\mathbf{e}^{*})$ is a market
equilibrium.

First, notice that $r^{*}_{j}>0$ for all $j\in\mathcal{M}$: for
$i\in\mathcal{N}$ such that $x^{*}_{ij}>0$, we have
$r^{*}_{j}=p\tilde{c}_{ij}\tilde{c}_{i}(\mathbf{x}^{*}_{i})^{p-1}$, where
$\tilde{c}_{ij}>0$ and and $\tilde{c}_{i}(\mathbf{x}^{*}_{i})^{p-1}>0$ (since
$x^{*}_{ij}>0$). Then, note that the _market clears_ (i.e.,
$\sum_{i=1}^{n}x^{*}_{ij}=1$, for all $j\in\mathcal{M}$): this is a direct
implication of Equation 12 and the fact that $r^{*}_{j}>0$ for all
$j\in\mathcal{M}$.

Second, $\mathbf{x}^{*}$ is a _maximum reward-per-cost_ allocation, i.e. if
$x^{*}_{ij}>0$, then Equation 16 implies that
$\mathrm{MRC}_{i}\coloneqq\frac{r^{*}_{j}}{\tilde{c}_{ij}}\geq\frac{r^{*}_{\ell}}{\tilde{c}_{i\ell}}$
for all $\ell\in\mathcal{M}$. Finally, every agent $i\in\mathcal{N}$ meets
their earning goal $e^{*}_{i}$. The amount of earned by agent $i$ is

$\displaystyle\sum_{j:\ x^{*}_{ij}>0}x^{*}_{ij}r^{*}_{j}$
$\displaystyle=\sum_{j:\
x^{*}_{ij}>0}x^{*}_{ij}\tilde{c}_{ij}\cdot\left(\frac{r^{*}_{j}}{\tilde{c}_{ij}}\right)$
$\displaystyle=\sum_{j:\
x^{*}_{ij}>0}x^{*}_{ij}\tilde{c}_{ij}\cdot\left(p\,\tilde{c}_{i}(\mathbf{x}^{*}_{i})^{p-1}\right)$
(Equation 16)
$\displaystyle=p\,\tilde{c}_{i}(\mathbf{x}^{*}_{i})^{p-1}\sum_{j:\
x^{*}_{ij}>0}x^{*}_{ij}\tilde{c}_{ij}$
$\displaystyle=p\,\tilde{c}_{i}(\mathbf{x}^{*}_{i})^{p-1}\tilde{c}_{i}(\mathbf{x}^{*}_{i})$
$\displaystyle=e^{*}_{i}.$

∎

*

###### Proof.

To prove the first inequality, we consider the following difference,

$\displaystyle\left(a+\frac{\beta(1-a)}{1-\alpha}\right)^{2}+(1-\alpha-\beta)^{2}-(a^{2}+(1-\alpha)^{2})$
$\displaystyle=a^{2}+\frac{\beta^{2}(1-a)^{2}}{(1-\alpha)^{2}}+\frac{2a\beta(1-a)}{1-\alpha}+(1-\alpha)^{2}+\beta^{2}-2\beta(1-\alpha)-a^{2}-(1-\alpha)^{2}$
$\displaystyle=\beta\left(\frac{\beta(1-a)^{2}}{(1-\alpha)^{2}}+\frac{2a(1-a)}{1-\alpha}+\beta-2(1-\alpha)\right)$
$\displaystyle<\beta\left(\frac{(1-2\alpha)(1-a)^{2}}{(1-\alpha)^{2}}+\frac{2a(1-a)}{1-\alpha}+(1-2\alpha)-2(1-\alpha)\right)$
($\beta<1-2\alpha$)
$\displaystyle=\beta\left(\frac{(1-2\alpha)(1-a)^{2}}{(1-\alpha)^{2}}+\frac{2a(1-a)}{1-\alpha}-1\right)$
$\displaystyle=\frac{\beta}{(1-\alpha)^{2}}((1-2\alpha)(1-a)^{2}+2a(1-a)(1-\alpha)-(1-\alpha)^{2})$
$\displaystyle=\frac{\beta}{(1-\alpha)^{2}}(2a\alpha-a^{2}-\alpha^{2})$
(simplifying) $\displaystyle=-\frac{\beta}{(1-\alpha)^{2}}(a-\alpha)^{2}<0.$
($a<\alpha$)

