# Socioeconomic Clustering and Racial Segregation
on Lattices with Heterogeneous Sites

Zhanzhan Zhao<EMAIL_ADDRESS>Supported by Georgia Tech ARC-TRIAD
Fellowship and ARO MURI award W911NF-19-1-0233. School of Computer Science,
Georgia Institute of Technology.  Dana Randall<EMAIL_ADDRESS>Supported by ARO MURI award W911NF-19-1-0233. School of Computer Science,
Georgia Institute of Technology.

###### Abstract

The Schelling model of segregation moves colored agents (or particles) on the
vertices of a graph (typically a lattice), with unhappy agents trying to move
to new positions if the number of their neighbors with a different color
exceeds some threshold. A well-studied stochastic variant of this model
assumes that agents are geometrically happier with each additional neighbor of
the same color, moving with probability proportional to their unhappiness. The
stochastic version behaves like a fixed magnetization Ising model, so it is
known that we will have segregation at equilibrium if the homophily, or
preference for the neighbors of the same color, is large enough.

In this work, we consider race and socioeconomic status simultaneously to
understand how carefully architected placement of urban infrastructure that
mostly helps low-income individuals might affect segregation. In our model,
agents are both two-colored and also labeled as rich or poor. We designate
certain vertices on the graph as “urban sites,” providing civic infrastructure
that most benefits the poorer population, incentivizing occupation of these
vertices by poor agents of either color. Infrastructure that is centralized,
like a city center or mall, encourages poor agents to cluster centrally in
addition to their homophily preferences, while infrastructure that is well
distributed, like a large grid of inner-city bus routes, tends to disperse the
low-income agents. We ask what effect these two scenarios have on segregation.

We find that centralized infrastructure simultaneously causes segregation and
the “urbanization of poverty” (i.e., occupation of urban sites primarily by
poor agents) when the homophily and incentives drawing the poor to urban sites
are large enough. However, even when homophily preferences are very small, as
long as the incentives drawing the poor to urban sites is large, under income
inequality where one race has a significantly higher proportion of the poor,
we get racial segregation on urban sites but integration on non-urban sites.
However, we find there is an overall mitigation of segregation when the urban
sites are distributed throughout the lattice and the incentive drawing the
poor on the urban sites exceeds the homophily preference. We prove that in
this case, no matter how strong homophily preferences are, it will be
exponentially unlikely that a configuration chosen from stationarity will have
large, homogeneous clusters of similarly colored agents, thus promoting
integration in the city with high probability.

##  1 Introduction

Over fifty years ago, Nobel prize winning economist Thomas Schelling studied
segregation by modeling residents as colored particles on a chessboard. Each
particle is considered happy if its color agrees with more than a fixed
fraction of its neighbors and tries to move to a new site if it is not content
[16]. Simulations reveal the surprising fact that even mild biases favoring
one‘s own color, known as homophily, is sufficient to cause segregation on a
macroscopic scale [29]. Extensive work has been done by economists and
sociologists to expand and explain Schelling‘s model using statistical
analyses and simulation tools [2, 18, 20, 33]. This work typically focuses on
how the dynamics determine the limiting distribution and connecting the model
to the real world population dynamics [8, 9, 19, 30, 10, 25].

The first rigorous analyses of these dynamics appears in [5, 12], where the
authors consider the one dimensional Schelling model with two different races.
For a set of agents $I$ living in a ring, the utility function is a simple
threshold function computed as follows. There is a critical value
$\theta\in[0,1]$. For each agent $j\in I$, if the fraction of the neighboring
agents with the same race as that of $j$ is more than $\theta$ then $j$ is
happy. Otherwise, $j$ is unhappy. The convergence of the dynamics is
guaranteed with the stationary configuration being proved. Further rigorous
work has been done by extending the neighborhood interactions to a broad class
of utility functions. The exponential mixing time and the segregated
stationary configurations are proved under certain preference thresholds [1,
32, 26].

A natural generalization of Schelling‘s original model can be described as
follows. A set of agents $I$ occupy some of the vertices of a graph $G=(V,E)$.
Each agent $j\in I$ belongs to a race $R_{j}\in R$. Each configuration
$\sigma$ of the system is a placement of agents on the vertices of the graph.
Also, for $j$ there is the utility function
$u_{j}:R^{N_{\sigma}(j)}\to\\{\textrm{``happy", ``unhappy"}\\}$, where
$N_{\sigma}(j)$ is a small neighborhood around $j$ in $G$, which specifies the
happiness of $j$. It is direct that given the utility functions and the
configuration we can specify which agents are happy and which are not. In
Schelling‘s model, each agent is content or not depending on whether the
number of like neighbors exceeds a threshold, while in the particle model each
agent‘s happiness increases incrementally with more neighbors of the same spin
[31].

