# Quantum black holes, partition of integers and self-similarity

P. Castorina(a,b), A Iorio (a) and L. Smaldone(c) (a) Institute of Particle
and Nuclear Physics, Faculty of Mathematics and Physics, Charles University, V
Holešovičkách 2, 18000 Praha 8, Czech Republic
(b) INFN Sezione di Catania, Via Santa Sofia 64, I-95123 Catania, Italy
(c) Faculty of Physics, Institute of Theoretical Physics, University of
Warsaw, Ul. Pasteura 5, 02-093, Warsaw, Poland<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We take the view that the area of a black hole’s event horizon is quantized,
$A=l_{P}^{2}\,(4\ln 2)\,N$, and the associated degrees of freedom are finite
in number and of fermionic nature. We then investigate general aspects of the
entropy, $S_{BH}$, our main focus being black-hole self-similarity. We first
find a two-to-one map between the black hole’s configurations and the ordered
partitions of the integer $N$. Hence we construct from there a composition law
between the sub-parts making the whole configuration space. This gives meaning
to black hole self-similarity, entirely within a single description, as a
phenomenon stemming from the well known self-similarity of the ordered
partitions of $N$. Finally, we compare the above to the well-known results on
the subleading (quantum) corrections, that necessarily require different
(quantum) statistical weights for the various configurations.

##  I Introduction

In this paper we elaborate on the view that the entropy of a black hole,
$S_{BH}$, stems from the quantization of the area of the BH’s event horizon
[3, 4, 5]

$A\ =\ \alpha\,N\,l_{P}^{2}\,,$ (1)

where $N\in\mathbb{N}$, $l_{P}$ is the Planck length, and the “it from bit”
[6] choice for the proportionality factor, $\alpha=4\ln 2$, allows for a spin-
system description: one bit of information, 0 or 1, $\uparrow$ or
$\downarrow$, etc, per given Planck cell. In other words

$S_{BH}\ =\ \frac{A}{4\,l_{P}^{2}}\ =\ N\,\ln 2\,,$ (2)

that means that each of the $2^{N}$ configurations is accounted for in the
same way, through $S=\ln$(number of microstates)$=\ln(2^{N})$, see, e.g., [7].

Notice that, within the model of [8], where the ultimate degrees of freedom,
making both space and matter, and called $X$ons [9], the fermionic character
gives precisely $\alpha=4\ln 2$, as shown in [10]. According to different
physical considerations, different choices of $\alpha$ are found in the
literature [11, 12]. Such quantization of the area is also an explicit result
of Loop Quantum Gravity [13, 14].

This general picture was proposed by Mukhanov and Bekenstein long ago [3, 4,
5], and called “the quantum BH”. That view made it till our days, although
quantum corrections to the formula (2), such as $S_{BH}\sim A/4-\beta\ln A$,
with $\beta>0$, are often taken into account [15, 16, 17, 18, 19, 20]. Such
corrections mean, of course, that not all configurations have the same weight.
On this see, e.g., [7]. With this in mind, the study of the microscopic
configurations of a BH amounts to the study of the configurations of a spin
model, see, e.g. [7].

The (quantum) information-theoretical approach to BH entropy/information
evoked in the “it from (qu)bit” choice appears to clash with the latter,
because if BH is the ultimate quantum computer, see, e.g., [21], then one
expects that: first, all configurations are treated equally, at the start of
the computation; and second, the evolution of the quantum states through an
Hamiltonian is not fixed by a given spin model, but rather governed by
different Hamiltonians depending on the “computation” (for the factorization,
see, e.g., [22] and the semiclassical counterpart [23]). Furthermore, certain
arguments suggest that the fermionic degrees of freedom ($X$ons) are actually
free in the BH phase, see, e.g., [8] and [10], and this clashes with an
interacting spin model that is necessary to explain why certain configurations
dominate over others.

In [24, 25, 26] it was remarked that BHs, in the microcanonical description,
fulfill a bootstrap condition [27, 28, 29, 30] and should then exhibit self-
similarity. That is, a BH could be viewed as formed by smaller BHs, formed in
turn of even smaller BHs, and so on. Such picture becomes even more suggestive
if we recall that the Hawking temperature [31] of a Planck-sized BH
($T_{H}\approx l^{-1}_{P}$) could be viewed as the Hagedorn temperature in
string theory [32, 33, 34]. At that temperature, BH evaporation stops and a
phase transition is expected to occur, along the line of the transition
between the hadronic phase and the quark-gluon-plasma phase [35]. There, such
behavior is described by a bootstrap equation [27, 28, 29, 30]. Heavy hadronic
resonances behave as if they were made of lighter resonances, and then the
latter as if they were made of even lighter resonances, and so on, till the
Hagedorn temperature.

To our knowledge, a precise mathematical description of that type was not
proposed for BHs. Then, BH self-similarity is only a little-explored
landscape. In this paper, mainly motivated by such considerations and not
resorting to any specific spin-model, we propose a simple mathematical
construction which explicitly describes BH self-similarity, within the quantum
BH picture.

Let us recall that, the string Hagedorn temperature derives from the typical
dual resonance model spectrum [32], $M^{2}\simeq N$, where $N$ in the integer
labeling the states. This can be seen by evaluating the number of unordered
partition of the integer (UPI) $N$, which grows as $\exp\\{\gamma\sqrt{N}\\}$,
for some constant $\gamma$ [36]. An exponential mass spectrum can be obtained
by a bootstrap equation, as for the hadronic resonances [27], hence a similar
equation should hold in the string description.

On the other hand, in the purely gravitational picture described earlier, the
number of configurations one has to include into $S_{BH}$ grows as $2^{N}$.
The relation of this number, $2^{N}$, with the number of ordered partitions of
the integer (OPI) $N$, that is $2^{N-1}$, is our main concern here. We would
like to reconcile these two numbers, entirely within the quantum BH picture of
[3, 4, 5], and then, knowing that self-similarity is at work in the OPI [37],
move from there to extract from this connection some information on BH self-
similarity, based entirely on the gravitational description.

The paper is organized as follows. In Section II we construct a two-to-one map
between the BH configurations and the OPI. In Section III we use that map to
construct a composition law between the sub-parts making the whole
configuration space that is in one-to-one correspondence with the OPI. We are
then in the position to discuss how the self-similarity of OPI can find an
incarnation in the self-similarity of the BH, as we do in Section IV. Finally,
in Section V we compare the above to the well-known results on the subleading
$\ln A$ corrections, and offer our conclusions in Section VI.

##  II The spin-flip map

What in the physics literature is often referred to as OPI of the (positive)
integer $N$, in more precise mathematical terms, is usually called composition
of $N$ (where the UPIs are called decompositions). Thus compositions are
decompositions of $N$ as the sum of positive integers in which the order of
the terms matters. The number of compositions of $N$, $O(N)$, is such that, if
the first term of a composition of $N$ is $n$, the remaining terms are a
composition of $N-n$, and there are $O(N-n)$ such compositions. This leads to

$O(N)=1+O(N-1)+O(N-2)+\cdots+O(1)\,,$ (3)

therefore $O(N+1)=O(N)+O(N-1)+\cdots+O(1)+1=O(N)+O(N)=2O(N)$, which is solved
by $O(N)=2^{N-1}$, consistently with $O(1)=1$. Notice that the recursive
relation (3) is an instance of the OPI self-similarity.

Thus, on the one hand, $O(N)=2^{N-1}$, whereas the number of configurations,
$C(N)$, of $N$ copies of a two-level spin-$1/2$ system, for instance with one
spin accommodated into each of the $N$ (Planck-sized) plaquettes making the
area $A$, is given by $C(N)=2^{N}$. Therefore, if we want to relate the two
ways of counting configurations, we need to find a 2-to-1 map from the latter
to the former.

Of course, there are many ways to do that, but the crucial point is to
understand which spin configurations correspond to a specific OPI. The way we
find here has many advantages, especially the fact that the self-similarity
will then be obtained in a simple and consistent way.

As will be clarified later, by mapping the OPI of N to the corresponding spin
configurations, we need to distinguish between configurations differing not
only by how many spin are up and how many are down, but also by the position
of the spin. E.g., the configuration $(\uparrow,\uparrow,\downarrow)$ needs to
be distinct from the configuration $(\uparrow,\downarrow,\uparrow)$ and from
the configuration $(\downarrow,\uparrow,\uparrow)$. Other countings do not
distinguish between those configurations, hence carry a degeneracy that is
only a function of how many arrows are down (one here) and how many arrows are
up (two here), see, e.g., [7].

When we distinguish, we indeed have $2^{N}$ configurations, that is precisely
what we need to explain the result in equation (2), i.e., $S=\ln$(number of
microstates)$=\ln(2^{N})$. Now we want to find how this number is related to
the number of ways one can make an OPI of $N$, that is known to be $2^{N-1}$.
Thus, we need to halve the number of BH configurations, and we need to do that
in a consistent way, in order to associate spin states, on the one hand, with
partitions of $N$, on the other hand.

One way to do that is to identify the BH configurations that are related by a
full “spin-flip”, that is: to any one given OPI of $N$ we associate two BH
states that can be obtained one from the other when all the spins that
identify the given configuration change their orientation,
$\uparrow\leftrightarrow\downarrow$. For instance, a particular partition of
$N=3$ (1+1+1, in this case, see later) we make it correspond to
$(\uparrow,\downarrow,\uparrow)$ and to $(\downarrow,\uparrow,\downarrow)$.

This identification is motivated, a posteriori, by the fact that later we
shall be able to propose a composition law among sub-systems of the
configuration space, consistently related to the integers composing $N$, that
gives back precisely the configuration space we started with and this will
enable us to discuss self-similarity.

The other ingredient of the map we are constructing is a rule that relates a
given pair of BH (spin) configurations to a given OPI. We found that the
following rule does the job.

When a spin is next to an opposite spin, i.e., when $\uparrow$ is next to
$\downarrow$ or when $\downarrow$ is next to $\uparrow$, in the OPI this
corresponds to 1 + 1. E.g., $(\uparrow,\downarrow,\uparrow,...)$ and the spin-
flipped $(\downarrow,\uparrow,\downarrow,...)$ both correspond in the OPI to
the partition $1+1+...$. When the spin are likewise they contribute with an
integer that is the sum of how many times the spin does not flip. E.g.,
$(\uparrow,\uparrow,\downarrow,...)$ and
$(\downarrow,\downarrow,\uparrow,...)$ correspond in the OPI to the partition
$2+...$.

Together with the spin-flip, with this rule one takes into account all
possibilities. These make the wanted two-to-one map from the BH configurations
to the OPI. We call it the “spin-flip map”.