This establishes the first inequality. For the second inequality, we have

$a+\frac{1-a}{1-\alpha}\cdot\beta<1-\alpha\Longleftarrow\
\alpha<\frac{1-a}{1-\alpha}(1-\alpha-\beta).$

The last inequality above is true because $a<\alpha<1$ (i.e.,
$1<\frac{1-a}{1-\alpha}$ ) and $\alpha<1-\alpha-\beta$. ∎

######  Lemma 4.

For the divisible chores allocation instance below and $p\in(1,2)$, the
allocation that minimizes the $p$-mean of normalized disutilities is such
that, when $\delta>\varepsilon$, chore 2 is fully allocated to agent 1.

$\begin{pmatrix}\frac{1}{2}+\delta&\frac{1}{2}-\delta\\\
\frac{1}{2}+\varepsilon&\frac{1}{2}-\varepsilon\end{pmatrix}$

###### Proof.

First, consider an allocation in which each agent has a nonzero fraction of
each chore. Then, there exists $t_{1}$ and $t_{2}$ such that transferring a
$t_{1}$ fraction of chore 1 to agent 2 and $t_{2}$ fraction of chore 2 to
agent 1 results in a strictly smaller $p$-mean. To achieve this, pick $t_{1}$
and $t_{2}$ so that the cost of agent 2 does not change, i.e.
$t_{1}(1/2+\varepsilon)-t_{2}(1/2-\varepsilon)=\frac{1}{2}(t_{1}-t_{2})+\varepsilon(t_{1}+t_{2})=0$.
Then, consider the change in cost for agent 1.

$\displaystyle t_{2}(1/2-\delta)-t_{1}(1/2+\delta)$
$\displaystyle=\frac{1}{2}(t_{2}-t_{1})-\delta(t_{1}+t_{2})$
$\displaystyle=\varepsilon(t_{1}+t_{2})-\delta(t_{1}+t_{2})$
($\frac{1}{2}(t_{1}-t_{2})+\varepsilon(t_{1}+t_{2})=0$) $\displaystyle<0.$
($\delta>\varepsilon$)

Since performing this transfer is giving a Pareto improvement, the allocation
could not have been $p$-mean optimal. If agent $1$ receives all of chore $2$
we are done. We show that if chore 1 is fully allocated to agent 2, then
transferring the entirety of chore 2 to agent 1 results in a smaller $p$-mean.
Let $x$ be the fraction of chore 2 allocated to agent 1. Then,
$\tilde{c}_{1}(\mathbf{x}_{1})=(1/2-\delta)x$ and
$\tilde{c}_{2}(\mathbf{x}_{2})=1/2+\varepsilon+(1-x)(1/2-\varepsilon)=1-x(1/2-\varepsilon)$.
We then write the normalized $p$-mean as follows:

$\displaystyle
w_{p}(x)=\left(x\left(\frac{1}{2}-\delta\right)\right)^{p}+\left(1-x\left(\frac{1}{2}-\varepsilon\right)\right)^{p}.$

We then consider the derivative with respect to $x$ and show that it is
negative (thus larger $x$ implies smaller $p$-mean).