Bhakta et al. [3] considered a reversible variant of the Schelling dynamics
using tools from statistical physics. Each configuration $\sigma$ has a $\it
Gibbs$ measure based on the happiness of each agent, parameterized by
$\lambda$, the racial bias parameter. Specifically, each agent $j$ contributes
the number of same-colored neighbors ${N_{\sigma}(j)}$ to the Hamiltonian, or
energy of the system. The probability of a configuration is given by
$\pi(\sigma)=\Pi_{j\in I}\sqrt{\lambda}^{N_{\sigma}(j)}/Z,$ where $Z$ is the
normalizing constant. A reversible Markov chain can be constructed that
converges to this Gibbs distribution in which every agent has some probability
of moving, and those with more neighbors of the other color move with
increasing probability. The dynamics are similar to the well-studied Ising
model from statistical physics, and the model inherets many of its properties
such as a phase transition from a disordered phase to an ordered one when
$\lambda$ surpasses a threshold.

The Schelling model and its variants typically assume all agents of each race
are homogeneous and have the same homophily, regardless of location. However,
it has been observed that the Schelling model can be used to explain other
social phenomena where like individuals are inclined to prefer similar
individuals over those who differ from them, including an empirical study of
both segregation of ethnicity and wealth [28, 11, 24].

Here we explore the impact of wealth disparity and potential civic
interventions to mitigate segregation, taking into account two widely observed
phenomena: centralization, or the degree to which a group is spatially located
near the the city center, and urbanization of poverty, whereby city centers
and other areas dense with civic infrastructure disproportionately attract the
poor to aggregate. Centralization is widely-used to measuring segregation in
metropolitan areas [22, 17]. Urban economists show that urbanization of
poverty results from better access to public transportation in central cities
and other resources [13].

Segregation by race and wealth: To better understand these socioeconomic
distintinctions and the effects of economic disparity within a city, we
introduce a new Scioeconomic Schelling Model, where individuals of each race
are also designated rich or poor. We label some vertices to be urban sites if
they provide useful infrastructure (or resources) that are most beneficial to
poor citizens. For instance, the urban sites can be grouped centrally,
representing a metropolitan city center, or distributed in a large spanning
grid, representing a netwrok of public transportation (see Fig 1). While all
individuals have uniform homophily preferences, like the standard Schelling
model, we add additional incentives that benefit poor people residing on urban
sites, so more racial intolerance can be justified if more of the poor people
are benefitting from proximity to urban sites. We are interested in
understanding when urban infrastructure can help mitigate racial biases and
lessen segregation for various typologies of urban sites.

Specifically, we represented the city by a finite torus on the triangular
lattice, with each site accommodating exactly one person. Each person (or
agent) is either blue or red, representing race, and rich or poor,
representing wealth and $\mathcal{U}\subseteq\mathcal{V}$ are the vertices
that are designated urban sites. Each pair of neighbors has a homophily (or
racial) bias $\lambda$, representing how much they each prefer neighbors of
their own color. Setting $\lambda>1$ corresponds to the ’’ferromanetic
setting,‘‘ where agents prefer same-colored neighbors. Further, poor agents
have an affinity for urban sites with a wealth bias parameter $\gamma$, in
addition to their homophily biases. Setting $\gamma>1$ biases poor agents to
prefering residing on urban sites. When $\gamma=1,$ we recover the pure
homophily model where wealth of individuals is not considered. The stationary
probability of any configuration $\sigma$ is given by

$\pi(\sigma)=\lambda^{-h(\sigma)}\gamma^{p(\sigma)}/Z,$

where $h(\sigma)$ is the number of racially heterogeneous edges (whose
endpoints do not share the same color), $p(\sigma)$ is the number of poor
agents on urban sites, and $Z$ is the normalizing constant.

A randomized algorithm $\mathcal{M}$ for sampling from $\pi$ can be described
as follows. At each time step, two random agents are selected, and they swap
locations with the appropriate Metropolis probabilities so as to converge to
$\pi$. In particular, they are more likely to swap if they are each in less
homogeneous neighborhoods, as previously studied in [3, 6], with an additoinal
bias toward keeping poor agents on urban sites, so happier individuals are
less likely to move.