As an example, let us write the spin configurations corresponding, through the
spin-flip map, to a specific partition of $N=7$:

$1+3+2+1\to\begin{array}[]{ccccccc}(\uparrow,&\downarrow,&\downarrow,&\downarrow,&\uparrow,&\uparrow,&\downarrow)\\\
(\downarrow,&\uparrow,&\uparrow,&\uparrow,&\downarrow,&\downarrow,&\uparrow)\\\
\end{array}\,.$ (4)

Let us also show how the spin-flip map works for all the OPI of $N=4$

$\displaystyle 1+1+1+1$ $\displaystyle\to$
$\displaystyle(\uparrow,\downarrow,\uparrow,\downarrow)$ $\displaystyle 2+1+1$
$\displaystyle\to$ $\displaystyle(\uparrow,\uparrow,\downarrow,\uparrow)$
$\displaystyle 1+2+1$ $\displaystyle\to$
$\displaystyle(\uparrow,\downarrow,\downarrow,\uparrow)$ $\displaystyle 1+1+2$
$\displaystyle\to$ $\displaystyle(\uparrow,\downarrow,\uparrow,\uparrow)$
$\displaystyle 2+2$ $\displaystyle\to$
$\displaystyle(\uparrow,\uparrow,\downarrow,\downarrow)$ $\displaystyle 3+1$
$\displaystyle\to$ $\displaystyle(\uparrow,\uparrow,\uparrow,\downarrow)$
$\displaystyle 1+3$ $\displaystyle\to$
$\displaystyle(\uparrow,\downarrow,\downarrow,\downarrow)$ $\displaystyle 4$
$\displaystyle\to$ $\displaystyle(\uparrow,\uparrow,\uparrow,\uparrow)\,.$ (5)

The number of OPI $O(N)$ we obtain is precisely $8=2^{3}=O(4)$, while, since
the spin-flip is understood everywhere, on the right side we count
$16=2^{4}=C(4)$ configurations.

Notice that in [3], the multiplicity $2^{N}$ has been interpreted as the
number of ways in which a BH, in the $N$th area level, can be made by first
making a BH in the ground state, and then going up in the ladder of all
possible levels. It has been also discussed in [38] that such multiplicity may
rather represent the number of ways in which the BH can “decay” down the
staircase of levels to the ground state. For both interpretations see also [4]
and [5].

However, in those papers, no specific correlation has been discussed with the
spin configurations. Here, instead, we do so, even though we do not commit
ourselves to any specific physical model that would give meaning to the spin-
flip map. Actually the spin-flip symmetry (degeneracy) may even be solely
instrumental to the construction of the map with the OPI and it should be as
well lifted-up, when it comes to compute the actual information content of a
given configuration. If we do so, then a proper information-theory approach
becomes possible, giving meaning to the “it for qubit” choice discussed
earlier. Indeed, the possibility to distinguish among configurations that have
the same number of $\downarrow$s and of $\uparrow$s, but arranged in different
positions in the string, e.g.
$(\downarrow,\downarrow,\uparrow)\neq(\uparrow,\downarrow,\downarrow)$, allows
such information-theoretical approach, where the position in the string of
values is crucial. E.g., for the example above, we are saying that 1 is
different from 8, in binary notation, $001\neq 100$ (we simply use
$\uparrow\equiv 1$ and $\downarrow\equiv 0$).

##  III The composition law

Having established the spin-flip map, between BH configurations and OPIs, we
want now to see how the self-similarity patterns of the OPI can be
imported/translated into the self-similarity of BHs.

This is less easy than it might seem, because we need to find a way to
construct the given configuration space, of dimension $2^{N}$ starting from
any combination of the smaller configurations spaces, of dimensions
$2^{N_{i}}$s, with $\sum_{i}N_{i}=N$, but, at the same time, we must keep the
correspondence with the partition of integers.

To start, let us notice that, in the example (5) we can read the partition of
the smaller number encoded into the lager, here the partition of $N=3$ into
that of $N=4$

$\displaystyle 1+1+1+1$ $\displaystyle\to$
$\displaystyle(\underbrace{\uparrow,\downarrow,\uparrow},\overbrace{\downarrow})\to(1+1+1)+1$
$\displaystyle 2+1+1$ $\displaystyle\to$
$\displaystyle(\underbrace{\uparrow,\uparrow,\downarrow},\overbrace{\uparrow})\to(2+1)+1$
$\displaystyle 1+2+1$ $\displaystyle\to$
$\displaystyle(\underbrace{\uparrow,\downarrow,\downarrow},\overbrace{\uparrow})\to(1+2)+1$
$\displaystyle 1+1+2$ $\displaystyle\to$
$\displaystyle(\overbrace{\uparrow},\underbrace{\downarrow,\uparrow,\uparrow})\to
1+(1+2)$ $\displaystyle 2+2$ $\displaystyle\to$
$\displaystyle(\underbrace{\uparrow,\uparrow},\underbrace{\downarrow,\downarrow})\to(2)+(2)$
$\displaystyle 3+1$ $\displaystyle\to$
$\displaystyle(\underbrace{\uparrow,\uparrow,\uparrow},\overbrace{\downarrow})\to(3)+1$
$\displaystyle 1+3$ $\displaystyle\to$
$\displaystyle(\overbrace{\uparrow},\underbrace{\downarrow,\downarrow,\downarrow})\to
1+(3)$ $\displaystyle 4$ $\displaystyle\to$
$\displaystyle(\uparrow,\uparrow,\uparrow,\uparrow)\quad{\rm NEW}\,.$ (6)

On the right side we read in parentheses the partitions of $N=3$, obtained on
the left side by grouping spins with the underbrace. That is indeed a way to
extract a smaller partition from a bigger partition, but it is easy to see
that there is some unwanted degeneracy on the right side. This is evident in
the fifth line, where we have twice the partition of 2, but it is also true
that we could have grouped the spins of the first line to obtain $1+(1+1+1)$,
or the spins of the third line to obtain $1+(2+1)$, etc. The degeneracies,
clearly, grow when we include the partitions of the next smaller number, 2
here, and, in general, they grow with $N$. So, we can indeed trace the
partitions of the smaller numbers within the partitions of the given number
$N$, but we have a problem with a heavy over-counting.

To avoid that, we shall work with the spin configurations, and define an
operation, $\hat{+}$, that allows to obtain the configuration space of the
given BH made of $N$ plaquettes, of dimension $C(N)=2^{N}$, only once for any
given partition. In other words, if we indicate with $\mathbf{N}$ such
configuration space, and $N_{1}+N_{2}+\cdots=N$ is a given OPI of N, we want
that $\mathbf{N_{1}}\hat{+}\mathbf{N_{2}}\hat{+}\cdots=\mathbf{N}$. Doing so
we would establish a one-to-one correspondence between the OPI of $N$, and the
way to combine the subspaces of $\mathbf{N}$, corresponding to the OPI. The
operation is constructed as follows:

We first list all the OPIs of $N$. Then take each partition in the list, say
$N_{1}+N_{2}=N$, and write the spin configurations for $\mathbf{N_{1}}$, the
configuration space associated to the first number of the sum. (To simplify
the operation we can consider even just one representative per each spin-
flipped pair, even though, to obtain all the configurations, at the end of the
procedure the spin-flip needs be applied). Then, we take the tensor product of
each of those representatives with all the spin configurations of
$\mathbf{N_{2}}$ explicitly including the spin-flipped configurations. The
result of such operation, $\mathbf{N_{1}}\hat{+}\mathbf{N_{2}}$, are all the
spin configurations of $\mathbf{N}$, with no redundant nor missed
configuration. (Clearly, if the spin-flip was understood at the start, then so
will be in the final result). The operation will give the same result for each
OPI of $N$, including those with more than two terms. For the latter, one must
start from the first term on the left, act with the second as just described,
and the result of this needs be acted upon with the next term, and so on, till
the end.

The trivial example is $\mathbf{N}=\mathbf{N}$, where no composition is
performed. The first non-trivial operation is $\mathbf{1}\hat{+}\mathbf{1}$,
that originates from the partition 1+1=2, so it must give $\mathbf{2}$:

$\mathbf{1}\hat{+}\mathbf{1}=\uparrow\otimes\begin{array}[]{c}{\color[rgb]{1,0,0}\uparrow}\\\
{\color[rgb]{1,0,0}\downarrow}\end{array}=\begin{array}[]{c}\uparrow{\color[rgb]{1,0,0}\uparrow}\\\
\uparrow{\color[rgb]{1,0,0}\downarrow}\end{array}=\mathbf{2}\,.$ (7)

Here, in the second-last term, the first line is one spin representative of 2,
$(\uparrow,\uparrow)$, while the second line is one spin representative of
1+1, $(\uparrow,\downarrow)$. As said, the four-dimensional ($2^{N}=2^{2}$),
full configuration space, $\mathbf{2}$, is obtained when we spin-flip each
final configuration: $(\uparrow,\uparrow)$,$(\downarrow,\downarrow)$ and
$(\uparrow,\downarrow)$,$(\downarrow,\uparrow)$.

Let us now show how to act with three addends, e.g.,
$\mathbf{1}\hat{+}\mathbf{1}\hat{+}\mathbf{1}$ that is one of the four ways
($2^{N-1}=2^{2}$) to obtain $\mathbf{3}$

$(\mathbf{1}\hat{+}\mathbf{1})\hat{+}\mathbf{1}=\left(\uparrow\otimes\begin{array}[]{c}{\color[rgb]{1,0,0}\uparrow}\\\
{\color[rgb]{1,0,0}\downarrow}\end{array}\right)\hat{+}\mathbf{1}=\left(\begin{array}[]{c}\uparrow\uparrow\\\
\uparrow\downarrow\end{array}\right)\otimes\begin{array}[]{c}{\color[rgb]{1,0,0}\uparrow}\\\
{\color[rgb]{1,0,0}\downarrow}\end{array}=\begin{array}[]{c}\uparrow\uparrow{\color[rgb]{1,0,0}\uparrow}\\\
\uparrow\uparrow{\color[rgb]{1,0,0}\downarrow}\\\
\uparrow\downarrow{\color[rgb]{1,0,0}\uparrow}\\\
\uparrow\downarrow{\color[rgb]{1,0,0}\downarrow}\end{array}=\mathbf{3}\,.$ (8)

Here, in the second-last term, the first line is the spin representative of 3,
$(\uparrow,\uparrow,\uparrow)$, the second line is the spin representative of
2+1, $(\uparrow,\uparrow,\downarrow)$, the third line is the spin
representative of 1+1+1, $(\uparrow,\downarrow,\uparrow)$, and the third line
is the spin representative of 1+2, $(\uparrow,\downarrow,\downarrow)$. Again,
the full configuration space $\mathbf{3}$, of dimension eight, is obtained
when we spin-flip each of the four final configuration.

It is crucial to notice again that this operation gives $\mathbf{N}$, the full
configuration space, of dimension $2^{N}$, in a non degenerate way, each time
we use any of the $2^{N-1}$ OPIs of $N$. Therefore, the 8-dimensional
$\mathbf{3}$ ($8=2^{3}$) is obtained 4 times ($4=2^{3-1}$) as
$\mathbf{1}\hat{+}\mathbf{1}\hat{+}\mathbf{1}$, $\mathbf{2}\hat{+}\mathbf{1}$,
$\mathbf{1}\hat{+}\mathbf{2}$ and $\mathbf{3}$.