$\displaystyle\frac{\partial w_{p}(x)}{\partial x}$
$\displaystyle=p\left(\left(\frac{1}{2}-\delta\right)^{p}x^{p-1}-\left(1-x\left(\frac{1}{2}-\varepsilon\right)\right)^{p-1}\left(\frac{1}{2}-\varepsilon\right)\right)<0$
$\displaystyle\Longleftrightarrow\left(\frac{1}{2}-\delta\right)^{p}x^{p-1}<\left(1-x\left(\frac{1}{2}-\varepsilon\right)\right)^{p-1}\left(\frac{1}{2}-\varepsilon\right)$
$\displaystyle\Longleftrightarrow
x<\frac{\left(\frac{1}{2}-\varepsilon\right)^{\frac{1}{p-1}}}{\left(\frac{1}{2}-\delta\right)^{\frac{p}{p-1}}+\left(\frac{1}{2}-\varepsilon\right)^{\frac{p}{p-1}}}.$

Showing that the stated upper bound on $x$ is larger than $1$ would imply that
the derivative is negative everywhere in the interval $[0,1]$, and therefore
the fraction of chore $2$ allocated to agent $1$ should be equal to $1$. To
this end:

$\displaystyle\frac{\left(\frac{1}{2}-\varepsilon\right)^{\frac{1}{p-1}}}{\left(\frac{1}{2}-\delta\right)^{\frac{p}{p-1}}+\left(\frac{1}{2}-\varepsilon\right)^{\frac{p}{p-1}}}$
$\displaystyle>\frac{\left(\frac{1}{2}-\varepsilon\right)^{\frac{1}{p-1}}}{2\left(\frac{1}{2}-\varepsilon\right)^{\frac{p}{p-1}}}$
($\delta>\varepsilon$) $\displaystyle=\frac{1}{1-2\varepsilon}>1.$

∎


# Many subalgebras of $\mathcal{P}(\omega)/\mathit{fin}$

Klaas Pieter Hart  Faculty EEMCS
TU Delft
Postbus 5031
2600 GA Delft
the Netherlands<EMAIL_ADDRESS>https://fa.ewi.tudelft.nl/~hart

###### Abstract.

In answer to a question on Mathoverflow we show that the Boolean algebra
$\mathcal{P}(\omega)/\mathit{fin}$ contains a family
$\\{\mathcal{B}_{X}:X\subseteq\mathfrak{c}\\}$ of subalgebras with the
property that $X\subseteq Y$ implies $\mathcal{B}_{Y}$ is a subalgebra of
$\mathcal{B}_{X}$ and if $X\not\subseteq Y$ then $\mathcal{B}_{Y}$ is not
embeddable into $\mathcal{B}_{X}$. The proof proceeds by Stone duality and the
construction of a suitable family of separable zero-dimensional compact
spaces.

###### Key words and phrases:

$\mathcal{P}(\omega)/\mathit{fin}$, subalgebra, non-embeddability, Alexandrov
double arrow, Bernstein set

###### 1991 Mathematics Subject Classification:

Primary 06E05; Secondary: 06E15, 54C05, 54D65, 54G20

## Introduction

The purpose of this note is to give a more leisurely presentation, complete
with definitions and references, of an answer to a question on MathOverflow
[overflow442111]:

> Is there a strictly decreasing chain of subalgebras of the Boolean algebra
> $\mathcal{P}(\omega)/\mathit{fin}$?

The answer to the question as stated is an obvious “yes”, but the poser of the
question asked for a sequence $\langle{B}_{n}:n\in\omega\rangle$ of
subalgebras such that $B_{n+1}\subseteq B_{n}$ _and_ $B_{n}$ is _not_
embeddable into $B_{n+1}$, for all $n$.

We shall show that the family of subalgebras of
$\mathcal{P}(\omega)/\mathit{fin}$ is rich enough to contain such a sequence;
in fact, there is a family $\\{B_{X}:X\subseteq\mathfrak{c}\\}$ of subalgebras
with the property that for all subsets $X$ and $Y$ of $\mathfrak{c}$ we have:
if $X\subseteq Y$ then $B_{Y}\subseteq B_{X}$ _and_ if $X\notsubseteq Y$ then
$B_{Y}$ is _not_ embeddable into $B_{X}$. This more than answers the question
and shows that one can even have a decreasing chain of length $\mathfrak{c}$
or a chain or order type that of the real line.

