# Hamming distance as a measure of spatial chaos in evolutionary games

Gaspar Alfaro<EMAIL_ADDRESS>Nonlinear Dynamics, Chaos and Complex
Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán
s/n, Móstoles, 28933, Madrid, Spain  Miguel A.F. Sanjuán
<EMAIL_ADDRESS>Nonlinear Dynamics, Chaos and Complex Systems Group,
Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, Móstoles,
28933, Madrid, Spain

###### Abstract

From a context of evolutionary dynamics, social games can be studied as
complex systems that may converge to a Nash equilibrium. Nonetheless, they can
behave in an unpredictable manner when looking at the spatial patterns formed
by the agents’ strategies. This is known in the literature as spatial chaos.
In this paper we analyze the problem for a deterministic prisoner’s dilemma
and a public goods game and calculate the Hamming distance that separates two
solutions that start at very similar initial conditions for both cases. The
rapid growth of this distance indicates the high sensitivity to initial
conditions, which is a well-known indicator of chaotic dynamics.

evolutionary dynamics social games prisoner’s dilemma public goods numerical
simulations Hamming distance

††preprint: APS/123-QED

##  I Introduction

Evolutionary Game Theory has granted us a better knowledge of social behavior,
for example understanding of cooperation, which has a great importance in
social games like the prisoner’s dilemma [1, 2], PD, or the public goods game
[3, 4], PGG. We think of an evolutionary game as a dynamical problem of
natural selection, where the payoff of each agent, acting as a fitness,
depends on its strategy and on that of its neighbors. This field has attracted
physicists [5] that have included ideas from nonlinear dynamics, like the
discovery of spatial, pattern-forming chaos arising from a simple
deterministic game as in the research work of Martin A. Nowak and Robert M.
May [6].

The rules of the game are simple: there are cooperators, C, and defectors, D,
which play games with one neighbor. They gain a payoff according to the payoff
matrix:

|  C |  D
---|---|---
C | R | S
D | T | P

The game is a prisoner’s dilemma when $T>R>P>S$. This is called a dilemma
because even though cooperation benefits all players in the group, defection
is the preferable strategy. This is known as tragedy of the commons [7]. But
all is not lost for cooperation, since it can prosper under several conditions
like reputation effects [8, 9] and network reciprocity [10, 11, 12].

The evolutionary mechanism used is a pairwise imitation process where each
agent copies the strategy of the one with maximum payoff among its neighbors.
This game can be viewed as a cellular automaton, but one with $2^{25}$ rules,
that’s because there are $25$ neighbors (with Moore neighborhood) that affect
an agent strategy at the next iteration, see Fig 1.

xy Figure 1: Agent $x$’s strategy is affected by the payoff of all agents in
his Moore neighborhood (inside the dashed line); for example agent $y$, whose
payoff depends of his own Moore neighborhood (shaded region). Therefore, the
strategy of the agent $x$ depends on all $25$ agents in his second Moore
neighborhood (inside the bold line).

Even though the “ever-changing sequences of spatial patterns” that May and
Nowak saw in [6] for a parameter regime of the game have an obvious chaotic
nature, no method has been provided to evaluate or quantify the stability of a
less obvious case.

Lyapunov’s exponents have been used broadly in the context of nonlinear
dynamics and chaotic systems to quantify chaos. Some efforts to bring this
chaos indicator to the context of cellular automata have been done in [13,
14]. Indeed our PD game is basically a cellular automata as we have previously
mentioned. We could use this ”Lyapunov’s exponent for cellular automata”, in
fact, we tried, but no reasonable results were obtained. We think that may be
because the exponent developed in these works does not assess chaotic
behaviour, but a different measure.

To assess correctly the chaotic behaviour of the spatial patterns arising in
the PD game we have used the Hamming distance. This is a measure of the
difference between two ordered sets of equal length. It is defined as the
number of positions at which the corresponding symbols are different in the
two ordered sets. In the context of social games, Hamming distance has been
used with various purposes. Authors of [15] use Hamming distance to measure
the separation between strategies that are binary strings. In a similar
manner, in [16] Hamming distance is used in order to measure the separation
between a tag which agents use to discriminate other agents and either
cooperate if their tag is close by or defect if its too different.

Moreover, the Hamming distance is used to asses chaotic behaviour in [17, 18],
where they calculate the distance between two initially close configurations
of generalized rock-paper-scissors models that describe stochastic network
simulations of the May and Leonard type. They find that Hamming distance
converges to a certain value, but, if a parameter representing the mobility is
above a critical point, after some time, the distance oscillates increasing
its amplitude of oscillation until it goes to zero or to its maximum value,
the size of the population. This means that changing only one specie, or
strategy, can drastically change the outcome. From a final state where one
specie completely dominates the population to another final state where it is
another specie that wins. This shows that the system is susceptible to initial
perturbations and behaves chaotically. Our study imitates this algorithm using
the Hamming distance between two initially close configurations.

In [18] and in our context, the Hamming distance is the number of agents that
have a different strategy in each configuration, which we call mismatches. We
have replicated the game in [6] and have measured the Hamming distance versus
time from two relaxed configurations only differing by one agent. For most
parameter values, the Hamming distance grows to a very small constant, a
periodic value, or is zero. But, for parameter values within the spatial-chaos
regime discovered by the authors, the distance grows rapidly towards the
Hamming distance that two random configurations would have. This rapid
increase of the Hamming distance is proof of the sensitivity to the initial
conditions, which affects the spatial structure of the final state. However,
the mean final ratio between cooperators and defectors does not change from
the two configurations, so the system only presents spatial chaos.

We have also studied a public goods game (PGG), which presents a broader
regime of parameters that present spatial chaos. This game is a prisoner’s
dilemma under some parameter values, but one that is played in groups of more
than just two agents. The agents are either cooperators or defectors.
Cooperators enrich the public goods investing a cost of $1$ unit which is
reduced from their payoff. Then, for each cooperator in the group, each agent
receives a quantity $r/G$, being $r$ a parameter called enhancement or
multiplication factor and $G$ the number of agents in the group, which is
added to their payoff either if they are cooperators or defectors. The dilemma
is, therefore, that defectors leech off cooperators, reducing the global
payoff. Nonetheless, cooperation can also be promoted through reputation and
network reciprocity, as in the prisoner’s dilemma. Additionally, cooperation
can be sustained through punishment [19, 20] and reward [21]. We applied the
imitation rule to the game but this time only one random agent is selected to
change its strategy to the one of a random neighbor if its strategy is lower
than the neighbor’s. By setting all agents in a square lattice and making them
play games with its immediate neighbors, a spatial structure appears. These
structures are not static, but evolve in time, and so we want to measure how
chaotic this pattern evolution is. Authors of [22] have studied the ecological
PGG and have found chaotic dynamics similar to the patterns of [6] and they
have verified the irregular dynamics qualified as spatial chaos.