To show how $\hat{+}$ acts when none of the addends is $\mathbf{1}$ let us
show the case $\mathbf{2}\hat{+}\mathbf{2}$ that should give $\mathbf{4}$

$\mathbf{2}\hat{+}\mathbf{2}=\begin{array}[]{c}\uparrow\downarrow\\\
\uparrow\uparrow\end{array}\otimes\begin{array}[]{c}{\color[rgb]{1,0,0}\uparrow\downarrow}\\\
{\color[rgb]{1,0,0}\downarrow\uparrow}\\\
{\color[rgb]{1,0,0}\uparrow\uparrow}\\\
{\color[rgb]{1,0,0}\downarrow\downarrow}\end{array}=\begin{array}[]{c}\uparrow\downarrow{\color[rgb]{1,0,0}\uparrow\downarrow}\\\
\uparrow\downarrow{\color[rgb]{1,0,0}\downarrow\uparrow}\\\
\uparrow\downarrow{\color[rgb]{1,0,0}\uparrow\uparrow}\\\
\uparrow\downarrow{\color[rgb]{1,0,0}\downarrow\downarrow}\\\
\uparrow\uparrow{\color[rgb]{1,0,0}\uparrow\downarrow}\\\
\uparrow\uparrow{\color[rgb]{1,0,0}\downarrow\uparrow}\\\
\uparrow\uparrow{\color[rgb]{1,0,0}\uparrow\uparrow}\\\
\uparrow\uparrow{\color[rgb]{1,0,0}\downarrow\downarrow}\end{array}=\mathbf{4}\,.$
(9)

Here, in the second-last term, scrolling from the first to the last line, we
recognise one spin representative of: 1+1+1+1, 1+2+1, 1+1+2, 1+3, 3+1, 2+1+1,
4 and 2+2, i.e, of all the 8 OPIs of 4. As said, the full 16-dimensional
$\mathbf{4}$ is obtained via spin-flip.

Once we have exploited the first simple cases, the general proof follows by
induction. In fact, suppose the above composition rule holds for $\mathbf{N}$.
Then

$\mathbf{1}\,\hat{+}\,\mathbf{N}\ =\
\mathbf{1}\,\hat{+}\,\mathbf{N_{1}}\,\hat{+}\,\mathbf{N_{2}}\,\hat{+}\,\cdots\,\hat{+}\mathbf{N_{M}}\,,$
(10)

where $N_{1}+N_{2}+\cdots N_{M}=N$, with $M\leq N$. Because $N_{1}\leq N$, we
can apply the composition rule to get

$\mathbf{1}\,\hat{+}\,\mathbf{N}\ =\
\left(\mathbf{N_{1}+1}\right)\,\hat{+}\,\mathbf{N_{2}}\,\hat{+}\,\cdots\,\hat{+}\mathbf{N_{M}}\
=\ \mathbf{N+1}\,,$ (11)

which concludes our proof.

##  IV The self-similarity

If the configuration space $\mathbf{N}$ is that of a quantum BH a la
Bekenstein [5], then we have found a clean and consistent way to see BH self-
similarity:

Keeping the BH fixed, its configuration space $\mathbf{N}$ is made of the
configuration spaces of smaller BHs, that are made of configuration spaces of
even smaller BHs, and again and again, until we reach $N$ copies of
$\mathbf{1}$, the configuration space of the tiniest (elementary) BH.

That is, since to any of the $2^{N-1}$ OPIs of $N$ we can associate one of the
$2^{N-1}$ OPIs of $\mathbf{N}$

$\sum_{i}N_{i}\ =\
N\to\hat{\sum}_{i}\mathbf{N_{i}}=\mathbf{N}\,,\,\,\sum_{j}M_{j}=N\to\hat{\sum}_{j}\mathbf{M_{j}}=\mathbf{N}\,,...\,,$
(12)

where
$\hat{\sum}_{i}\mathbf{N_{i}}=\mathbf{N_{1}}\hat{+}\mathbf{N_{2}}\hat{+}\cdots$,
whatever pattern we found in the OPI of $N$, it is found in the configuration
space $\mathbf{N}$ of the BH, and then repeated for the smaller numbers, until
we reach the “quantum” of the BH space, $\mathbf{1}$.

For instance, a suggestive pattern is given by

$\mathbf{N}\ =\ \mathbf{1}\hat{+}(\mathbf{N-1})\ =\
\mathbf{1}\hat{+}(\mathbf{1}\hat{+}(\mathbf{N-2}))=\mathbf{1}\hat{+}(\mathbf{1}\hat{+}(\mathbf{1}\hat{+}(\mathbf{N-3})))=\cdots=\hat{\sum}^{N}_{i=1}\mathbf{1}\,.$
(13)

Here one can say that when the configuration space of the tiniest BH,
$\mathbf{1}$, is isolated from the rest, than this can be repeated again and
again till the complete splitting.

One crucial comment is that, as wanted, in this picture the self-similarity
does not require any change of description of the degrees of freedom. We keep
the BH fixed and we are just finding patterns within the configuration space
of that given BH. In other words, we do not deal here neither with BH
evaporation [31] nor we deal with BHs merging [39].

In the first case, we have a configuration space that “decays” till complete
evaporation [10]: $\mathbf{N}\to\mathbf{N-1}\to\cdots\to\mathbf{0}$, and the
self-similarity is something that should emerge from the Hagedorn spectrum, as
discussed in [33, 32], but that was never explicitly found, e.g., by solving
the related bootstrap equation. An approximate self-similarity in BH
evaporation in the semiclassical limit was also discussed in Ref. [40].

In the second case, when we merge two BHs, one of area $A_{1}$ and mass
$M_{1}$, with $A_{1}=\alpha N_{1}\sim M_{1}^{2}$, one of area $A_{2}$ and mass
$M_{2}$, with $A_{2}=\alpha N_{2}\sim M_{2}^{2}$, and with
$\alpha=4l_{P}^{2}\ln 2$, the area of the resulting BH of mass $M=M_{1}+M_{2}$
is $A\sim(M_{1}+M_{2})^{2}$, i.e.,

$A=\alpha N=\alpha(N_{1}+N_{2}+2\sqrt{N_{1}N_{2}})\,,$ (14)

which points to

$S\ =\ S_{1}\ +\ S_{2}\ +\ 2\sqrt{S_{1}S_{2}}\,.$ (15)

In other words, the configuration spaces $\mathbf{N_{1}}$ and
$\mathbf{N_{2}}$, of dimensions $2^{N_{1}}$ and $2^{N_{2}}$, respectively,
when combined through the operation that we have constructed, $\hat{+}$, give
a configuration space of dimension $2^{N_{1}+N_{2}}$, which is much smaller
than the configuration space of the BH obtained by merging the two BHs, that
is

$2^{N}\ =\ 2^{N_{1}+N_{2}}\times\left(2^{\sqrt{N_{1}N_{2}}}\right)^{2}\,.$
(16)

##  V Configurations Analysis

In the previous Sections, consistently with a quantum information-theoretical
approach and with the picture of a BH as a free (non-interacting) gas of
fermions ($X$ons), we have treated all configurations in $\mathbf{N}$ equally.
Let us compare here our analysis to the literature [7], where interesting
questions are posed on weighting differently the different configurations, and
how this can be related to the subleading corrections to the Bekenstein
formula, $S_{BH}\sim A/4-\beta\ln A$. Such corrections are usually referred to
as “quantum corrections”, as indeed they are obtained in perturbative quantum
theories [15, 16, 17, 18, 19, 20]. Here we do not call them so, because we are
within a quantum BH model [3, 4, 5], in the first place.

As discussed, in our case the actual position of the spin within the given
configuration matters, so that, e.g., $8\,(=100)\neq 1\,(=001)$, where in
parentheses we indicate the binary number. This is not the point of view of
interacting spin systems, taken elsewhere [7]. There, in general, the various
configurations, characterized by how many spins are up and how many are down,
are labeled by the energy.

However, to reproduce the BH entropy in that approach, one cannot select a
unique configuration. A large entropy requires that the lowest energy state
corresponds to a large subset of the possible $2^{N}$ configurations.
Therefore, in that picture, the spin-spin coupling between spins in positions
$i$ and $j$, say it $J_{i,j}$, should actually be insensitive to such
positions, i.e., $J_{i,j}\equiv J$. For example, one can consider the Ising
model in an antiferromagnetic state, with constant spin-spin coupling, $J<0$,
and energy $E$.

The energy levels of such system do not depend on the position of the spin. If
we have $N$ total spins, and $n$ of those are flipped with respect to the
remaining $N-n$, the energy levels are given by [41]

$E_{n}\ =\ \frac{J}{4}[n-\frac{1}{2}(N-n/2)(N-n/2-1)]\,,$ (17)

where, for simplicity we take $N$ even and where $n=0,1,2,..N$. The total
number of energy levels is $N+1$.

The degeneracy of level $n$ is

$d_{n}=B(N,n)\ =\ \frac{N!}{n!(N-n)!}\,.$ (18)

The lowest energy level is for $n=N/2$, with degeneracy

$d_{N/2}\ =\
\frac{N!}{[(N/2)!]^{2}}\simeq\frac{2^{N}}{\sqrt{N}}\sqrt{2/\pi}\,,$ (19)

where the Stirling approximation has been used. Therefore, the resulting
entropy of this particular subset of the whole set of configurations is

$S(N/2)\ =\ N\ln 2-\frac{1}{2}\ln N+constant\,,$ (20)

which gives a logarithmic subleading correction, as calculated with various
approaches with different coefficients of the $\ln N$ term [15, 16, 17, 18,
19, 20]. This may be considered a point in favour of this approach, but it
also has evident drawbacks.

First, this state has negative energy $E=JN/8<0$ proportional to the total
number of spins, i.e. to the number of Planck plaquettes, $N$, whereas the BH
mass is proportional to $\sqrt{N}$.

Second, by an ad hoc choice of the coupling, $J\simeq 1/\sqrt{N}$, a fine-
tuning with the large amount of positive energy is required to get the BH
mass. This problem can be overcome only by assuming the specific coupling
$J=K/N$, i.e the Kac model [42], since in this case the BH mass should be [41]

$M\ =\ M_{P}\sqrt{N}\,+\,E_{N/2}\ =\ M_{P}\sqrt{N}\,+\,K/8\,.$ (21)

Finally, the coefficient of the $\ln N$ correction crucially depends on the
specific configuration, henceforth, if some other configuration contributes
too, the coefficient changes accordingly. Indeed the total degeneracy in the
general case is given by

$D(E)\ =\ \Sigma_{n\neq
N/2}\frac{B(N,n)}{|n-N/2|}[\delta(n-n_{+}(E))+\delta(n-n_{-}(E))]\,+\,d_{N/2}\delta(E-E_{N/2})\,,$
(22)

where

$n_{\pm}\ =\ \frac{1}{2}[N\pm\sqrt{(N-8E/J)}]$ (23)

Let us consider a configuration with $n=N/2+m$ with $m<<N/2$. From the
previous equation one obtains

$m\ =\ \frac{1}{2}[\sqrt{(N-8E/J)}]$ (24)

The corresponding term in the degeneracy is

$d_{m}\ =\ \frac{1}{m}B(N,N/2+m)$ (25)

which, after some algebra by using Stirling formula, turns out to be

$d_{m}\ \simeq\ 2^{N}\frac{1}{m}\frac{1}{\sqrt{2\pi
N}}\frac{1}{(1-4m^{2}/N^{2})^{N/2+1}}\frac{1}{1-m^{2}/2N^{2}}$ (26)

For large $N$, the most relevant correction to the entropy associated to
$d_{N/2}$ comes from the $1/m$ term ($c$ and $c^{\prime}$ being numerical
constants or $O(1/N)$ terms) by Eqs.(17)-(21), i.e.