The construction of the family proceeds via Stone duality: rather than
constructing subalgebras of $\mathcal{P}(\omega)/\mathit{fin}$ we construct a
family $\\{K_{X}:X\subseteq\mathfrak{c}\\}$ of separable compact zero-
dimensional spaces with the dual property that there is a set
$\\{h_{X,Y}:X\subseteq Y\subseteq\mathfrak{c}\\}$ of continuous maps, where
$h_{X,Y}:K_{X}\to K_{Y}$ is a continuous surjection _and_ if $X\notsubseteq Y$
then $K_{X}$ is _not_ a continuous image of $B_{Y}$. In addition all triangles
in the set of maps will commute.

Then $K_{\emptyset}$ is a continuous image of $\omega^{*}$, the Stone space of
$\mathcal{P}(\omega)/\mathit{fin}$, and hence so are all other spaces $K_{X}$.
The maps $h_{\emptyset,X}$ embed the algebras of clopen sets of the $K_{X}$
into $\mathcal{P}(\omega)/\mathit{fin}$, the commutativity of the triangles in
the family of continuous surjections yields the desired inclusions, and the
nonexistence of further continuous surjections dualizes to the nonexistence of
further embeddings.

##  1\. Preliminaries

###  1.1. Stone duality

Stone’s duality for Boolean algebras and compact zero-dimensional spaces
associates with every compact zero-dimensional space $X$ its Boolean algebra
$\mathcal{C}_{X}$ of closed-and-open subsets and conversely with every Boolean
algebra $B$ a compact zero-dimensional space $\operatorname{St}(B)$, its
_Stone space_. The associations are each others inverses and they dualize
various notions; the most important for us is that an embedding $B\to C$ of
Boolean algebras becomes a continuous surjection
$\operatorname{St}(C)\to\operatorname{St}(B)$, and vice versa.

The book [MR991565]*Chapter 3 contains further information on Stone’s duality
for Boolean algebras and compact zero-dimensional spaces.

###  1.2. Bernstein sets

In our construction we shall uses Bernstein sets in $[0,1]$. We say $A$ is a
_Bernstein set_ in $[0,1]$ if $A$ and its complement both intersect every
uncountable closed set in $[0,1]$. These are also called totally imperfect
sets ([zbMATH02640876]) because if a set is closed in $[0,1]$ and contained in
$A$ then it must be countable.

For other topological material we refer to [MR1039321]. The fact used here,
that separable compact spaces are continuous images of the remainder
$\omega^{*}$ can be proved using Theorem 3.5.13 and Exercise 3.5.H of that
book.

##  2\. The spaces

The spaces are variations on Alexandroff’s double-arrow space $\A$, called the
two arrows space in [MR1039321]*Exercise 3.10.C.

The underlying set is
$D=\bigl{(}[0,1]\times\\{0,1\\}\bigr{)}\setminus\bigl{\\{}\langle
0,0\rangle,\langle 0,1\rangle\bigr{\\}}$, ordered lexicographically and
endowed with the order topology. (We drop the points $\langle 0,0\rangle$ and
$\langle 0,1\rangle$ because they would be (the only) isolated points of
$\A$.)

Pictorially we have taken the unit interval $[0,1]$ and split each point $x$
of the open interval $(0,1)$ into two copies, $\langle{x},{0}\rangle$ and
$\langle{x},{1}\rangle$. The space $\A$ is compact and separable, hence a
continuous image of $\omega^{*}$.

The variations will be obtained by specifying a subset $X$ of $(0,1)$ and
taking $\A_{X}=\\{\langle x,i\rangle\in D:x\in X\to i=0\\}$; that is, by
splitting the points of $(0,1)\setminus X$ only. Thus we can write
$\A=\A_{\emptyset}$, and $[0,1]=\A_{(0,1)}$ for example. In all our examples
the complement of $X$ will be dense in $(0,1)$ and this will ensure that
$\A_{X}$ is zero-dimensional.