For both cases, we have computed the Hamming distance over time, and
normalized to the statistical Hamming distance of two random configurations of
cooperators and defectors with the proportions matching the studied system at
each time. Our main finding indicates that the divergence of two initially
close solutions, measured using the Hamming distance, serves as a reliable
indicator of chaotic dynamics in the deterministic game under study.
Furthermore, in the case of the game exhibiting a degree of stochasticity, the
distance increases more rapidly in regimes displaying spatial-chaotic
behavior.

The organization of this manuscript is as follows. On Section II, we replicate
the results of Novak and May [6] and calculate the normalized Hamming distance
fitting it with the Weibull ”stretched exponential” function. On Section III,
we explain the model used for the public goods game and also fit the
normalized Hamming Distance. Finally, we present the main conclusions at the
end.

##  II Prisoner’s dilemma

$D_{g}$$D_{r}$CHPDSHH Figure 2: Diagram representing the four types of
pairwise social games according to dilemma strength. CH stands for the Chicken
game, PD for the Prisoner’s Dilemma, SH stands for the Stag-hunt game and H
stands for Harmony game.  Figure 3: Color map of the prisoner’s dilemma with a
Moore neighborhood, $R=1$, $P=S=0$, and $1.8<T<2$. We have represented
cooperators in blue and defectors in red for a grid size of $L=99$ and fixed
boundary conditions after relaxation of the prisoner’s dilemma. The box
counting dimension of the boundary between the region of cooperators and
defectors is $2$.  Figure 4: Color map of the prisoner’s dilemma with a Moore
neighborhood, $R=1$, $P=S=0$, and $1.8<T<2$. We have represented cooperators
in blue and defectors in red for a grid size of $L=999$ and fixed boundary
conditions after relaxation of the prisoner’s dilemma. Box counting dimension
of the frontier between defectors and cooperators is $2$¿. There is not seen
clusters of the same strategy at different scales because all clusters have
similar size. In other words, the boundary is not fractal.

The game in [6] is the spatial prisoner’s dilemma (PD). It is an agent-based
model where the authors put agents in the nodes of a square lattice and each
agent played a game with each of its Moore neighbors. The total payoff is the
sum of the payoff gained in each game which was $R=1$ if both players
cooperate, $P=0$ if both players defect, and if an agent defects and another
cooperates, the first gains $T=b>1$ and the second $S=0$. Even though the
prisoner’s dilemma condition $T>R>P>S$ does not hold strictly, the conclusions
are the same for $P=\epsilon\to 0$ positive, which satisfies the condition.
The concept of dilemma strength [23, 24, 25] is a key classification of
pairwise social games. This is defined by the risk-averting dilemma strength
$D_{r}=P-S$ and the gamble-intending dilemma strength $D_{g}=T-R$. The sign of
these two values distinguishes the social games in four types of games as seen
on the diagram of Fig. 2.

The settings of the game on [6] are that of a boundary game, since $D_{r}=0$.
This type of game is not a well representation of the general PD since it has
none of Stag-Hunt type dilemma, but has the advantage of having a single
parameter representing dilemma extent. A more suited dilemma setting that also
has the advantage of having a single parameter representing dilemma extent is
the Donor & Recipient game, presumed in the community of theoretical biologist
as the standard template representing PD [26]. Besides this, we have chosen to
adopt the boundary game from Nowak and May to replicate the same results at
the same parameter values, and thus, see if our tools correctly measure
spatial chaos.

At each iteration of the game, each agent copies the strategy of its Moore
neighbor that has the highest payoff, including themselves. The process
happens simultaneously for all agents. Therefore, the game is deterministic,
and its rules are symmetrical. Granting that the initial conditions have
symmetry, e.g., all agents are cooperators except one defector at the center,
the spatial distribution of cooperators and defectors will be symmetrical.
Nonetheless, the generated patterns are not fractals as the authors claim. In
Figs. 3 and 4 we plot cooperators in blue and defectors in red for different
population sizes $N=L^{2}$, being $L$ the grid size. Observing Fig. 3 that
corresponds to the case $L=99$, one could think that the structure is fractal
as the authors claim in [6]. However, the pattern is not fractal as we can see
on Fig. 4 for $L=999$, instead we can observe near homogeneous size clusters.
Furthermore, we computed the box counting dimension of the boundary between
cooperators and defectors and it is $2$ for both figures.

Figure 5: Proportion of cooperators for the last $50$ iterations after
relaxation of a deterministic prisoner’s dilemma game with Moore neighborhood
and a population size of $999\times 999$ in function of the temptation payoff
parameter $T$. We can distinguish 14 regions corresponding with shifts at the
values of the temptation payoff that induce a change in the game. At
$1.8<T\leq 2$ the dynamics is chaotic. As we can see, the line is bold, i.e.,
there is a large number of possible states with different proportion of
cooperators, whereas for the other regions only a few periodic states are
present.

In Fig. 5 we show the proportion of cooperators for the last $50$ generations
after relaxation of a deterministic PD game with a Moore neighborhood and
$L=999$ in function of the temptation payoff $T$. We can see that the
$1.8<T<=2.0$ regime presents a different behavior, than the rest of the
parameter scan, a chaotic one. There is a sudden transition to chaos due to
the discrete nature of the temptation payoff $T$, i.e., although $T$ is a real
number, the system is only altered at some thresholds by the nature of the
game rules, and not continuously. Assuming that two neighbor agents, $i$ and
$j$, have a different strategy, if one of them, e.g., $j$ changes its strategy
to be the one of agent $i$, is because the payoff of $i$, $\Pi_{i}$ is greater
than any of the agent $j$’s neighbors. Assume that the agent $i$ has $x$
neighbouring cooperators and that the neighbor of agent $j$ with maximum
payoff besides agent $i$, which we will call agent $k$ with payoff $\Pi_{k}$,
has $y$ neighbouring cooperators. The thresholds are given by the possible
values of $y/x$. This is because $\Pi_{k}=y$ if agent $k$ is a cooperator or
$\Pi_{k}=yT$ if it is a defector. Likewise $\Pi_{i}=xT$ if agent $x$ is a
defector or $\Pi_{i}=x$ if it is a cooperator, so $\Pi_{i}>\Pi_{k}$ is held by
$T>y/x$ or $T<x/y$. The value of $T$ has to be greater than one for the game
to be a prisoner’s dilemma, and $x,y\leq 9$ since there are only $9$ agents
that play each game. Therefore there are $27$ thresholds, but in fact, in Fig.
5 defectors completely dominate the game for values beyond $T>2$, so we would
only see $15$ different regimes. We only see $14$ regimes in Fig. 5 since
there is no change in the frequency between threshold $7/6$ and $6/5$ for
unknown reasons.

###  II.1 Hamming distance

The Hamming distance between two configurations $s$ and $s^{\prime}$ of
cooperators and defectors is

$H(t)=\sum_{i}^{N}|s_{i}-s^{\prime}_{i}|,$ (1)

where $s_{i}$ and $s^{\prime}_{i}$ is the strategy of agent $i$ in the
configuration $s$ or $s^{\prime}$ which can be $0$, a defector, or $1$, a
cooperator.