$S(m)\ \simeq\ N\ln 2-\frac{1}{2}\ln N-\ln m+c=N\ln 2-\ln N+c^{\prime}$ (27)

with a change in the coefficient of the $\ln N$ correction, that, for large
$N$ does not depend on the different choices of the coupling $J$ (
$J\sim$constant, $J\sim O(1/\sqrt{N})$ or $J\sim O(1/N)$). Therefore, the
coefficient of the subleading contributions originates from a specific,
although large, subset of all the possible configurations.

##  VI Conclusions

Within a quantum model of BHs, whose associated degrees of freedom are finite
in number and fermionic, we have studied general aspects of its entropy. In
particular, we have established a precise correspondence between the OPI of a
natural number $N$ and the configuration (Hilbert) space of the BH, of
dimension $2^{N}$. This is done by constructing a two-to-one map from the
former to the latter, and then a composition law, $\hat{+}$, based on the
tensor product, between sub-parts of the configuration space.

Through this composition law, we have been able to account for BH self-
similarity through the well-known self-similarity of OPI. In our approach,
such self-similar structure of the BH’s configuration space emerges in a
transparent way, and the operation $\hat{+}$ allows to always obtain all
possible $2^{N}$ configurations.

The point of view taken here considers non-interacting (free) fermionic
degrees of freedom (spins), along the lines of [8, 9, 10], with equally
probable configurations, identified by number and position of the spin in the
(information theoretical) string.

If we want to use an information-theoretical approach, the selection of a
specific configuration over the others is not appropriate. To use the BH as
the ultimate quantum computer [21], then one expects that all configurations
are treated equally, and the evolution of the quantum states should not be
fixed by a given spin model, but should rather be governed by a specific
Hamiltonian that “implements” the given “computation”.

## Acknowledgements

P.C. and A.I. gladly acknowledge support from Charles University Research
Center (UNCE/SCI/013). L.S. was supported by the Polish National Science
Center grant 2018/31/D/ST2/02048.

## References

  * [1]
  * [2]

## References

* [3] V. Mukhanov, JEPT Lett. 44 (1986) 63.
* [4] J. D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum black hole, Phys. Lett. B 360 (1995) 7.
* [5] J. D. Bekenstein, Quantum Black Holes as Atoms, Proc. Eight Marcel Grossmann Meeting, T. Piran and R. Ruffini, eds. (World Scientific Singapore 1999), pp. 92-111.
* [6] J. A. Wheeler, Proc. III International Symposium on Foundations of Quantum Mechanics, Tokyo, 1989, p. 354-368.
* [7] C. Kiefer, J. Phys.: Conf. Series 1612 (2020) 012017.
* [8] G. Acquaviva, A. Iorio and M. Scholtz, Ann. Phys. 387 (2017) 317.
* [9] A. Iorio, J. Phys.: Conf. Series 1275 (2019) 012013.
* [10] G. Acquaviva, A. Iorio and L. Smaldone, Phys. Rev. D 102 (2020) 106002.
* [11] S. Hod, Phys. Rev. Lett. 81 (1998) 4293.
* [12] M. Maggiore, Phys. Rev. Lett. 100 (2008) 141301.
* [13] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80 (1998) 904.
* [14] C. Rovelli, Quantum gravity, (Cambridge University Press, Cambridge, 2004); C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity : An Elementary Introduction to Quantum Gravity and Spinfoam Theory, (Cambridge University Press, Cambridge, 2014).
* [15] R. K. Kaul and P. Majumdar, Phys. Rev. Lett. 84 (2000) 5255.
* [16] K. S. Gupta and S. Sen, Phys. Lett. B 526 (2002) 121.
* [17] A. Ghosh and P. Mitra, Phys. Rev. D 71 (2005) 027502.
* [18] A. Majhi and P. Majumdar, Class. Quant. Grav. 31 (2014) 195003.
* [19] D. Singleton, E. C. Vagenas and T. Zhu, JHEP 05 (2014) 074 .
* [20] Y. C. Ong, Phys. Lett. B 785 (2018) 217.
* [21] S. Lloyd, Nature 406 (2000) 1047.
* [22] P. Shor, Proc. of the 35th Annual Symposium of the Foundations of Computer Science (IEEE, New York, 1994); SIAM J. Comput. 26 (1997) 1484; arXiv:quant-ph/0005003.
* [23] P. Giorda, A. Iorio, S. Sen, S. Sen, Phys. Rev. A 70 (2004) 032303.
* [24] B. Harms and Y. Leblanc, Phys. Rev. D 46 (1992) 2334.
* [25] B. Harms and Y. Leblanc, Phys. Rev. D 47 (1993) 2438.
* [26] W. H. Huang, Phys. Rev. D 62 (2000) 043002.
* [27] R. Hagedorn, Nuovo Cim. Suppl. 3 (1965) 147.
* [28] S. C. Frautschi, Phys. Rev. D 3 (1971) 2821.
* [29] W. Nahm, Nucl. Phys. B 54 (1972) 525.
* [30] R.Hagedorn and J. Rafelski, Comm. Math.Phys. 83 (1982) 563.
* [31] S.W. Hawking, Nature 248 (1974) 30; Comm. Math. Phys. 43 (1975) 199.
* [32] G. Veneziano, in Melting Hadrons, Boiling Quarks – From Hagedorn Temperature to Ultra-Relativistic Heavy-Ion Collisions at CERN, J. Rafelski ed., Springer open.
* [33] L. Susskind, L. Thorlacius and J. Uglum, Phys. Rev. D 48 (1993) 3743.
* [34] L. Susskind and J. Lindesay, An introduction to black holes, information, and the string theory revolution, The Holographic Universe,( World Scientific, 2005).
* [35] N. Cabibbo and G. Parisi, Phys. Lett. B 59 (1975) 67.
* [36] B. Zwiebach, A first course in string theory, (Cambridge University Press, 2006).
* [37] P. Blanchard, S. Fortunato and H. Satz, Eur. Phys. J. C 34 (2004) 361.
* [38] U. H. Danielsson and M. Schiffer, Phys. Rev. D 48 (1993) 4779.
* [39] B. P. Abbott, R. Abbott , T. D. Abbott., M. B. Abernathy, F. Acernese, K. Ackley et al., Phys. Rev. Lett. 116 (2016) 061102.
* [40] G. Dvali and C. Gomez, Phys. Lett. B 716 (2012) 240.
* [41] A. Czachor, Acta Phys. Pol. A 113 (2008) 1161.
* [42] M. Kac, in Statistical Physics, Phase Transitions and Superfluidity, Eds. M. Chretien, E. P. Gross, S. Desser, Vol. 1, Gordon and Breach, New York 1968, p. 241.


1.6 K, neglecting non-linearity of the Hall coefficient in $B$ (typically
10-20% in our devices). The temperature-dependent Hall mobility
$\mu_{\text{H}}$ was calculated as $\mu_{\text{H}}(T)=(eN(\text{1.6
K})R(T))^{-1}$, i.e. a $T$-independent carrier density is assumed.

Figure S31:  High mobility 2DEGs in additional devices. Device A: (a) Hall
density at 1.6 K, tun by $V_{\text{GIL}}$ above 220 K, (b) corresponding Hall
mobility. Different traces are for different channel widths along the device.
(c) Temperature dependence of Hall mobility in devices A, B, C and in typical
devices without a barrier layer. Devices labeled in order of low temperature
mobility.

Fig. S30a,b shows an example of systematic carrier density tuning in the
2-8$\times 10^{13}$ cm-2 range (measured at 1.6 K) by adjusting
$V_{\text{GIL}}$ above 220 K. Gradual non-uniformity of measured Hall density
over tens of microns was usually present, especially in larger devices. The
density shown for each Hall bar region was taken to be an average between the
two adjacent pairs of Hall contacts. The Hall mobility in optimized conditions
typically reached several thousands of cm2/Vs (see Fig. S31b,c). The general
trend of increasing $\mu_{\text{H}}$ at high $N_{\text{H}}$ was common in
studied devices.

Between 1.6K and near room temperatures, metallic behavior was observed for
carrier densities that were high enough to get reliable ohmic contacts
(usually above $\approx 10^{13}$ cm-2 ). Extrapolated mobility at room
temperature was always close to 10 cm2/Vs, as typical for electron-doped
SrTiO3 [80]. Typical traces for SrTiO3 Hall bar devices without HfOx barrier
layers are also shown for comparison in Fig. S31c. Such devices have
mobilities of order 100-1000 cm2/Vs at base temperature, see also [81, 82,
26]. Comparison of temperature dependence also showcases the much larger
residual resistivity ratio in high mobility SrTiO3/HfOx devices (up to
$\approx$ 500).

Consequently, despite significant statistical scatter between devices, the
insertion of a thin HfOx barrier layer consistently improves Hall mobility
from 102-103 cm2/Vs into the 103-104 cm2/Vs range. Correspondingly, the mean
free path is improved from tens of nm into the range of hundreds of nm to a
few microns.