If $X\subseteq Y$ then there is a natural continuous surjection
$s:\A_{X}\to\A_{Y}$, given by

  * •

$s(x,i)=\langle{x},{i}\rangle$ if $x\notin Y$;

  * •

$s(x,i)=\langle{x},{0}\rangle$ if $x\in Y\setminus X$; and

  * •

$s(x,0)=\langle{x},{0}\rangle$ if $x\in X$.

Our goal will be to create a family $\\{S_{X}:X\subseteq\mathfrak{c}\\}$ of
subsets of $(0,1)$ such that with $K_{X}=\A_{S_{X}}$ for all $X$ we get our
family $\\{K_{X}:X\subseteq\mathfrak{c}\\}$.

We shall construct a family $\\{A_{\alpha}:\alpha\in\mathfrak{c}\\}$ of
subsets of $(0,1)$ (all disjoint from $\Q$) and put
$S_{X}=\Q\cup\bigcup_{\alpha\in X}A_{\alpha}$ for $X\subseteq\mathfrak{c}$.

Clearly then $X\subseteq Y$ implies $S_{X}\subseteq S_{Y}$ and hence that
$K_{X}$ maps onto $K_{Y}$ by $h_{X,Y}:A_{S_{X}}\to\A_{S_{Y}}$ as described
above. It is readily seen that $h_{X,Z}=h_{Y,Z}\circ h_{X,Y}$ all triangles in
this family commute, as described in the introduction.

It remains to construct the sets $A_{\alpha}$ in such a way that whenever
$X\notsubseteq Y$ each of the $A_{\alpha}$ with $\alpha\in X\setminus Y$ will
prohibit the existence of a continuous surjection from $K_{Y}$ onto $K_{X}$.

To see how this may be accomplished note that since $A_{\alpha}\subseteq
S_{X}$ the points of $A_{\alpha}$ are not split in $\A_{S_{X}}$. In that case
the subspace topology that $A_{\alpha}$ inherits from $\A_{S_{X}}$ is the same
as the subspace topology that it inherits from $[0,1]$.

If $s:K_{X}\to K_{Y}$ is continuous then the composition $t\circ s$, where
$t:K_{Y}\to[0,1]$ is the map that send $\langle{x},{i}\rangle$ to $x$, is
continuous as well and its restriction $g$ to $A_{\alpha}$ is also continuous.
We shall arrange matters in such a way that the only maps that can appear in
this way will force the range of the map $s$ to be countable.

##  3\. The sets $A_{\alpha}$

We can obtain our sets $A_{\alpha}$ by a direct application of Theorem 2.0 in
[MR808722] but to keep this note reasonably self-contained we shall repeat the
construction for the special case that we need. The method goes back to
[zbMATH03006477] and is occasionally referred to as “Sierpiński’s technique of
killing homeomorphisms” [MR1173263], but it can be used to eliminate other
maps as well.

In our case we consider the set $\mathcal{F}$ of all maps $f$ that satisfy:
$\operatorname{dom}f$ is a co-countable subset of $[0,1]$ and
$f:\operatorname{dom}f\to[0,1]$ is continuous. For every $f\in\mathcal{F}$ we
let $S(f)=\\{x\in\operatorname{dom}f:f(x)\neq x\\}$ and
$E(f)=\operatorname{dom}f\setminus S(f)$. We choose a subset $C(f)$ of
$\operatorname{dom}f$ such that the restriction $f:C(f)\to
f\bigl{[}S(f)\bigr{]}$ is a bijection.

Before we continue we make some remarks that will be useful later. For each
$f\in\mathcal{F}$ the domain is completely metrizable as it is a
$G_{\delta}$-subset of $[0,1]$. As $S(f)$ is open in $\operatorname{dom}f$,
and $E(f)$ is closed, both sets are completely metrizable as well.
Furthermore, the image $f\bigl{[}S(f)\bigr{]}$ is an analytic subset of
$[0,1]$. By familiar results from Descriptive Set Theory it follows that each
of these sets either is countable or contains a topological copy of the Cantor
set. One can modify the construction outlined in Problem 4.5.5 in [MR1039321]
to prove these results.

This means that in each case we can check whether the set is countable by
looking at its intersection with some Bernstein set.