We calculate the final Hamming distance that separates two configurations that
begun at the same initial conditions, where there is a random 50%-50%
cooperator-defector setting, with only one random-agent’s strategy changed
from one configuration to the other. We normalize this distance to the
statistical Hamming distance, which is the Hamming distance that separates two
random configurations with proportion of cooperators $p_{C}(t)$ and defectors
$p_{D}(t)$, which can be calculated as

$H_{stat}(t)=p_{C}(t)p_{D}(t)+p_{D}(t)p_{C}(t)=2p_{C}(t)(1-p_{C}(t)).$ (2)

Our normalized Hamming distance is $H^{\prime}(t)=H(t)/H_{stat}(t)$, and must
not be confused with the usual normalization of the Hamming distance, which is
usually normalized to the size of the set, in this case, the size of the
population.

Figure 6:  Normalized Hamming distance of the two solutions versus time
(number of generations). Multiple curves are shown with different colors,
representing the different grid size $L$ values. The curves grow in a sigmoid-
like curve towards one. They are normalized to the statistical Hamming
distance which depends on L. The larger $L$ is, the longer it takes for the
normalized Hamming distance to reach $1$. Figure 7: Hamming distance of the
the two solutions versus time (number of generations). Different colors
represent different values of the grid size $L$. The larger the grid size, the
longer it takes for the Hamming distance to reach $H_{stat}$.

We have seen that the final normalized distance is zero (or close to zero) for
all values except for the parameter regime specified by Novak and May in [6],
where it is close to one. This indicates that this parameter regime presents
spatial chaos, whereas the others do not. In Fig. 6 we plot $H^{\prime}(t)$
and it grows as a sigmoid-like curve. For larger populations sizes it takes
more time to reach the maximum value of the distance, but it gets closer to
$H_{stat}$ for large times. This happens because $H_{stat}$ grows as the
population size increases, but the growing rate of the Hamming distance is not
altered by the population size, as seen on Fig. 7, where all curves have a
similar growing rate. Discrepancies in the growing rate are due to the random
initial conditions and the choice of the first mismatch, i.e., the first agent
that is different in both configurations, which is also randomly chosen.

We have fitted these curves to a sigmoid function. We have chosen the Weibull
“stretched exponential” function [27], of the form

$F(t;k,a)=1-e^{-(t/a)^{k}},$ (3)

where $a$ sets the timescale, and $k$ indicates how abruptly the curve grows.
We choose this fit since it is sigmoid-like, it starts at the origin and has
only two relevant parameters.

Figure 8: Parameter $a$ from the Weibull “stretched exponential” function
$F(t;k,a)=1-e^{-(t/a)^{k}}$ fitted to the normalized Hamming distance of the
solutions for the prisoner’s dilemma versus grid size $L$. It shows a linear
regression (dashed blue line) where $a$ grows proportionally to $L$. Error
bars, derived from the fitting, are smaller than marker size. Figure 9:
Parameter $k$ from the Weibull “stretched exponential” function
$F(t;k,a)=1-e^{-(t/a)^{k}}$ fitted to the normalized Hamming distance of the
solutions for the prisoner’s dilemma versus grid size $L$. All points have
similar values of $k\approx 2.4\pm 0.4$. Error bars are derived from the
fitting.

Figure 8 shows that $a$ grows linearly with $L$ whith a slope of $0.56\pm
0.05$ and the intercept is $0\pm 30$. It is reasonable that $a$ grows linearly
with $L$, since the mismatches can propagate at a velocity limited by the
range of action of an agent, i.e., the distance at which an agent can affect
another agent. We can see in Fig. 1 that this distance is two agents at each
direction, so the time it takes a mismatch to propagate to the whole
population should be no less than $0.5L$ iterations. Since the slope is near
this maximum of velocity, it shows that the propagation of mismatches is very
fast and uninterrupted. As we see in Fig. 9, the value of $k$ is independent
of $L$, all values are around $2.45\pm 0.4$.

##  III Public Goods game

###  III.1 Model and initial results

We have studied the PGG by means of an agent-based model of $N$ total agents
where each one is a node on a square lattice that plays $5$ games with $G=5$
von Neumann second nearest neighbors, those agents at a Manhattan distance of
two nodes, which form cross-like patterns like that of Fig. 10. In each game
$g=1,...,G$, cooperators gain a payoff $\Pi^{g}_{C}$ and defectors gain a
payoff $\Pi^{g}_{D}$ that depends on the number of cooperators $N^{g}_{C}$
among their neighbors

$\begin{split}&\Pi_{C}^{g}=\frac{r}{G}N_{C}^{g}-1\\\
&\Pi_{D}^{g}=\frac{r}{G}N_{C}^{g}.\end{split}$ (4)

i Figure 10: Von Neumann neighborhood at a distance $2$ of agent $i$. The
individual $i$ plays $5$ games with agents in cross-like patterns like the one
shaded.

The accumulated payoff $\Pi=\sum_{g}^{G}\Pi^{g}$ of each agent, calculated as
the sum of the payoff gained in each game played, determines how likely each
agent survives and reproduces. The evolutionary mechanism we have adopted is a
pairwise imitation process in which, through Monte Carlo simulations, an agent
is randomly chosen to adopt the strategy of a random nearest neighbor if its
payoff is lower than the neighbor’s. Cooperation is chosen if there is a tie
in payoffs.

Once $N$ iterations has passed, all agents have had the possibility on average
to adopt a neighbor’s strategy. Hence our Monte Carlo Step (MCS), which
accounts for one generation, will be $N$ iterations.

We have started every simulation with each agent strategy chosen at random
with the same probability of being cooperators or defectors.

Figure 11: Proportion of cooperators (in blue) and defectors (in red) after
running a simulation of the public goods game starting with a random strategy
in each of the $N=90000$ agents for a relaxation time of $2000$ generations.
We can see a series of steps matching the values of $r$ that, for the given
configurations, the surrounding cooperators have greater payoff than the
surrounded defectors. For the first configuration this holds strictly for the
white-marked cooperator and the defector below it at the first shift at
$r=25/7\approx 3.57$.

After running the simulation through a relaxation time of $2000$ generations,
we present the proportion of cooperators and defectors versus the
multiplication factor $r$ in Fig. 11. In this figure, we see a series of
shifts discontinuously increasing cooperation proportion, and horizontal steps
where the proportion remains constant. We are able to predict the $r$ values
for which the proportion of cooperators shifts by calculating the payoffs of
the surrounding cooperators and defectors in the shown configurations.

We can get the value of the $r$ value of the first shift calculating the
payoff of the white-marked cooperator and the defector below it. The payoff of
this cooperator is

$\Pi_{C}=\frac{1}{5}(5r+4r+2\times 3r+r)-5=\frac{16}{5}r-5,$ (5)

The payoff of the defector is

$\Pi_{D}=\frac{1}{5}(4r+2\times 2r+r+0)=\frac{9}{5}r.$ (6)

The shifts in Fig. 11 occur when there is a change in the behavior of the
system. The shifts are met when $\Pi_{C}=\Pi_{D}$, since cooperation gets more
profitable. For the first configuration this shift occurs at $r\geq
25/7\approx 3.57$. The rest of shifts occur at $r\geq 25/6=4.1\hat{6}$, $r\geq
25/5=5$ and $r\geq 25/3=8.\hat{3}$ all well in accordance with the figure. The
last shift has not been plotted on the figure because is barely visible, but
at this point cooperators completely dominate the game.