[SMrefs]


| strongly violated | 0.874 | 0.166 | 4.17 | 4.44 | 0.854 | 0.143 | 3.28 | 0.50
|  | Federated II (ours) | Target site only
| holds | 0.906 | 0.021 | 4.12 | 0.52 | 0.907 | 0.023 | 4.18 | 0.60
ASR | weakly violated | 0.904 | 0.019 | 4.08 | 0.49 | 0.906 | 0.022 | 4.12 | 0.58
| strongly violated | 0.904 | 0.021 | 4.09 | 0.50 | 0.905 | 0.023 | 4.14 | 0.60
| holds | 0.921 | 0.033 | 3.84 | 0.49 | 0.924 | 0.043 | 3.98 | 0.73
Local ASR | weakly violated | 0.924 | 0.035 | 3.87 | 0.56 | 0.924 | 0.043 | 3.96 | 0.72
| strongly violated | 0.923 | 0.032 | 3.87 | 0.52 | 0.923 | 0.047 | 3.97 | 0.76
| holds | 0.879 | 0.143 | 3.76 | 1.59 | 0.854 | 0.177 | 3.48 | 0.81
CQR | weakly violated | 0.884 | 0.141 | 3.77 | 1.67 | 0.849 | 0.183 | 3.46 | 0.81
| strongly violated | 0.889 | 0.143 | 4.03 | 3.53 | 0.849 | 0.177 | 3.46 | 0.82
|  | Federated III | Equal weights
| holds | 0.908 | 0.021 | 4.18 | 0.55 | 0.915 | 0.025 | 4.43 | 0.77
ASR | weakly violated | 0.908 | 0.019 | 4.17 | 0.51 | 0.915 | 0.023 | 4.44 | 0.73
| strongly violated | 0.907 | 0.020 | 4.16 | 0.53 | 0.914 | 0.024 | 4.43 | 0.73
| holds | 0.929 | 0.031 | 3.98 | 0.52 | 0.942 | 0.035 | 4.38 | 0.96
Local ASR | weakly violated | 0.932 | 0.031 | 4.03 | 0.67 | 0.947 | 0.032 | 4.58 | 2.28
| strongly violated | 0.930 | 0.032 | 3.99 | 0.56 | 0.947 | 0.033 | 4.49 | 1.24
| holds | 0.942 | 0.079 | 4.02 | 1.08 | 0.964 | 0.064 | 5.00 | 3.18
CQR | weakly violated | 0.937 | 0.088 | 4.01 | 1.54 | 0.960 | 0.074 | 4.81 | 2.30
| strongly violated | 0.940 | 0.080 | 4.07 | 1.72 | 0.964 | 0.065 | 5.11 | 2.77

  * •

CFS: conformal score; CCOD: common conditional outcome distribution

  * •

CP: coverage probability; wd: width; s.d.: standard deviation (over 500
replications)

Table 8: $n_{k}=1000$, strongly heterogeneous covariate distribution

CFS | CCOD | CP | s.d.(CP) | wd | s.d.(wd) | CP | s.d.(CP) | wd | s.d.(wd)
---|---|---|---|---|---|---|---|---|---
|  | Homoscedasticity where $\sigma(x)=1$
|  | Federated I | Pooled sample
| holds | 0.898 | 0.017 | 3.28 | 0.15 | 0.899 | 0.008 | 3.28 | 0.06
ASR | weakly violated | 0.897 | 0.016 | 3.27 | 0.14 | 1.000 | 0.000 | 8.74 | 0.15
| strongly violated | 0.897 | 0.017 | 3.27 | 0.15 | 1.000 | 0.000 | 23.78 | 0.49
| holds | 0.898 | 0.018 | 3.28 | 0.15 | 0.899 | 0.011 | 3.29 | 0.09
Local ASR | weakly violated | 0.898 | 0.019 | 3.28 | 0.17 | 0.835 | 0.015 | 2.78 | 0.08
| strongly violated | 0.898 | 0.018 | 3.28 | 0.15 | 0.732 | 0.022 | 2.22 | 0.09
| holds | 0.900 | 0.024 | 3.31 | 0.22 | 0.899 | 0.011 | 3.29 | 0.10
CQR | weakly violated | 0.899 | 0.026 | 3.31 | 0.24 | 0.899 | 0.015 | 3.29 | 0.13
| strongly violated | 0.899 | 0.024 | 3.31 | 0.21 | 0.899 | 0.016 | 3.30 | 0.15
|  | Federated II (ours) | Target site only
| holds | 0.899 | 0.015 | 3.28 | 0.13 | 0.900 | 0.014 | 3.30 | 0.13
ASR | weakly violated | 0.898 | 0.014 | 3.28 | 0.12 | 0.900 | 0.014 | 3.30 | 0.13
| strongly violated | 0.898 | 0.014 | 3.28 | 0.12 | 0.900 | 0.014 | 3.30 | 0.12
| holds | 0.898 | 0.016 | 3.28 | 0.14 | 0.900 | 0.018 | 3.30 | 0.17
Local ASR | weakly violated | 0.899 | 0.017 | 3.29 | 0.15 | 0.901 | 0.019 | 3.31 | 0.18
| strongly violated | 0.899 | 0.016 | 3.29 | 0.13 | 0.900 | 0.018 | 3.31 | 0.17
| holds | 0.901 | 0.020 | 3.32 | 0.19 | 0.901 | 0.020 | 3.32 | 0.18
CQR | weakly violated | 0.900 | 0.022 | 3.31 | 0.20 | 0.903 | 0.021 | 3.34 | 0.19
| strongly violated | 0.900 | 0.020 | 3.31 | 0.18 | 0.903 | 0.020 | 3.34 | 0.18
|  | Federated III | Equal weights
| holds | 0.901 | 0.015 | 3.30 | 0.13 | 0.903 | 0.017 | 3.33 | 0.16
ASR | weakly violated | 0.900 | 0.015 | 3.30 | 0.13 | 0.903 | 0.018 | 3.33 | 0.17
| strongly violated | 0.901 | 0.013 | 3.30 | 0.12 | 0.902 | 0.017 | 3.33 | 0.16
| holds | 0.902 | 0.016 | 3.32 | 0.15 | 0.907 | 0.019 | 3.37 | 0.19
Local ASR | weakly violated | 0.903 | 0.017 | 3.33 | 0.15 | 0.907 | 0.019 | 3.37 | 0.19
| strongly violated | 0.903 | 0.014 | 3.33 | 0.13 | 0.906 | 0.018 | 3.37 | 0.17
| holds | 0.910 | 0.018 | 3.41 | 0.21 | 0.918 | 0.022 | 3.51 | 0.28
CQR | weakly violated | 0.908 | 0.020 | 3.39 | 0.20 | 0.915 | 0.025 | 3.48 | 0.28
| strongly violated | 0.909 | 0.017 | 3.41 | 0.17 | 0.915 | 0.022 | 3.48 | 0.25
|  | Heteroscedasticity where $\sigma(x)=-\log(x)$
|  | Federated I | Pooled sample
| holds | 0.906 | 0.017 | 4.09 | 0.40 | 0.908 | 0.007 | 4.11 | 0.13
ASR | weakly violated | 0.905 | 0.015 | 4.06 | 0.37 | 0.983 | 0.003 | 9.04 | 0.18
| strongly violated | 0.904 | 0.016 | 4.07 | 0.38 | 1.000 | 0.000 | 23.98 | 0.51
| holds | 0.925 | 0.024 | 3.83 | 0.37 | 0.932 | 0.014 | 3.91 | 0.20
Local ASR | weakly violated | 0.924 | 0.024 | 3.85 | 0.35 | 0.866 | 0.019 | 3.11 | 0.15
| strongly violated | 0.924 | 0.028 | 3.85 | 0.37 | 0.773 | 0.023 | 2.42 | 0.14
| holds | 0.855 | 0.141 | 3.28 | 0.53 | 0.886 | 0.060 | 3.19 | 0.22
CQR | weakly violated | 0.861 | 0.142 | 3.30 | 0.53 | 0.877 | 0.078 | 3.19 | 0.25
| strongly violated | 0.863 | 0.134 | 3.29 | 0.49 | 0.868 | 0.098 | 3.19 | 0.32
|  | Federated II (ours) | Target site only
| holds | 0.907 | 0.014 | 4.10 | 0.33 | 0.908 | 0.014 | 4.14 | 0.34
ASR | weakly violated | 0.905 | 0.013 | 4.07 | 0.31 | 0.907 | 0.014 | 4.11 | 0.33
| strongly violated | 0.905 | 0.014 | 4.08 | 0.32 | 0.907 | 0.013 | 4.13 | 0.34
| holds | 0.927 | 0.021 | 3.85 | 0.32 | 0.930 | 0.027 | 3.93 | 0.44
Local ASR | weakly violated | 0.925 | 0.021 | 3.86 | 0.32 | 0.927 | 0.026 | 3.92 | 0.41
| strongly violated | 0.926 | 0.023 | 3.87 | 0.33 | 0.930 | 0.025 | 3.95 | 0.44
| holds | 0.866 | 0.123 | 3.27 | 0.44 | 0.850 | 0.144 | 3.26 | 0.50
CQR | weakly violated | 0.870 | 0.124 | 3.29 | 0.45 | 0.863 | 0.127 | 3.30 | 0.46
| strongly violated | 0.876 | 0.113 | 3.29 | 0.42 | 0.863 | 0.132 | 3.30 | 0.50
|  | Federated III | Equal weights
| holds | 0.909 | 0.014 | 4.17 | 0.34 | 0.912 | 0.017 | 4.27 | 0.45
ASR | weakly violated | 0.908 | 0.014 | 4.15 | 0.35 | 0.911 | 0.017 | 4.24 | 0.47
| strongly violated | 0.908 | 0.013 | 4.16 | 0.32 | 0.910 | 0.015 | 4.22 | 0.41
| holds | 0.933 | 0.021 | 3.96 | 0.36 | 0.939 | 0.024 | 4.11 | 0.49
Local ASR | weakly violated | 0.931 | 0.022 | 3.97 | 0.37 | 0.936 | 0.026 | 4.10 | 0.52
| strongly violated | 0.933 | 0.020 | 3.98 | 0.33 | 0.937 | 0.023 | 4.07 | 0.43
| holds | 0.914 | 0.090 | 3.49 | 0.43 | 0.933 | 0.083 | 3.75 | 0.60
CQR | weakly violated | 0.916 | 0.085 | 3.50 | 0.43 | 0.929 | 0.090 | 3.74 | 0.60
| strongly violated | 0.921 | 0.082 | 3.51 | 0.41 | 0.934 | 0.082 | 3.73 | 0.53

  * •

CFS: conformal score; CCOD: common conditional outcome distribution

  * •

CP: coverage probability; wd: width; s.d.: standard deviation (over 500
replications)

Table 9: $n_{k}=3000$, strongly heterogeneous covariate distribution Figure 9:
Boxplots of coverage probability, under strongly heterogeneous covariate
distributions Figure 10: Boxplots of prediction interval width, under strongly
heterogeneous covariate distributions

###  B.3 Local coverage over covariate values and scatterplots of federated
weights

In this section, we provide the conditional coverage and federated weights
plots.

Figure 11 shows the plots of local coverage of the constructed prediction
intervals over a grid of $X\in[0,1]$, where the sample size is set to be
$n_{k}=3000$ for all sites. We used the smoothing method and published R code
by Lei et al. (2018) for these plots. We can see that under homoscedasticity,
the local coverage is constant (a horizontal line) over the covariate values
by a given conformal score. Most of these horizontal lines are close to $0.9$,
except for the pooled sample. The three federated weights consistently
performed well under homoscedasticity. Furthermore, under heteroscedasticity,
we can see the local coverage when the value of $X$ is too small always
deviates from the nominal level by all methods and conformal scores, which
makes sense as $-\log x\to\infty$ when $x\to 0$. When $X$ is sufficiently
larger than $0$, the local coverage increases. Among the three conformal
scores, ASR is the most sensitive one to the change in variance, and does not
have coverage close to $0.9$ almost everywhere. This confirms findings in Lei
et al. (2018). The other two conformal scores are more robust against the
heteroscedastic variance. When $X\in[0.1,0.6]$, their local coverages are
close to $0.9$, except for the pooled sample method.