The following proposition yields the family
$\\{A_{\alpha}:\alpha\in\mathfrak{c}\\}$.

######  Proposition 3.1.

There is a pairwise disjoint family
$\\{V\\}\cup\\{A_{\alpha}:\alpha\in\mathfrak{c}\\}$ of Bernstein sets in
$(0,1)$ with the following properties. All are disjoint from $\Q$, and for
every $f\in\mathcal{F}$: if $f\bigl{[}S(f)\bigr{]}$, and hence $C(f)$, has
cardinality $\mathfrak{c}$ then for all $\alpha$ the intersections $C(f)\cap
A_{\alpha}$ and $f\bigl{[}C(f)\cap A_{\alpha}\bigr{]}\cap V$ both have
cardinality $\mathfrak{c}$.

###### Proof.

Since $[0,1]$ has cardinality $\mathfrak{c}$ it also has $\mathfrak{c}$ many
co-countable subsets. Since each subset of $[0,1]$ is separable every co-
countable set has $\mathfrak{c}$ many continuous functions to $[0,1]$. Hence
we may enumerate the subfamily of $\mathcal{F}$ consisting of those $f$ for
which $f\bigl{[}S(f)\bigr{]}$ is uncountable as
$\langle{f}_{\beta}:\beta\in\mathfrak{c}\rangle$, in such a way that every $f$
occurs $\mathfrak{c}$ many times in the sequence. We take a similar
enumeration $\langle{F}_{\beta}:\beta\in\mathfrak{c}\rangle$ of the family of
uncountable closed subsets of $[0,1]$ (each set is listed $\mathfrak{c}$
times).

To facilitate the construction we replicate both enumerations $\mathfrak{c}$
times and turn them into $\mathfrak{c}\times\mathfrak{c}$-matrices:
$\\{f_{\alpha,\beta}:\langle{\alpha},{\beta}\rangle\in\mathfrak{c}^{2}\\}$ and
$\\{F_{\alpha,\beta}:\langle{\alpha},{\beta}\rangle\in\mathfrak{c}^{2}\\}$,
where $f_{\alpha,\beta}=f_{\beta}$ and $F_{\alpha,\beta}=F_{\beta}$ for all
$\alpha$ and $\beta$. We also take a well-order $\prec$ of $\mathfrak{c}^{2}$
in order type $\mathfrak{c}$,

By recursion on the well-order $\prec$ we will choose points
$a_{\alpha,\beta}$, $b_{\alpha,\beta}$, $u_{\alpha,\beta}$, and
$v_{\alpha,\beta}$, as follows.

When the points have been found for
$\langle{\gamma},{\delta}\rangle\prec\langle{\alpha},{\beta}\rangle$ collect
them and the rational numbers in a set:
$P=\Q\cup\bigcup_{\langle{\gamma},{\delta}\rangle\prec\langle{\alpha},{\beta}\rangle}\\{a_{\gamma,\delta},b_{\gamma,\delta},u_{\gamma,\delta},v_{\gamma,\delta}\\}$.
Note that the cardinality of $P$ is strictly smaller than $\mathfrak{c}$.
Therefore we can find $a_{\alpha,\beta}\in C(f_{\alpha,\beta})\setminus P$
such that $u_{\alpha,\beta}=f_{\alpha,\beta}(a_{\alpha,\beta})\notin P$; and
note that $u_{\alpha,\beta}\neq a_{\alpha,\beta}$. Next take points
$b_{\alpha,\beta}$ and $v_{\alpha,\beta}$ in
$F_{\alpha,\beta}\setminus(P\cup\\{a_{\alpha,\beta},v_{\alpha,\beta}\\}$ such
that $b_{\alpha,\beta}\neq v_{\alpha,\beta}$.