In the PGG cooperation is a Nash equilibrium for $r\geq 5$. Nash equilibrium
occurs when no agent can increase its payoff when changing strategy if the
others keep theirs unchanged. Nonetheless since the payoff of each agent is
affected by the strategies of its neighbours, a change of one’s strategy can
lower the payoff of its neighbours more than what it lowers his own payoff,
making Nash equilibrium not being a sufficient condition to eliminate all
defectors. For a cooperative Nash equilibrium with $r\geq 5$ even a defector
surrounded by cooperators would have greater payoff if it were a cooperator,
but since its surrounding cooperators have lesser payoff, it cannot adopt the
cooperation strategy given our evolutionary model and therefore frequency is
lower than 1, see Fig.11. This holds until $r=8.\hat{3}$ as the theoretical
study predicts, when cooperators have more payoff than the isolated defector.
We have not plotted this shift because it can hardly be appreciated at the
current scale and the computational times for all defectors to be gone are
large.

###  III.2 Hamming distance

We wanted to know how fast two solutions for the public goods game that
differed only on one agent’s strategy at the beginning would separate
calculating their Hamming distance, so we could get an idea of when the system
presents spatial chaos. The election of the agent subjected to adopt a new
strategy is chosen at random by the Monte Carlo simulation. This
stochasticity, could be manifest in a growth of the Hamming distance. We try
to minimize this effect by setting all random interactions to be a set of
predetermined values, the same for both configurations but different at each
iteration.

We fit the normalized Hamming distance with the Weibull ”stretched
exponential” function as in the previous section. In Fig. 12, we plot the
normalized Hamming distance for different $r$ values. We see two slightly
separated regimes. Those in $25/7<r<25/6$, which have lesser $a$ values and
those in $25/6<r<5$, with larger $a$ as fitted in Fig. 13. This is well
observed in Fig. 15. However, the value of $k$ seems to be independent of $r$
as can be seen in Fig. 16. All curves have similar $k$ values of $3\pm 1$. In
[18] they refer to the time it takes for the Hamming distance to reach $1/4$
of the maximum, which is similar as the value of $a$, and compared it to the
Lyapunov time, but never developed further investigation.

In Fig. 14, we see that for $r=5$, $a$ is broadly larger than at the other
regimes. However, the Hamming distance still grows towards $H_{stat}$, even
though we did not expect spatial chaos for this parameter value because of the
sparse density of defectors. This could be a sign that the Hamming distance
also tracks the randomness of the Monte Carlo simulations. Since we set all
random interactions at the beginning, the agent that adopts a new strategy is
the same for the two solutions, but since one solution initially differs from
the other, the outcomes become different and diverse, diverging more at each
iteration. However, we get a much larger $a$ value, meaning that it takes much
more time for the Hamming distance to reach its maximum so we perceive this as
a sign that the system is less unstable. This gives us the idea that the
system with this parameter value is less spatially chaotic, if chaotic at all.

Figure 12: Normalized Hamming distance between the solutions for the public
goods game versus time (number of generations). Multiple curves are shown with
different colors, representing the different $r$ values. The curves grow in a
sigmoid-like curve towards one. They are normalized to the statistical Hamming
distance, which depends on $r$, so curves ranging from $25/7\leq r<25/6$ are
normalized to a different value than those at $25/6\leq r<5$. There is a
distinction between the two regimes, the curves of the first one (blue ones)
reach its midpoint after than than the second regime (orange ones). Figure 13:
Normalized Hamming distance between the solutions for the public goods game
versus time (number of generations). Different colors represent the different
$r$ values. The curves grow towards one in a sigmoid-like form. They are
normalized to the statistical Hamming distance, which depends on $r$, so the
two curves are normalized to a different value. They are fitted with the
Weibull “stretched exponential” function $F(t;k,a)=1-e^{-(t/a)^{k}}$. The
$r=4$ curves (blue and left ones), representing the regime $25/7\leq r<25/6$
has a lower value for the $a$ parameter than the curves with $r=4.65$ (orange
and right ones), meaning that for this parameter, the system is more
sensitive. Figure 14: Normalized Hamming distance between the solutions for
the public goods game versus time (number of generations). The curve grows in
a sigmoid-like curve towards one. It is normalized to the statistical Hamming
distance and it is fitted with the Weibull “stretched exponential” function
$F(t;k,a)=1-e^{-(t/a)^{k}}$. This curve for $r=5$ is plotted separated from
the previous ones because it spans farther into the x axis since the value of
$a$ is far greater, meaning the system is less unstable for this parameter
value. Figure 15: Value of the fitted parameter $a$ in the Weibull “stretched
exponential” function $F(t;k,a)=1-e^{-(t/a)^{k}}$ fitted to the normalized
Hamming distance of two solutions for the public goods game for different $r$
values. Notice the rightmost point at $r=5$ and $a\simeq 3600$. We can see the
shift between the 3 regimes $25/7\leq r<25/6\approx 4.17$, $25/6\leq r<5$ and
$r\geq 5$. Error bars, derived from the fitting, are smaller than marker size.
Figure 16: Value of the fitted parameter $k$ in the Weibull “stretched
exponential” function $F(t;k,a)=1-e^{-(t/a)^{k}}$ fitted to the normalized
Hamming distance of two solutions for the public goods game with different $r$
values. All points have similar values of $k\approx 3.2$ and are indistinctant
to the different regimes. Error bars are derived from the fitting.

##  IV Conclusions

We have studied the public goods game to see how two relaxed almost identical
configurations diverged in its spatial configuration through the means of the
Hamming distance in order to categorize if there was spatial chaos. Under the
study of a well-known spatial chaos example, the game of Nowak and May [6], we
have seen that the Hamming distance grows as a sigmoid-like curve towards the
statistical Hamming distance that two random configurations would have for
those cases where spatial chaos was observed, and the distance is zero or
close to zero for the cases where the system falls into a fixed point or
periodic solutions. The settings of the prisoner’s dilemma game were that of a
boundary game where $D_{r}=0$ which is not representing of general PD, but we
used it nonetheless since we wanted to replicate the same results as in [6].
We have no reasons to think the main conclusions will not hold for a more
general case.

For the public goods game, the Hamming distance grows towards the statistical
value for all the studied parameter values, even for those we did not expect
the system to be spatially chaotic, that is, with $r>5$. This behavior may be
explained due to the randomness of the Monte Carlo simulation, making the
instability test less precise even though we have tried to minimize this by
setting the same random values in both configurations. Moreover, the algorithm
is not completely useless since the distance grows much slower for those
cases. Therefore our tool gives a broad idea of how unstable are the spatial
configurations, even though is not completely efficient in discerning a
spatially chaotic behavior in a model with some sort of stochasticity,
therefore further research should be done.