In addition, Figure 12 shows three federated weights vs. $\chi^{2}_{k}$ values
using data of $n_{k}=3000$ and under heteroscedasticity, where we only plotted
weights corresponding to $\chi_{k}^{2}\in[0,0.5]$ for illustration purposes,
i.e., some weights corresponding to $\chi_{k}^{2}>0.5$ are not shown. As can
be seen from the upper 9 panels when CCOD holds, in every case, all weights
are clustered more or less around $0.2$. When covariate distributions are
heterogeneous, the weights distributions become more complex, but generally
when $\chi_{k}^{2}$ is smaller, there are larger weights in each panel. Also,
there are obviously some larger weights ($>0.2$, i.e., above the red dashed
lines) in site 1; about half of the weights are below $0.2$ for both sites 2
and 3, and most weights for site 4 are close to $0$. Although site 3 has some
surprisingly large weights, it also shows a more unstable pattern of weights,
which might be a reflection of its heterogeneity to the target site. Overall,
the trend of weights fits the expectation of our method: the bigger the
difference to the target site, the smaller (or the less stable) the weights.

Figure 11: Local coverages, under CCOD is strongly violated and strongly
heterogeneous covariate distributions and $n_{k}=3000$

Figure 12: Weights vs. $\chi_{k}^{2}$ values, using $n_{k}=3000$ data under
heteroscedasticity. The green points are by Federated I, the orange points are
by Federated II (ours), the blue points are by Federated III, and the red
dashed lines are for a reference line weights $=0.2$.  Figure 13: Comparison
of coverage probabilities and average interval width when modifying the
propensity score of observing the outcome between $(0.4,0.6)$ (panel (a)) and
$(0.1,0.9)$ (panel (b)).


# Anisotropic Cage Evolution in Quasi-two-dimensional Colloidal Fluids

Noman Hanif Barbhuiya  Department of Physics, Indian Institute of Technology
Gandhinagar, Palaj, Gandhinagar, 382055, Gujarat, India  Chandan K. Mishra
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Gandhinagar, Palaj, Gandhinagar, 382055, Gujarat, India

###### Abstract

Employing video microscopy, we explore the cage dynamics for colloidal
particles confined in quasi-two dimensions (q2D). Our experiments reveal that
while ensemble-averaged dynamics of cages are isotropic in the laboratory
frame, its evolution in the displacement frame of the caged particle is
anisotropic and asymmetric. In turn, this leads to particles in specific
regions of the cage contributing either to cage persistence or breaking,
influencing the structural relaxation of the fluid. Our findings, thus,
provide microscopic insights into cage evolution and dynamics for colloidal
fluids in q2D, with direct potential implications for the flow of complex
fluids, structural relaxation in dense suspensions, and collective motion in
active matter in confined geometries.

Cage formation, whether transient or long-lived, is a fundamental feature of
condensed matter systems such as (supercooled) liquids [1, 2, 3, 4], glasses
[5, 6, 7, 8] and crystals [9, 10, 11, 12]. Microscopic insights into the
structural configuration and dynamical evolution of cages have offered crucial
understandings of diverse phenomena, such as the rheological [13] and
mechanical properties [5] of the materials as well as their phase behavior
[14]. For example, while the shape and geometry of the cage can distinguish
between supercooled liquid and glassy states of passive glass formers [15],
the cage length and persistence elucidate the phase behavior of active glasses
[14]. Notably, the distinct nature of glass transition and melting of crystal
in reduced spatial dimensions like two-dimensions (2D), compared to three-
dimensions (3D), wherein long-wavelength Mermin-Wagner fluctuations influence
the dynamics, have become dimension-agnostic once the dynamics of the
particles relative to its cage, instead of their self-dynamics, are considered
[16, 17, 18].

Interestingly, recent studies suggest that long-wavelength Mermin-Wagner
fluctuations are also present in 2D colloidal liquids, leading to the
violation of the ubiquitous Stokes-Einstein relation, which connects the
microscopic diffusivity of the tracers and the bulk viscosity of the liquid.
Once again, the cage-relative dynamics of the particles have been shown to
restore the usual behavior of the Stokes-Einstein relation for 2D liquids
[19]. However, despite the extensive exploration of cage relative dynamics,
the evolution of the cages themselves in 2D colloidal fluids $-$ their
transient nature leading to possible changes in their shape over time, which
consequently dictates the local structural relaxation of the fluid$-$ has
never been explored. It is particularly intriguing as long-wavelength
fluctuations have been alluded to stem from hydrodynamic interactions [19,
20], which have been revealed to be direction-dependent in the body frame of
particle-pair for quasi-two-dimensional (q2D) colloidal fluids [21]. The
influence of these emergent hydrodynamic interactions in q2D colloidal fluids
on the dynamic evolution of the cages is anything but obvious.

In this Letter, we employ video microscopy to experimentally investigate the
temporal evolution of cages, both in the laboratory and in the displacement
frame of the reference caged-colloid in q2D colloidal fluids. The cage of each
particle at the initial time, $t_{0}$, is determined using its Voronoi cell
[15]. As expected, with time, the size of the cages grows, leading to the
local structural relaxation of the fluid [Figure 1] [22, 23]. Interestingly,
ensemble-averaged dynamics of particles forming the cage reveal that while
they diffuse isotropically in the laboratory frame of reference, the
distribution of the particles forming the cage is observed to be anisotropic
and asymmetric in the displacement frame of the caged-particle [Figure 2].
Finally, consistent with dipolar hydrodynamic motional modes, we identify
distinct hotspots for particles of the cage at the initial time that either
contribute to cage persistence or cage relaxation at a later time and
demonstrate their anisotropic evolution [Figure 3].

Aqueous suspension of charge-stabilized polystyrene colloidal particles,
diameter: $\sigma=1.04$ $\mu$m, polydispersity $\sim 3\%$, were loaded in a
wedge-shaped cell, which was then allowed to stand vertically for particles to
sediment under gravity and form a monolayer in the quasi-two-dimensional
region of the cell [21]. Once a desired area fraction was achieved, the cell
was equilibrated for several hours under the microscope. After equilibration,
video microscopy was performed for 20 minutes using 100x oil objective (1.4
Numerical aperture) at 10 frames per second (fps) using the Hamamatsu ORCA-
Flash4.0 camera. We have captured data in the liquid regime at five area
fractions, $\phi$, in the range, $0.15\leq\phi\leq 0.35$, in the same region
of the cell. The trajectory of each particle was determined using standard
image processing and tracking algorithms [24]. The dynamic spatial resolution
in our experiment was 20 nm.

The identity of cage particles, $C(i,t)$, for $i^{th}-$particle at time, $t$,
is determined using Voronoi tesselation, which partitions the 2D space into
neighborhoods (cells) around each particle such that every point within a cell
is closer to the particle than to any other particle [inset to Fig. 1(a)]
[25]. The order of $C(i,t)$, $n(C(i,t))$, represents the number of particles
forming the cage of the $i^{th}-$particle that is also equal to the number of
vertices in its Voronoi cell. With time, the identity of particles contained
in $C(i,t)$ changes, leading to the relaxation of the cage and, consequently,
the local structural relaxation of the fluid. The relaxation (or persistence)
time of the cage is measured by calculating the ensemble-averaged fraction of
common cage particles, $\langle f(\Delta t)\rangle$, with lag time, $\Delta
t$, $\langle f(\Delta t)\rangle={1\over
N}{\langle\sum_{i}^{N}\frac{n(C(i,t_{0})\cap C(i,t_{0}+\Delta
t))}{n(C(i,t_{0}))}\rangle}_{t_{0}}$. Here, ${\langle\cdot\rangle}_{t_{0}}$
represents initial time averaging over $t_{0}$, corresponding to a given
$\Delta t$, and $N$ is the total number of particles. $\langle f(\Delta
t)\rangle$ exhibits a stretched-exponential decay and yields the persistence
time of the cage [Fig. 1(a)]. For the range of $\phi$s studied, the
persistence time was observed to be around $10$ s, which is comparable to the
timescales over which effects of hydrodynamics are conspicuous. Thus, the cage
evolution and dynamics may be closely connected to the local structural
relaxation of the fluid governed by hydrodynamic interactions.

Figure 1: Cage characterization in quasi-two-dimensional colloidal fluids. (a)
Average fraction of common cage particles, $\langle f(\Delta t)\rangle$,
versus lag time, $\Delta t$, for area fraction, $\phi=0.22$. Inset shows the
rendering of the Voronoi cells (polygons) for a representative subset of
particles (spheres) in the experimental field of view at initial, $t_{0}$
(left), and later time, $t_{0}+\Delta t$ (right). The representative reference
particle’s (red sphere) Voronoi cell with its adjoining cells at $t_{0}$ (cyan
polygons) and $t+\Delta t$ (green polygons), determines the identity of the
particles forming the cage (shown as numerals on the spheres). (b) The average
radius of gyration, $R_{g}(\Delta t)$, of the particles forming the cage
versus lag time, $\Delta t$, for $\phi=0.20$ (blue circles) and $\phi=0.35$
(red squares). The inset shows the distribution of $R_{g}$ for $\phi=0.20$ at
$\Delta t=2.5$ s.

Moreover, as the local structure of the fluid relaxes, the particles of the
cage will randomly diffuse, leading to an increase in average cage size, which
can be measured using the radius of gyration, $R_{g}(i,\Delta t)$, of the
particles forming the cage of $i^{th}-$particle over $\Delta t$.
$\displaystyle R_{g}(i,\Delta t)=\sqrt{{1\over n(C(i,\Delta t))}\sum_{c_{j}\in
C(i,\Delta t)}(\textbf{r}_{j}(t_{0}+\Delta
t)-{\langle\textbf{r}_{j}(t_{0}+\Delta t)\rangle}_{j}})^{2}$. Here,
$\textbf{r}_{j}(t_{0}+\Delta t)$ is the position of the $j^{th}-$particle
belonging to the set of particles that form the cage of $i^{th}-$particle at
$t_{0}$. Note, throughout the Letter, the sets and their elements are
represented using upper and lowercase letters, respectively. The ensemble-
averaged distribution of $R_{g}(i,\Delta t)$, $R_{g}(\Delta t)$, is found to
be Gaussian [inset to Fig. 1(b)]. As expected, $R_{g}(\Delta t)$ grows with
$\Delta t$. Over the persistent time of the cages ($\sim 10$ s), it is found
to be about 2-3$\sigma$ [Fig. 1(b)], implying that the temporal evolution of
cages should be strongly governed by the nature of near-field hydrodynamic
interactions in q2D spatial confinement, which is direction-dependent
(anisotropic) in the body frame of colloid-pairs [21].