Now, by construction all points chosen in this way are distinct. For every
$\alpha\in\mathfrak{c}$ we let

$A_{\alpha}=\\{a_{\alpha,\beta}:\beta\in\mathfrak{c}\\}\cup\\{b_{\alpha,\beta}:\beta\in\mathfrak{c}\\}$

we also let

$V=\\{u_{\alpha,\beta}:\langle{\alpha},{\beta}\rangle\in\mathfrak{c}^{2}\\}\cup\\{v_{\alpha,\beta}:\langle{\alpha},{\beta}\rangle\in\mathfrak{c}^{2}\\}$

Because all points chosen are distinct the family
$\\{V\\}\cup\\{A_{\alpha}:\alpha\in\mathfrak{c}\\}$ is pairwise disjoint.

The sets are Bernstein sets because $A_{\alpha}\cap
F\supseteq\\{b_{\alpha,\beta}:F=F_{\alpha,\beta}\\}$ and $V\cap
F\supseteq\\{v_{\alpha,\beta}:F=F_{\alpha,\beta}\\}$, both intersections have
cardinality $\mathfrak{c}$.

Likewise, if $f\bigl{[}S(f)\bigr{]}$ has cardinality $\mathfrak{c}$ then
$A_{\alpha}\cap C(f)\supseteq\\{a_{\alpha,\beta}:f=f_{\alpha,\beta}\\}$ and
$V\cap f[A_{\alpha}\cap
C(f)]\supseteq\\{u_{\alpha,\beta}:f=f_{\alpha,\beta}\\}$; again both sets have
cardinality $\mathfrak{c}$. ∎

It now remains to show that the resulting family
$\\{K_{X}:X\subseteq\mathfrak{c}\\}$ of compact zero-dimensional spaces has
the desired properties. We already know that $K_{X}$ maps onto $K_{Y}$ if
$X\subseteq Y$. We prove the other implication in the next section. There it
will become clear what the function of the set $V$ is.

##  4\. Non-existence of continuous surjections

The following lemma implies that if $X$ and $Y$ are subsets of $\mathfrak{c}$
such that $X\notsubseteq Y$ then there is no continuous surjection from
$K_{X}$ onto $K_{Y}$.

######  Lemma 4.1.

Let $X$ and $Y$ be subsets of $(0,1)$ such that $\Q\subseteq X$ and such that
there is an $\alpha$ for which $A_{\alpha}\subseteq X$ and
$Y\cap(A_{\alpha}\cup V)=\emptyset$. Then every continuous map
$s:\A_{X}\to\A_{Y}$ has a countable range.

###### Proof.

Let us write $A$ for $A_{\alpha}$ and let $t:\A_{Y}\to[0,1]$ be the natural
surjection. Also, we identify $x$ and $\langle{x},{0}\rangle$ when $x\in X$.

As observed before the topology on $A$ in $\A_{X}$ is the same as its subspace
topology in $[0,1]$. Let $g$ be the restriction of $(t\circ s)$ to $A$.

By one half of Lavrentieff’s theorem (Theorem 4.3.20 in [MR1039321]) we can
find a $G_{\delta}$-set $G$ that contains $A$ and a continuous map
$f:G\to[0,1]$ that extends $g$. The complement, $C$, of $G$ in $[0,1]$ is a
countable union of closed sets, each of which is countable because closed sets
that are disjoint from a Bernstein set are countable; and $A$ is a Bernstein
set. It follows that $f$ belongs to the family $\mathcal{F}$.

The set $A$ is dense in $[0,1]$, and the maps $f$ and $t\circ s$ agree on $A$.
This implies that $f$ determines much of the behaviour of $s$ on $G$, in the
following way.

  * •

If $x\in G\cap X$ then $x$ is not split and $(t\circ s)(x)=f(x)$ and this
implies that
$s(x)\in\bigl{\\{}\langle{f(x)},{0}\rangle,\langle{f(x)},{1}\rangle\bigr{\\}}$.

  * •

If $x\in G\setminus X$ then $x$ is split and the continuity of $f$ implies
that $(t\circ s)(x,0)=(t\circ s)(x,1)=f(x)$ and so
$\bigl{\\{}s(x,0),s(x,1)\bigr{\\}}\subseteq\bigl{\\{}\langle{f(x)},{0}\rangle,\langle{f(x)},{1}\rangle\bigr{\\}}$.