Our study does not replicate the evolution of the Hamming distance equally as
in the case of [18], where initial perturbation changes which species would
win changing drastically the frequency of species value. In contrary, our case
keeps the same values of strategy frequencies on average. This means that the
models studied does not behave chaotically concerning frequencies since they
converge to an equilibrium, although they do present spatial chaos, and our
studies have brought a way to discern it.

Although we have only studied the spatial prisoner’s dilemma and the spatial
public goods game on a square lattice, we think the Hamming distance can be
held against other lattice geometries, in networks or even against other
games, to measure instability.

## Acknowledgments

This work has been supported by the Spanish State Research Agency (AEI) and
the European Regional Development Fund (ERDF, EU) under Project No.
PID2019-105554GB-I00 (MCIN/AEI/10.13039/501100011033)

## References

  * [1] R. Axelrod and W. D. Hamilton, The evolution of cooperation. Science 211, 1390–1396 (1981)
  * [2] G. Szabó and C. Tőke, Evolutionary prisoner’s dilemma game on a square lattice. Phys. Rev. E. 58, 69 (1998).
  * [3] M. Archetti and I. Scheuring, Game theory of public goods in one-shot social dilemmas without assortment. J. Theor. Biol. 299, 9–20 (2012).
  * [4] H. Brandt, C. Hauert, and K. Sigmund, Punishment and reputation in spatial public goods games. Proc. Royal Soc. B 270, 1099–1104 (2003)
  * [5] M. Jusup, P. Holme, K. Kanazawa, M. Takayasu, B. Podobnik, L. Wang, W. Luo, T. Klanjšček, J. Fan, S. Boccaletti, and M. Perc, Social physics. Phys. Rep. 948, 1–148 (2022).
  * [6] M. A. Nowak and R. M. May, Evolutionary games and spatial chaos. Nature 359, 826–829 (1992).
  * [7] G. Hardin, The tragedy of the commons: the population problem has no technical solution; it requires a fundamental extension in morality. Science 162 1243–1248 (1968)
  * [8] V. Buskens, Social networks and the effect of reputation on cooperation. In: Proc. of the 6th Intl. Conf. on Social Dilemmas 167–202 (1998)
  * [9] K. Sylwester and G. Roberts, Reputation-based partner choice is an effective alternative to indirect reciprocity in solving social dilemmas. Evol. Hum. Behav. 34, 201–206 (2013)
  * [10] E. Lieberman, C. Hauert, and M. A. Nowak Evolutionary dynamics on graphs. Nature 443, 312–316 (2005)
  * [11] H. Ohtsuki, C. Hauert, E. Lieberman, and M. A. Nowak A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505 (2006)
  * [12] B. Allen, G. Lippner, Y. T. Chen, B. Fotushi, N. Momeni, Y. Shing-Tung, and M. A. Nowak, Evolutionary dynamics on any population structure. Nature 544, 227–230 (2017)
  * [13] J. M. Baetens and B. De Baets, Phenomenological study of irregular cellular automata based on Lyapunov exponents and Jacobians. Chaos 20 (2010).
  * [14] J. M. Baetens and B. De Baets, A Lyapunov view on the stability of two-state cellular automata. In: Irreducibility and Computational Equivalence. Springer 2, 25–33 (2013).
  * [15] R. D’hulst and G. J. Rodgers, The hamming distance in the minority game. Physica A 270, 514–525 (1999)
  * [16] J.-W. Kim, A tag-based evolutionary prisoner’s dilemma game on networks with different topologies. JASSS, 13, 2 (2010)
  * [17] D. Bazeia, M. B. P. N. Pereira, A. V. Brito, B. F. Oliveira, and J. G. G. S. Ramos, A novel procedure for the identification of chaos in complex biological systems. Sci. Rep. 7, 44900 (2017)
  * [18] D. Bazeia, J. Menezes, B. F. De Oliveira, and J. G. G. S. Ramos, Hamming distance and mobility behavior in generalized rock-paper-scissors models. EPL 119, 58003 (2017)
  * [19] E. Fehr and K. M. Schmidt, A theory of fairness, competition, and cooperation. Q. J. Econ. 114, 817–868 (1999)
  * [20] A. Szolnoki, G. Szabó, and M. Perc, Phase diagrams for the spatial public goods game with pool punishment. Phys. Rev. E 83, 036101. (2011)
  * [21] Y. Li, S. Chen, and B. Niu, Reward depending on public funds stimulates cooperation in spatial prisoner’s dilemma games. Chaos Solitons Fractals 114, 38–45 (2018)
  * [22] J. Y. Wakano and C. Hauert, Pattern formation and chaos in spatial ecological public goodsgames. J. Math. Biol. 268, 30–38 (2011).
  * [23] J. Tanimoto and H. Sagara, Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game. BioSystems 90, 105–114 (2007)
  * [24] Z. Wang, S. Kokubo, M. Jusup, and J. Tanimoto, Universal scaling for the dilemma strength in evolutionary games. Phys. Life Rev. 14, 1–30 (2015).
  * [25] H. Ito and J. Tanimoto, Scaling the phase-planes of social dilemma strengths shows game-class changes in the five rules governing the evolution of cooperation. R. Soc. Open Sci. 5, 181085 (2018)
  * [26] U. Berger, Simple scaling of cooperation in donor-recipient games. BioSystems 97, 165–167 (2009)
  * [27] W. Weibull, A statistical distribution function of wide applicability. J. Appl. Mech. (1951).


Shubin invariants presented in Corollary 2.3, followed by the induction
principle from Lemma 2.1 and finally Propositon 2.4. For $\partial T_{1}$, we
replace $T_{1}^{j}$ by $\partial T_{1}^{j}=T_{1}^{j}\cap\partial
T_{1}\cong\operatorname{\mathbb{R}}^{n-1}$ and proceed completely analogously
as above (using that $\Gamma_{0}^{j}$ restricts to a cocompact lattice on
$\partial T_{1}^{j}$). ∎

Notice that the previous proof does not relate the different positive Novikov-
Shubin invariants from the different complexes
$\Omega_{(2)}^{*}(T_{R},E^{\rho}_{T_{R}})$. A priori, these do depend on $R$.
Showing that there exists a positive uniform lower bound requires a little
more work:

######  Proposition 5.8.

There exists constants $\epsilon,\alpha>0$, such that for all $R\geq 1$ and
all $0\leq p\leq m$ the following hold true

  1. 1.

For all $R\geq 1$ we have

$F(\Delta_{p}[E^{\rho}_{T_{R}}],\lambda)\leq
F(\Delta_{p}[E^{\rho}_{T_{1}}],\lambda).$ (5.18)

  2. 2.