As the fluid relaxes, particles of the cage, as defined at $t_{0}$, will also
diffuse, which may lead to a change in the shape of the cage. Thus, to capture
the microscopic evolution in the shape of the cages with $\Delta t$, we plot
the probability distribution, $P({\textbf{r}_{j}(t_{0}+\Delta t)}|{c_{j}\in
C(i,t_{0})})$, of the positions of the particles of the cage, defined at
$t_{0}$, over $\Delta t$ [Fig. 2(a)-(e)]. Here, the argument on the right of
the “$|$” represents the condition applied for the quantity being measured on
the left. Note that the probabilities are calculated from the ensembles of
particles (and their cages) and averaged over $\Delta t$. Particles forming
the cage spread radially, both outward and inward, from their initial
position, with $P(\textbf{r}_{j}(t_{0}+\Delta t)|c_{j}\in C(i,t_{0}))$ being
symmetric about the center of mass of the cages at $t_{0}$. The diffused
annular ring signifies the caging of the particle by its neighbors in
colloidal fluids [Fig. 2(b)-(e)], and as expected, the annular ring spreads
with $\Delta t$. For a given $\Delta t$, with an increase in $\phi$, as caging
effects strengthen, the annular ring becomes more localized [Fig. 2(b)-(i) &
Supplemental Video 1].

Figure 2: Cage evolution in quasi-two-dimensional colloidal fluids. (a)
Rendering from experiments illustrating the laboratory and displacement frames
of reference. A portion of the field of view at times $t_{0}$ (left) and
$t_{0}+\Delta t$ (middle), a representative particle (red sphere) and its cage
particles (cyan spheres) are determined from Voronoi cells (cyan polygons) at
$t_{0}$. The right panel demonstrates the transformation of particle
coordinates to the displacement frame, where the displacement of the
reference-caged particle (red) is aligned with the x-axis. Ensemble-averaged
colormap of the probability distribution of the positions of the particles
forming the cage, $P(\textbf{r}_{j}(t_{0}+\Delta t)|c_{j}\in C(i,t_{0}))$, in
the (b)-(e) laboratory frame, and (f)-(i) displacement frame,
$P(\textbf{r}_{j}^{disp}(t_{0}+\Delta t)|c_{j}\in C(i,t_{0}))$, from all
reference particles, for two distinct $\phi$ and $\Delta t$. The white cross
and red solid circle in (b)-(i) represent the mean of the probability
distribution at $t_{0}$ and $t_{0}+\Delta t$, respectively. The yellow arrows
in (g)-(i) show the displacement direction of the reference caged particle.
(j) The net probability within an annular region with an inner and outer
radius of $1\sigma$ and $1.25\sigma$, centered on the mean of the probability
distribution, as a function of angle $\theta$ with respect to the x-axis (as
illustrated in (e)) and $\Delta t$ for $\phi=0.35$, in the displacement frame
(main subfigure) and laboratory frame (inset). (k) The aspect ratio, $\alpha$,
of the fitted ellipse along the inner edge of the annular space of the
probability distribution as a function of $\Delta t$ for $\phi=0.20$ (blue
circles) and $\phi=0.35$ (red triangle) in the laboratory (open symbols) and
displacement frames (solid symbols).

$P(\textbf{r}_{j}(t_{0}+\Delta t)|c_{j}\in C(i,t_{0}))$ is found to be
symmetric. However, note that the laboratory frame of measurements of the
positions of particles forming the cage for the analysis of
$P(\textbf{r}_{j}(t_{0}+\Delta t)|c_{j}\in C(i,t_{0}))$ will be unable to
capture any possible influence of the unique nature of hydrodynamic
interactions in q2D colloidal fluids on the evolution of the shape of the
cages. Hence, to discern the effects of hydrodynamics on the evolution of the
shape of the cages, we calculate $P(\textbf{r}_{j}^{disp}(t_{0}+\Delta
t)|c_{j}\in C(i,t_{0}))$ in the displacement frame of the $i^{th}-$reference
particle. Here, we rotate the positions of all the particles of the cages such
that the displacement of reference-caged particle aligns along the $x-$axis
[Fig. 2(a)]. In essence, we transform $\textbf{{r}}_{i}$ in the lab frame to
$\textbf{r}_{i}^{disp}$ in displacement frame by
$\textbf{r}_{i}^{disp}=\textbf{R}(-\psi)\textbf{r}_{i}$, where $\psi$ is angle
that the displacement of the reference-caged particle substends with $x-$axis
and $\textbf{R}(-\psi)$ is the rotation matrix with elements
$R_{lm}(\psi)=\cos(\psi)\delta_{lm}+\sin(\psi)\delta_{lm}$.

Interestingly, unlike $P(\textbf{r}_{j}(t_{0}+\Delta t)|c_{j}\in C(i,t_{0}))$,
$P(\textbf{r}_{j}^{disp}(t_{0}+\Delta t)|c_{j}\in C(i,t_{0}))$ deviates from
the symmetric annular shape and is no longer isotropic with some preferred
regions with higher probability density [Fig. 2(f)-(i) & Supplemental Video
2]. To quantify this localization of the probabilities, we calculate the net
probability within an annular region with an inner and outer radius of
$1\sigma$ and $1.25\sigma$, which is centred on the mean of the probability
distribution, as a function of angle $\theta$ (for example, see Fig. 2(e)).
For a fixed $\phi$, while the net probability is constant for all $\theta$ for
$P(\textbf{r}_{j}(t_{0}+\Delta t)|c_{j}\in C(i,t_{0}))$, however, it displays
two maxima at $\theta\sim\pm 90$ for $P(\textbf{r}_{j}^{disp}(t_{0}+\Delta
t)|c_{j}\in C(i,t_{0}))$ [Fig. 2(j)]. It indicates an accumulation of
particles forming the cage in regions perpendicular to the motion of the caged
particle, which is in concordance with the dipolar nature of hydrodynamics in
q2D colloidal fluid.

To further quantify the anisotropy in $P(\textbf{r}_{j}^{disp}(t_{0}+\Delta
t)|c_{j}\in C(i,t_{0}))$, we fit an ellipse along the inner edge of the
annular space, which outlines the contour of the probability distribution’s
maxima [26]. We define the aspect ratio of the fitted ellipsoid, $\alpha$, the
ratio of the major to the minor axis, as the anisotropy parameter associated
with the probability distribution. In the laboratory frame, irrespective of
the $\phi$ and $\Delta t$, $\alpha$ remains closer to unity, suggesting the
isotropic spread of the particles of the cages around the reference caged-
particle [Fig. 2(k))]. However, intriguingly, in the displacement frame, as
already visually inferred from the colormaps in Figure 2(f)-(i), the shape
anisotropy parameter $\alpha$ increases with $\Delta t$. The angle-dependent
localization of $P(\textbf{r}_{j}^{disp}(t_{0}+\Delta t)|c_{j}\in C(i,t_{0}))$
and its time-dependent elongation, thus suggests a strong role of
hydrodynamics governing the evolution of cages, potentially dictating the
dynamics of cage breaking (and persistence).

Figure 3: Distinct anisotropic spatial probability distribution of the cage
particles leading to cage persistence and relaxation. (a) Rendering from the
experiments of a portion of the field of view illustrating the classification
of cage particles (cyan spheres) at time $t_{0}$ into common,
$K(i,t_{0},\Delta t)$, and uncommon, $\Lambda(i,t_{0},\Delta t)$, subsets
based on their presence or absence in the cage defined for the same reference
particle (red sphere) a later time, $t_{0}+\Delta t$. The common and uncommon
cage particles are circled in purple and blue, respectively. (b) Colormap of
the probability distribution of the positions of the common cage particles,
$P(\textbf{r}_{j}^{disp}(t_{0})|k_{j}\in K(i,t_{0},\Delta t))$ and (c)
$P(\textbf{r}_{j}^{disp}(t_{0}+\Delta t):k_{j}\in K(i,t_{0},\Delta t))$, at
$t_{0}$ and $t_{0}+\Delta t$, respectively, for $\phi=0.35$ and $\Delta t=2.5$
s. (d) and (e) show the probability distribution of the positions of the
common cage particles at $t_{0}$ and $t_{0}+\Delta t$, respectively, for
$\phi=0.35$ and $\Delta t=2.5$ s, considering only the subset of particles
within an angular region of $\pm 15^{\circ}$ with respect to the direction of
displacement of the reference-caged particle. (f) and (g) depict the same as
(b) and (c) but for the uncommon cage particles, $\Lambda(i,t_{0},\Delta t)$.
The white cross and red solid circle in (b)-(g) represent the mean of the
probability distribution at $t_{0}$ and $t_{0}+\Delta t$, respectively. The
yellow arrows in (b)-(g) show the displacement direction of the reference
caged particle.

Finally, having established that the shape of the cage is anisotropic in the
displacement frame of the reference caged-particle, we now delve into how the
particles of the cage rearrange, which either leads to the persistence of the
cage or its breaking and local structural relaxation of the fluid [22, 23]. In
other words, are there any specific regions in the cage that predominantly
help with cage relaxation and persistence? Note, for a reference caged-
particle, say $i^{th}-$particle, and its cage particles $C(i,t_{0})$ defined
at time $t_{0}$, a subset of these particles, $K(i,t_{0},\Delta t)$, will
remain part of the cage $C(i,t_{0}+\Delta t)$ at later time $t_{0}+\Delta t$.
This common subset, $K(i,t_{0},\Delta t)=C(i,t_{0})\cap C(i,t_{0}+\Delta t)$,
contributes to cage persistence, while particles that were part of
$C(i,t_{0})$ but are not included in $C(i,t_{0}+\Delta t)$, denoted as
$\Lambda(i,t_{0},\Delta t)=C(i,t_{0})\setminus C(i,t_{0}+\Delta t)$,
contribute to cage breaking [Fig. 3(a)]. Here, “$\setminus$” denotes the set
difference.

To investigate if particles in certain spatial regions of the cage are more
likely to contribute to cage persistence, we analyze
$P(\textbf{r}_{j}^{disp}(t_{0})|k_{j}\in K(i,t_{0},\Delta t))$ and
$P(\textbf{r}_{j}^{disp}(t_{0}+\Delta t)|k_{j}\in K(i,t_{0},\Delta t))$. We
find that at $t_{0}$, particles that are located in the direction of the
displacement of the reference caged-particle have a higher likelihood of
remaining part of the cage of the same particle at $t_{0}+\Delta t$ [Fig. 3(b)
& Supplemental Video 3]. At $t_{0}+\Delta t$, however, the common particles of
the cage at $t_{0}$ and $t_{0}+\Delta t$ are more likely to be found in the
transverse direction of the reference-cage particle [Fig. 3(c) & Supplemental
Video 3]. To further corroborate these arguments, we have only considered a
subset of $K(i,t_{0},\Delta t)$ such that they lie within an angular region of
$\pm 15^{\circ}$ with respect to the direction of displacement of the
reference-cage particle. Figure 3(d) and (e) clearly show that common cage
particles rearrange in a distinctive “mushroom cloud” pattern, rearranging
their positions vertically and moving opposite to the direction of the
reference caged-particle’s displacement. Conversely, the uncommon particles of
the cage, initially distributed opposite to the displacement direction at
$t_{0}$, become more diffused at $t_{0}+\Delta t$ [Fig. 3(f)-(g) &
Supplemental Video 3]. This indicates that cage breaking and structural
relaxation are predominantly facilitated by particles of the cage that are in
the regions opposite the reference caged-particle’s motion.