It follows that the range of $s$ is contained in the union of
$f[G]\times\\{0,1\\}$ and the image of the countable set of points whose first
coordinates are in the countable set $C$.

We finish the proof by showing that $f[G]$ is countable.

Let $x\in E(f)\cap A$, then $f(x)=x$ and so $s(x)=\langle{x},{0}\rangle$ or
$s(x)=\langle{x},{1}\rangle$. We divide $E(f)\cap A$ into two sets:
$E_{0}=\\{x:s(x)=\langle{x},{0}\rangle\\}$ and
$E_{1}=\\{x:s(x)=\langle{x},{1}\rangle\\}$.

If $x\in E_{0}$ then by continuity of $s$ there is an interval $(p_{x},q_{x})$
with rational end points such that
$s[(p_{x},q_{x})]\subseteq[0,\langle{x},{0}\rangle]$. Here we use that
$\Q\subseteq X$: we can talk without ambiguity about intervals with rational
end points. It is clear that when $x<y$ in $E_{0}$ we have
$y\in(p_{y},q_{y})\setminus(p_{y},q_{y})$, and it follows that
$x\mapsto(p_{x},q_{x})$ is injective. We deduce that $E_{0}$ is countable.
Likewise one shows that $E_{1}$ is countable.

We see that $E(f)\cap A$ is countable, and because $A$ is a Bernstein set it
follows that $E(f)$ itself is countable

Next we let $x\in C(f)\cap A$ such that $f(x)\in V$. Then $f(x)\notin Y$ and
so $f(x)$ is split in $\A_{Y}$, and
$s(x)\in\\{\langle{f(x)},{0}\rangle,\langle{f(x)},{1}\rangle\bigr{\\}}$; we
split $\\{x\in C(f)\cap A:f(x)\in V\\}$ into
$C_{0}=\bigl{\\{}x:s(x)=\langle{f(x)},{0}\rangle\bigr{\\}}$ and
$C_{1}=\bigl{\\{}x:s(x)=\langle{f(x)},{1}\rangle\bigr{\\}}$.

As above we take for $x\in C_{0}$ an interval $(p_{x},q_{x})$ with rational
end points such that $s[(p_{x},q_{x})]\subseteq[0,\langle{f(x)},{0}\rangle]$.
If $x\neq y$ in $C_{0}$ then $f(x)\neq f(y)$ because $f$ is injective on
$C(f)$; if, say, $f(x)<f(y)$ then $y\in(p_{y},q_{y})\setminus(p_{y},q_{y})$
and it follows that $x\mapsto(p_{x},q_{x})$ is injective. We conclude, as
above, that $C_{0}$ is countable, as is $C_{1}$.

We see that $f[C(f)\cap A]\cap V$ is countable and hence, by the properties of
the family $\\{V\\}\cup\\{A_{\beta}:\beta\in\mathfrak{c}\\}$ in Proposition
3.1, that $f\bigl{[}S(f)\bigr{]}$ does not have cardinality $\mathfrak{c}$.
But as noted in the remarks before that proposition this means that
$f\bigl{[}S(f)\bigr{]}$ is countable.

Thus we see that $f[G]=f[E(f)]\cup f\bigl{[}S(f)\bigr{]}$ is countable. ∎

Now let $X$ and $Y$ be subsets of $\mathfrak{c}$ such that $X\notsubseteq Y$
and take $\alpha\in X\setminus Y$. Then $S_{X}$ and $S_{Y}$ satisfy the
conditions of Lemma 4.1. Indeed, by definition we have $\Q\cup
A_{\alpha}\subseteq S_{X}$ and $(A_{\alpha}\cup V)\cap S_{Y}=\emptyset$.

The lemma then tells us that every continuous map $s:K_{X}\to K_{Y}$ has a
countable range, so that $K_{Y}$ is not a continuous image of $K_{X}$.

## References