For all $\lambda<\epsilon^{-1}$, we have

$\displaystyle F_{p}(D_{p}[E^{\rho}],\lambda)<\epsilon\cdot\lambda^{\alpha},$
$\displaystyle
F_{p}(D_{p}[E^{\rho}_{T_{R}}],\lambda)<\epsilon\cdot\lambda^{\alpha}.$

In particular, both $D_{p}[E^{\rho}]$ and $D_{p}[E^{\rho}_{T_{R}}]$ are
Fredholm at $p$ and have vanishing $p$-th cohomology.

###### Proof.

  1. 1.

There is a flat bundle isometry $F_{R}:E^{\rho}_{T_{R}}\downarrow{T}_{R}\to
E^{\rho}_{T_{1}}\downarrow{T}_{1}$, as defined in Lemma 3.4. Consequently, by
Equation 1.31, we have for any $x\in T_{R}$ and any $\lambda\geq 0$, that

$\operatorname{tr}(\chi_{[0,\lambda^{2}]}(\Delta_{p}[{E}_{0}])(F_{R}(x),F_{R}(x)))=\operatorname{tr}(\chi_{[0,\lambda^{2}]}(\Delta_{p}[E^{\rho}_{T_{R}}])(x,x)).$
(5.19)

We may choose fundamental domains $\mathcal{D}_{R}\subseteq{T}_{R}$ and
$\mathcal{D}_{1}\subseteq{T}_{1}$ for the respective $\Gamma$-actions,
satisfying $F_{R}(\mathcal{D}_{R})\subseteq\mathcal{D}_{1}$. Therefore, we
have

$\displaystyle
F(\Delta_{p}[E^{\rho}_{T_{R}}],\lambda)=\operatorname{tr_{\Gamma}}(\chi_{[0,\lambda^{2}]}(\Delta_{p}[E^{\rho}_{T_{R}}]))=\int_{\mathcal{D}_{R}}\operatorname{tr}(\chi_{[0,\lambda^{2}]}(\Delta_{p}[E^{\rho}_{T_{R}}])(x,x)))dx$
$\displaystyle=\int_{\mathcal{F}_{R}(\mathcal{D}_{R})}\operatorname{tr}(\chi_{[0,\lambda^{2}]}(\Delta_{p}[E^{\rho}_{T_{1}}])(F_{R}(x),F_{R}(x)))dx$
$\displaystyle\leq\int_{\mathcal{D}_{1}}\operatorname{tr}(\chi_{[0,\lambda^{2}]}(\Delta_{p}[E^{\rho}_{T_{1}}])(F_{R}(x),F_{R}(x)))dx=F(\Delta_{p}[E^{\rho}_{T_{1}}],\lambda).$

  2. 2.

Define $\beta\coloneqq\min\\{\alpha(\Delta_{p}[X]):0\leq p\leq
m,X\in\\{E^{\rho},E^{\rho}_{T_{1}}\\}\\}$. Propositions 5.6 and 5.7 imply that
we have $\beta>0$, as well as both
$\Delta_{p}[E^{\rho}]=\Delta_{p}^{\perp}[E^{\rho}]$ and
$\Delta_{p}[E^{\rho}_{T_{1}}]=\Delta_{p}^{\perp}[E^{\rho}_{T_{1}}]$. For
$E^{\rho}$, we can apply Propositions 5.5 and 5.2 to find a constant $c\geq
1$, such that for all $0\leq p\leq m$, and all $\lambda<c^{-1}$ we have

$F_{p}(D_{p}[E^{\rho}],\lambda)\leq F_{p}(L_{p}[E^{\rho}],c\lambda)\leq
F(\Delta_{p}[E^{\rho}],\sqrt{c\lambda})<c^{\alpha}\lambda^{\alpha}.$

For $E^{\rho}_{T_{R}}$, we can use the same argument, along with assertion
$(2)$, similarly yielding

$\displaystyle F_{p}(D_{p}[E^{\rho}_{T_{R}}],\lambda)\leq
F_{p}(L_{p}[E^{\rho}_{T_{R}}],c\lambda)\leq
F_{p}(\Delta_{p}[E^{\rho}_{T_{R}}],\sqrt{c\lambda})$ $\displaystyle\leq
F_{p}(\Delta_{p}[E^{\rho}_{T_{1}}],\sqrt{c\lambda})\leq
c^{\alpha/2}\lambda^{\alpha/2}.$

Setting $\epsilon\coloneqq\max\\{c^{\alpha/2},c\\}$ and $\beta=\alpha/2>0$, we
obtain the result.

∎

For $R>1$, denote by

$\displaystyle i_{M_{R}}:M_{R}\to\operatorname{\mathbb{H}}^{n},$ (5.20)
$\displaystyle i_{C_{R}}:C_{R}\to\operatorname{\mathbb{H}}^{n},$ (5.21)
$\displaystyle i_{({T}_{R},-)}:T_{R}\to M_{R+1},$ (5.22) $\displaystyle
i_{({T}_{R},+)}:T_{R}\to C_{R}.$ (5.23)

the respective smooth inclusion maps. Each of these induces a
$\Gamma$-invariant map between the corresponding twisted de Rham complexes,
bounded with norm $1$. The proof for the scalar-values case of the following
important theorem, presented in the reference, carries over to our situation
of bundle-valued forms without further modification:

######  Lemma 5.9.

[Lueck:3-mflds, Lemma 5.14] For any $R>1$ and any $0\leq p\leq m$, the
sequence of morphisms of Hilbert $\operatorname{\mathcal{N}}(\Gamma)$-cochain
complexes

$0\rightarrow D_{p}[E^{\rho}]\xrightarrow{j_{R}}D_{p}[E^{\rho}_{M_{R}}]\oplus
D_{p}[E^{\rho}_{C_{R-1}}]\xrightarrow{q_{R}}D_{p}[E^{\rho}_{T_{R-1}}]\rightarrow
0$ (5.24)

is exact. Here, for smooth forms, we have

$\displaystyle j_{R}\omega\coloneqq i^{*}_{{M}_{R}}\omega\oplus
i^{*}_{C_{R-1}}\omega,$ (5.25) $\displaystyle
q_{R}(\omega_{1}\oplus\omega_{2})\coloneqq
i^{*}_{{T}_{(R-1,-)}}\omega_{1}-i^{*}_{{T}_{(R-1,+)}}\omega_{2}.$ (5.26)

######  Lemma 5.10.

There exists a constant $C>0$, such that for all $R>1$ and all $0\leq p\leq
m$, we have

$F_{p}(D_{p}[E^{\rho}_{M_{R}}],\lambda)\leq
F_{p}(D_{p}[E^{\rho}],C\cdot\lambda^{1/4})+F_{p}(D_{p}[E^{\rho}_{T_{R-1}}],C\cdot\lambda^{1/2})\hskip
8.5359pt\text{for}\;0\leq\lambda\leq C^{-1}.$ (5.27)