In summary, our experiments represent a first-of-its-kind unravelling of the
microscopic dynamics of cage evolution, consistent with unique features of
emergent near-field hydrodynamic interactions for q2D colloidal fluids. We
demonstrate cages evolve isotropically and symmetrically in the laboratory
frame of reference. However, ensemble-averaged analyses reveal anisotropic and
asymmetric evolution in the displacement frame of the reference caged-
particle. Colloids forming the cage, which are initially ahead of the
reference caged-particle, reorganize and diffuse transversely with respect to
the direction of displacement of the caged-particle, contributing to the
persistence of the cage. Conversely, particles behind the reference caged-
particle diffuse further away, predominantly leading to cage breaking and
structural relaxation of the fluid. While our study focussed on cage dynamics
for colloidal fluids in q2D rigid confinement, it would be worthwhile to
explore them for suspension of particles at interfaces [16, 27] as well as
ones confined on curved surfaces [28, 29]. Additionally, future studies may
explore systems with broken ergodicity, such as jammed and glassy states [30,
31, 32], active matter that may have non-trivial cage relaxation dynamics
[33], and systems with soft and anisotropic particles that will introduce
ruggedness in the energy landscapes, even in two dimensions [34, 35, 36].

We are grateful for the useful discussions with Sivasurender Chandran, B.
Prasanna Venkatesh, and K. Hima Nagamanasa. We gratefully acknowledge
financial support from the Department of Science and Technology (Government of
India), INSPIRE fellowship IF200274 (N.H.B.), research initiation grant from
IIT Gandhinagar through IP/IITGN/PHY/CM/2021/11 (C.K.M.) and the Start-up
Research Grant of Science and Engineering Research Board of Government of
India through SRG/2021/001077 (C.K.M.).

## References

  * Reis  _et al._ [2007] P. M. Reis, R. A. Ingale, and M. D. Shattuck, Caging dynamics in a granular fluid, Phys. Rev. Lett. 98, 188301 (2007).
  * Perakis  _et al._ [2018] F. Perakis, G. Camisasca, T. J. Lane, A. Späh, K. T. Wikfeldt, J. A. Sellberg, F. Lehmkühler, H. Pathak, K. H. Kim, K. Amann-Winkel, _et al._ , Coherent X-rays reveal the influence of cage effects on ultrafast water dynamics, Nat. Communications 9, 1917 (2018).
  * Giri  _et al._ [2015] N. Giri, M. G. Del Pópolo, G. Melaugh, R. L. Greenaway, K. Rätzke, T. Koschine, L. Pison, M. F. C. Gomes, A. I. Cooper, and S. L. James, Liquids with permanent porosity, Nature 527, 216 (2015).
  * Gallo  _et al._ [1996] P. Gallo, F. Sciortino, P. Tartaglia, and S.-H. Chen, Slow dynamics of water molecules in supercooled states, Phys. Rev. Lett. 76, 2730 (1996).
  * Li  _et al._ [2020] B. Li, K. Lou, W. Kob, and S. Granick, Anatomy of cage formation in a two-dimensional glass-forming liquid, Nature 587, 225 (2020).
  * Marty and Dauchot [2005] G. Marty and O. Dauchot, Subdiffusion and cage effect in a sheared granular material, Phys. Rev. Lett. 94, 015701 (2005).
  * Weeks and Weitz [2002] E. R. Weeks and D. Weitz, Properties of cage rearrangements observed near the colloidal glass transition, Phys. Rev. Lett. 89, 095704 (2002).
  * Parisi and Zamponi [2010] G. Parisi and F. Zamponi, Mean-field theory of hard sphere glasses and jamming, Rev. Mod. Phys. 82, 789 (2010).
  * Rabani  _et al._ [1999] E. Rabani, J. D. Gezelter, and B. Berne, Direct observation of stretched-exponential relaxation in low-temperature Lennard-Jones systems using the cage correlation function, Phys. Rev. Lett. 82, 3649 (1999).
  * White  _et al._ [2017] S. L. White, P. Banerjee, and P. K. Jain, Liquid-like cationic sub-lattice in copper selenide clusters, Nature Communications 8, 14514 (2017).
  * Quinn and Goree [2002] R. Quinn and J. Goree, Particle interaction measurements in a Coulomb crystal using caged-particle motion, Phys. Rev. Lett. 88, 195001 (2002).
  * Gerbode  _et al._ [2010] S. J. Gerbode, D. C. Ong, C. M. Liddell, and I. Cohen, Dislocations and vacancies in two-dimensional mixed crystals of spheres and dimers, Phys. Rev. E 82, 041404 (2010).
  * Mayer  _et al._ [2008] C. Mayer, E. Zaccarelli, E. Stiakakis, C. Likos, F. Sciortino, A. Munam, M. Gauthier, N. Hadjichristidis, H. Iatrou, P. Tartaglia, _et al._ , Asymmetric caging in soft colloidal mixtures, Nat. Materials 7, 780 (2008).
  * Debets  _et al._ [2021] V. E. Debets, X. M. De Wit, and L. M. Janssen, Cage length controls the nonmonotonic dynamics of active glassy matter, Phys. Rev. Lett. 127, 278002 (2021).
  * Zhang  _et al._ [2024] H. Zhang, Q. Zhang, F. Liu, and Y. Han, Anisotropic-isotropic transition of cages at the glass transition, Phys. Rev. Lett. 132, 078201 (2024).
  * Illing  _et al._ [2017] B. Illing, S. Fritschi, H. Kaiser, C. L. Klix, G. Maret, and P. Keim, Mermin–wagner fluctuations in 2D amorphous solids, Proc. Natl. Acad. Sci. USA 114, 1856 (2017).
  * Vivek  _et al._ [2017] S. Vivek, C. P. Kelleher, P. M. Chaikin, and E. R. Weeks, Long-wavelength fluctuations and the glass transition in two dimensions and three dimensions, Proc. Natl. Acad. Sci. USA 114, 1850 (2017).
  * Zheng and Earnshaw [1998] X. Zheng and J. Earnshaw, On the Lindemann criterion in 2D, Europhys. Lett. 41, 635 (1998).
  * Li  _et al._ [2019] Y.-W. Li, C. K. Mishra, Z.-Y. Sun, K. Zhao, T. G. Mason, R. Ganapathy, and M. Pica Ciamarra, Long-wavelength fluctuations and anomalous dynamics in 2-dimensional liquids, Proc. Natl. Acad. Sci. USA 116, 22977 (2019).
  * Shiba  _et al._ [2019] H. Shiba, T. Kawasaki, and K. Kim, Local density fluctuation governs the divergence of viscosity underlying elastic and hydrodynamic anomalies in a 2d glass-forming liquid, Phys. Rev. Lett. 123, 265501 (2019).
  * Barbhuiya  _et al._ [2023] N. H. Barbhuiya, A. Yodh, and C. K. Mishra, Direction-dependent dynamics of colloidal particle pairs and the Stokes-Einstein relation in quasi-two-dimensional fluids, Nat. Communications 14, 5109 (2023).
  * Chen  _et al._ [2023] Y. Chen, Z. Ye, K. Wang, J. Huang, H. Tong, Y. Jin, K. Chen, H. Tanaka, and P. Tan, Visualizing slow internal relaxations in a two-dimensional glassy system, Nat. Phys. 19, 969 (2023).
  * Cao  _et al._ [2017] X. Cao, H. Zhang, and Y. Han, Release of free-volume bubbles by cooperative-rearrangement regions during the deposition growth of a colloidal glass, Nat. Communications 8, 362 (2017).
  * Crocker and Grier [1996] J. C. Crocker and D. G. Grier, Methods of digital video microscopy for colloidal studies, J. Colloid Interface Sci. 179, 298 (1996).
  * Burrough  _et al._ [2015] P. A. Burrough, R. A. McDonnell, and C. D. Lloyd,  _Principles of geographical information systems_ (Oxford University Press, USA, 2015).
  * [26] Method for determining the inner contour of the probability distribution’s maxima through ray tracing: Rays are emanated in all directions and traced from the probability distribution’s center (mean) to identify points where the probability exceeds $\sim 95\%$ of the maximum probability. These points define the contour on which an ellipse is fitted.
  * van Baalen  _et al._ [2023] C. van Baalen, J. Vialetto, and L. Isa, Tuning electrostatic interactions of colloidal particles at oil-water interfaces with organic salts, Phys. Rev. Lett. 131, 128202 (2023).
  * Bausch  _et al._ [2003] A. Bausch, M. J. Bowick, A. Cacciuto, A. Dinsmore, M. Hsu, D. Nelson, M. Nikolaides, A. Travesset, and D. Weitz, Grain boundary scars and spherical crystallography, Science 299, 1716 (2003).
  * Singh  _et al._ [2022] N. Singh, A. Sood, and R. Ganapathy, Observation of two-step melting on a sphere, Proc. Natl. Acad. Sci. USA 119, e2206470119 (2022).
  * Charbonneau  _et al._ [2012] P. Charbonneau, A. Ikeda, G. Parisi, and F. Zamponi, Dimensional study of the caging order parameter at the glass transition, Proc. Natl. Acad. Sci. USA 109, 13939 (2012).
  * Charbonneau  _et al._ [2014] P. Charbonneau, Y. Jin, G. Parisi, and F. Zamponi, Hopping and the Stokes-Einstein relation breakdown in simple glass formers, Proc. Natl. Acad. Sci. USA 111, 15025 (2014).
  * Morse and Corwin [2014] P. K. Morse and E. I. Corwin, Geometric signatures of jamming in the mechanical vacuum, Phys. Rev. Lett. 112, 115701 (2014).
  * Ni  _et al._ [2013] R. Ni, M. A. C. Stuart, and M. Dijkstra, Pushing the glass transition towards random close packing using self-propelled hard spheres, Nat. Communications 4, 2704 (2013).
  * Yang  _et al._ [2016] X. Yang, R. Liu, M. Yang, W.-H. Wang, and K. Chen, Structures of local rearrangements in soft colloidal glasses, Phys. Rev. Lett. 116, 238003 (2016).
  * Zheng  _et al._ [2011] Z. Zheng, F. Wang, and Y. Han, Glass transitions in quasi-two-dimensional suspensions of colloidal ellipsoids, Phys. Rev. Lett. 107, 065702 (2011).
  * Mishra and Ganapathy [2015] C. K. Mishra and R. Ganapathy, Shape of dynamical heterogeneities and fractional Stokes-Einstein and Stokes-Einstein-Debye relations in quasi-two-dimensional suspensions of colloidal ellipsoids, Phys. Rev. Lett. 114, 198302 (2015).