###### Proof.

Consider the exact sequence

$0\rightarrow D_{p}[E^{\rho}]\xrightarrow{j_{R}}D_{p}[E^{\rho}_{M_{R}}]\oplus
D_{p}[E^{\rho}_{C_{R-1}}]\xrightarrow{q_{R}}D_{p}[E^{\rho}_{T_{R-1}}]\rightarrow
0$ (5.28)

from Lemma 5.9. Because of Proposition 5.8, the outer two complexes are
Fredholm at $p$ and, moreover, have vanishing $p$-cohomology. In particular,
we can apply Proposition 1.2 to obtain that

$F_{p}(D_{p}[E^{\rho}_{M_{R}}],\lambda)\leq
F_{p}(D_{p}[E^{\rho}],c_{1}(R)\cdot\lambda)+F_{p}(D_{p}[E^{\rho}_{T_{R-1}}],c_{1}(R)\cdot\lambda),$
(5.29)

for all $\lambda<c_{2}(R)$, where $c_{1}(R)$ and $c_{2}(R)$ are constants
given by rational expressions of the norms of $q_{R},j_{R}$, the differential
on $D_{p}[E^{\rho}_{M_{R}}]\oplus D_{p}[E^{\rho}_{C_{R-1}}]$, and their
respective inverses. Using the flat bundle isometries given in Lemma 3.4, one
can now proceed analogously as in [Lueck:hyp, Lemma 6.6] to show that these
norms are bounded from above by universal constants independent of $R>1$, thus
proving the lemma. ∎

######  Proposition 5.11.

There exists constants $C,\beta>0$, such that for all $0\leq p\leq m$, the
following hold:

  1. 1.

The Hilbert $\operatorname{\mathcal{N}}(\Gamma)$-cochain complex
$\Omega^{*}_{(2)}(M_{R},E^{\rho}_{M_{R}})$ has vanishing $p$-th homology.
Equivalently, we have

$\Delta_{p}[E^{\rho}_{M_{R}}]=\Delta_{p}^{\perp}[E^{\rho}_{M_{R}}].$ (5.30)

  2. 2.

For all $R>1$, we have a uniform bound on the spectral density functions as
follows

$F_{p}(\Delta_{p}[E_{M_{R}}],\lambda)\leq C\lambda^{\beta}\hskip
8.5359pt\text{for}\;0\leq\lambda\leq C^{-1}.$ (5.31)

###### Proof.

First, observe that the $p$-th cohomology of
$\Omega^{*}(M_{R},E^{\rho}_{M_{R}})$ is isomorphic to the $p$-th cohomology of
$L_{p}[E^{\rho}_{M_{R}}]$. To show that the latter is trivial, we only need to
show that

$F_{p}(L_{p}[E^{\rho}_{M_{R}}],0)=0.$ (5.32)

Using Lemma 5.10, together with both Proposition 5.5 and Proposition 5.2, we
obtain constants $\alpha,c,C_{1},C_{2}>0$ independent of $R>1$, such that

$\displaystyle F_{p}(D_{p}[E^{\rho}_{M_{R}}],\lambda)\leq
F_{p}(D_{p}[E^{\rho}],c\lambda^{1/4})+F_{p}(D_{p}[E^{\rho}_{T_{R-1}}],c\lambda^{1/2})$
$\displaystyle\leq
F_{p}(L_{p}[E^{\rho}],cC_{1}\sqrt{2}\lambda^{1/4})+F_{p}(L_{p}[E^{\rho}_{T_{R-1}}],cC_{1}\sqrt{2}\lambda^{1/2})$
$\displaystyle\leq
F_{p}(\Delta_{p}^{\perp}[E^{\rho}],\sqrt{cC_{1}\sqrt{2}}\lambda^{1/8})+F_{p}(\Delta_{p}^{\perp}[E^{\rho}_{T_{R-1}}],\sqrt{cC_{1}\sqrt{2}}\lambda^{1/4}).$
$\displaystyle\leq 2(cC_{1}\sqrt{2})^{\alpha/2}\lambda^{\alpha/8}$

for $\lambda\leq\min\\{1,c^{-1},C_{1}^{-1},C_{2},\frac{1}{\sqrt{2}C_{2}}\\}$
and all $0\leq p\leq m$. In particular, since
$F(L_{p}[E^{\rho}_{M_{R}}],0)\leq F_{p}(D_{p}[E^{\rho}_{M_{R}}],0)$ by
Proposition 5.4, Equation 5.32 immediately follows from the above computation.
Applying now again Proposition 5.2 and Proposition 5.5 together with the above
inequality, we compute

$\displaystyle
F_{p}(\Delta_{p}[E^{\rho}_{M_{R}}],\lambda)=F_{p}(\Delta_{p}^{\perp}[E^{\rho}_{M_{R}}],\lambda)\leq
F_{p}(D_{p}[E^{\rho}_{M_{R}}],C_{1}\lambda^{2})+F_{p-1}(D_{p-1}[E^{\rho}_{M_{R}}],C_{1}\lambda^{2})$
$\displaystyle\leq 4(cC_{1}^{3/2}\sqrt{2})^{\alpha/2}\lambda^{\alpha/4}$

for
$\lambda\leq\min\\{1,c^{-1},C_{1}^{-1},C_{2},\frac{1}{\sqrt{2}C_{2}},C_{1}^{-1}C_{2},\frac{1}{C_{1}C_{2}\sqrt{2}}\\}\coloneqq
C^{\prime}$.
Setting
$C\coloneqq\max\\{(C^{\prime})^{-1},4(cC_{1}^{3/2}\sqrt{2})^{\alpha/2}\\}$ and
$\beta\coloneqq\alpha/4$, we obtain the desired result. ∎

Neatly summarized, we get the following result:

######  Corollary 5.12.

For each $R>0$, the pair $(M_{R},\rho|_{M_{R}})$ is $L^{2}$-acyclic and of
determinant class. In fact, there exists a uniform constant $\beta>0$
independent of $R$, so that for each $0\leq p\leq n$, one has
$\alpha_{p}^{An}(M_{R},\rho)\geq\beta$.

###  5.3 Proof of large-time convergence

For the next result, choose a nested sequence

$\dots\subset\mathcal{F}_{R-1}\subset\mathcal{F}_{R}\subset
F_{R+1}\subset\dots$ (5.33)

inside $\operatorname{\mathbb{H}}^{n}$, where $\mathcal{F}_{R}$ is a compact
fundamental domain for the $\Gamma$-action on $M_{R}$ and
$\mathcal{F}\coloneqq\bigcup_{R>1}\mathcal{F}_{R}$ is a finite-volume
fundamental domain for the $\Gamma$-action on $\operatorname{\mathbb{H}}^{n}$.
Thus, we have in particular

$\displaystyle\operatorname{Vol}(\mathcal{F}_{R})<\operatorname{Vol}(\mathcal{F})<\infty,$
(5.34)
$\displaystyle\lim_{R\to\infty}\operatorname{Vol}(\mathcal{F}\setminus\mathcal{F}_{R})=0.$
(5.35)

######  Proposition 5.13.

For all $t\geq 1$, we have

$\lim_{R\to\infty}\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}^{\perp}[E^{\rho}_{M_{R}}]})=\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}^{\perp}[E^{\rho}]}).$
(5.36)

###### Proof.

Proposititons 5.6 and 5.11 tell us that both
$\Delta_{p}^{\perp}[E^{\rho}_{M_{R}}]=\Delta_{p}[E^{\rho}_{M_{R}}]$ and
$\Delta_{p}^{\perp}[E^{\rho}]=\Delta_{p}[E^{\rho}]$. Moreover, since $R\geq
2$, we may apply Theorem 1.11 to find appropriate constants $c,C_{1},C_{2}>0$
independent of $R$ and $t$, so that

$|\operatorname{tr}(e^{-t\Delta_{p}[E^{\rho}_{M_{R}}]})(x,x)-\operatorname{tr}(e^{-t\Delta_{p}[E^{\rho}]})(x,x)|\leq
C_{1}e^{-\frac{R^{2}}{C_{2}t}}$

for all $x\in\mathcal{F}_{R/2}$. Together with 5.34 and 5.35, this implies
that

$\displaystyle|\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}[E^{\rho}_{M_{R}}]})-\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}[E^{\rho}]})|=|\int_{\mathcal{F}_{R}}\operatorname{tr}(e^{-t\Delta_{p}[E^{\rho}_{M_{R}}]})(x,x)dx-\int_{\mathcal{F}}\operatorname{tr}(e^{-t\Delta_{p}[E^{\rho}]})(x,x)|$
$\displaystyle\leq\int_{\mathcal{F}_{R/2}}|\operatorname{tr}(e^{-t\Delta_{p}[E^{\rho}_{M_{R}}]})(x,x)-\operatorname{tr}(e^{-t\Delta_{p}[E^{\rho}]})(x,x)|dx+\int_{\mathcal{F}_{R}-\mathcal{F}_{R/2}}|\operatorname{tr}(e^{-t\Delta_{p}[E^{\rho}_{M_{R}}]})(x,x)|dx$
$\displaystyle+\int_{\mathcal{F}-\mathcal{F}_{R/2}}|\operatorname{tr}(e^{-t\Delta_{p}[E^{\rho}]})(x,x)|dx\leq\operatorname{Vol}(\mathcal{F}_{R/2})C_{1}e^{-\frac{R^{2}}{C_{2}t}}+\operatorname{Vol}(\mathcal{F}\setminus\mathcal{F}_{R/2})c.\xrightarrow{R\to\infty}0.$

∎

######  Lemma 5.14.

There exists a positive function $G\in C^{0}(1,\infty)\cap L^{1}(1,\infty)$,
such that for all $t\geq 1$ and all sufficiently large $R>>0$, we have

$t^{-1}\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}^{\perp}[E_{M_{R}}]})\leq
G(t).$ (5.37)

###### Proof.

Throughout the proof, we will abbreviate

$F_{R}^{p}(\lambda)\coloneqq F_{p}(\Delta_{p}[E^{\rho}_{M_{R}}],\lambda).$
(5.38)

We compute that one has for any $R>0$, any $t\geq 1$ and arbitrary
$\epsilon>0$

$\displaystyle\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}^{\perp}}[E_{M_{R}}])=\int_{0}^{\infty}e^{-t\lambda}dF_{R}^{p}(\lambda)=\int_{0}^{\epsilon}e^{-t\lambda}dF_{R}^{p}(\lambda)+\int_{\epsilon}^{\infty}e^{-t\lambda}dF_{R}(\lambda)$
$\displaystyle\stackrel{{\scriptstyle
F_{R}^{p}(0)=0}}{{=}}t\int_{0}^{\epsilon}e^{-t\lambda}F_{R}^{p}(\lambda)d\lambda+e^{-t\epsilon}F_{R}^{p}(\epsilon)+\int_{\epsilon}^{\infty}e^{-t\lambda}dF_{R}(\lambda)$
$\displaystyle\stackrel{{\scriptstyle
t>1}}{{\leq}}t\int_{0}^{\epsilon}e^{-t\lambda}F_{R}^{p}(\lambda)d\lambda+e^{-t\epsilon}F_{R}^{p}(\epsilon)+e^{-t\epsilon}\int_{\epsilon}^{\infty}e^{-\lambda+\epsilon}dF_{R}(\lambda)$
$\displaystyle=t\int_{0}^{\epsilon}e^{-t\lambda}F_{R}^{p}(\lambda)d\lambda+e^{-t\epsilon}F_{R}^{p}(\epsilon)+e^{-t\epsilon}e^{\epsilon}\operatorname{tr_{\Gamma}}(e^{-\Delta_{p}^{\perp}[E_{M_{R}}]}),$

and therefore in particular

$t^{-1}\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}^{\perp}[E^{\rho}_{M_{R}}]})\leq\int_{0}^{\epsilon}e^{-t\lambda}F_{R}^{p}(\lambda)d\lambda+\frac{e^{-t\epsilon}}{t}(F_{R}^{p}(\epsilon)+e^{\epsilon}\operatorname{tr_{\Gamma}}(e^{-\Delta_{p}^{\perp}[{E^{\rho}_{M_{R}}]}})).$
(5.39)

By Proposition 5.13 we find some $\delta>0$, such that for all $R>>0$, we have
$\operatorname{tr_{\Gamma}}(e^{-\Delta_{p}^{\perp}[E^{\rho}_{M_{R}}]})\leq\operatorname{tr_{\Gamma}}(e^{-\Delta^{\perp}_{p}[E^{\rho}]})+\delta$.
Similarly, by Proposition 5.11, we can choose $\epsilon,\beta>0$ independently
of $R$ and small enough, such that
$F_{R}^{p}(\lambda)\leq\epsilon^{-1}\lambda^{\beta}$ for all
$\lambda\leq\epsilon$. From this, it becomes obvious that the function

$G(t)\coloneqq\epsilon^{-1}\int_{0}^{\epsilon}e^{-t\lambda}\lambda^{\beta}d\lambda+\frac{e^{-t\epsilon}}{t}(\epsilon^{-1+\beta}+e^{\epsilon}(\operatorname{tr_{\Gamma}}(e^{-\Delta^{\perp}_{p}[E^{\rho}]})+\delta))$
(5.40)

satisfies the assertions of our lemma. ∎

Using Proposition 5.13, Lemma 5.14 and Lebesgue’s theorem of dominated
convergence, we finally obtain the main result of this section:

######  Theorem 5.15 (Large-time convergence).

We have

$\lim_{R\to\infty}\int_{1}^{\infty}t^{-1}\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}[E^{\rho}_{M_{R}}]})dt=\int_{1}^{\infty}t^{-1}\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}[E^{\rho}]})dt.$
(5.41)

In particular, we obtain

$\lim_{R\to\infty}\sum_{p=0}^{n}\int_{1}^{\infty}t^{-1}\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}[E^{\rho}_{M_{R}}]})dt=\sum_{p=0}^{n}\int_{1}^{\infty}t^{-1}\operatorname{tr_{\Gamma}}(e^{-t\Delta_{p}[E^{\rho}]})dt.$
(5.42)

## References