We are interested in how these dynamics might be influenced by the placement
of urban sites based on the racial and wealth demographics of the population.
We note that when there are no urban sites (or all vertices are urban sites),
then the wealth of individuals becomes irrelevant and we recover the racial
segregation model studied in [6]. When it comes to the size and topology of
the urban sites, the case of no urban-sites at all or full urban sites have
been studied in [6], where the dichotomy of the phase change between
integration and segregation has been proved. Here we are interested in the
effects in heterogeneous cases where both urban and non-urban sites are
present. We also require the size of the urban sites to be of a constant
fraction of the total sites. For topology, we study the impact of the
centralized or distributed placement of the urban sites to segregation.

Our results: First, we show that our model yields urbanization of poverty when
$\gamma$ is sufficiently large, with all but an arbitrarily small fraction of
urban sites being occupied by poor agents. When the urban sites are
centralized (i.e., occupying a dense hexagon in the lattice), the urban sites
will be densely occupied with poor agents for any value of $\lambda$ as long
as $\gamma$ is large enough. Moreover, we will see a similar urbanization of
poverty even for arbitrarily distributed urban sites as long as the wealth
bias $\gamma$ exceeds the racial bias $\lambda$ sufficiently. In the
centralized case $\lambda$ does not play a role, while in the distributed case
we need the incentive for the poor to be on urban sites to outweigh their
racial biases. Conversely, we show that for any $\lambda>1$, if $\gamma>1$ is
small enough, then it is exponentially unlikely that poor agents will be
concentrated on urban sites for any constant fraction of poor agents in the
city.

Moreover, when the urban sites are centralized and both $\lambda$ and $\gamma$
are large enough, urbanization of poverty and racial segregation will occur
simultaneously. However, even when $\lambda$ is small, as long as $\gamma$ is
large, under severe income inequality where most of the poor are from one
race, we will have racial segregation on urban sites and racial integration on
the non-urban sites. This suggests that the urbanization of poverty can
enhance segregation when the infrastructure is centralized, such as with a
dense city center with civic services and perhaps subsidized housing,
providing a primary location that incentivizes occupation by the poor.

However, we find a dramatically different outcome when the urban sites are
distributed throughout the city, such as with public transportation evenly
partitions the entire city. First, we prove under income inequality, where one
race has a higher proportion of poor people, no matter how large $\lambda$ is,
as long as $\gamma$ exceeds $\lambda$ suffiently, both the urban and nonurban
sites will be integrated. That is, the probability of large clusters that are
predominantly one race forming is exponentially small. This suggests that
distributing urban infrastructure equitably throughout the city will have a
better effect for mitigating segregation when the incentives are large enough
compared to the inherent racial biases.

Techniques: Our proofs build on analysis for the integration and separation of
heterogenous particles in the context of programmable matter [6]. Although not
formulated as segregation, their proofs show that when there are no urban
sites, slight or no local homophily produces integration, whereas stronger
homophily produces segregation. The proofs use mappings known as Peierls
arguments to show that configurations not in the target sets are exponentially
unlikely.

Peierls arguments were developed in physics to prove the existence of phase
transitions, such as the sudden change in magnetization in the two-dimensional
Ising model [14, 15], and have been applied in computer science to study slow
mixing of Markov chains [4, 27]. The essential idea is to map the set not
satisfying a target property to configurations that have exponentially larger
probability at stationarity so that the inverse maps do not require
significant information, thus proving that configurations outside of the
target set must have small probability by evaluating ’’energy/entropy‘‘
balancing the probabilities and the number of preimages.

These techniques form the backbone of our proofs when the racial bias is high
and we get segregation, but the introduction of urban sites and the wealth
bias make the proofs significantly more challenging as we need to maintain the
same number of rich and poor members of both races before and after the
mapping $\nu=f(\sigma).$ When there are only two types of agents, red and
blue, we are able to allow the sizes to deviate and restore the appropriate
numbers of each race in a designed way to increase the weight significantly
through the map. Here, all four groups may deviate under the maps and it
requires careful arguments to be able to restore the cardinalities of all the
sets without losing too much information. In particular, we extend Peierls and
bridging arguments used in [6, 23] to suit the multi-dimensional information
about race and wealth embedded in each agent accounting for more complex
tradeoffs between the homophily and wealth biases, and the number of
configurations with each weight, requiring much more delicate mappings and
counts. We note that in addition to the typical energy-entropy tradeoffs of
Peierls arguments, here we additionally have to balance several tradeoffs
within the energy term $\frac{\pi(\sigma)}{\pi(\nu)}$, due to the different
relationships between wealth and racial biases induced by different placements
of the urban sites.

##  2 The Socioeconomic Schelling Model

In our proposed model, a city is represented by a finite toroidal region of
the triangular lattice $G_{\triangle}=(\mathcal{V},\mathcal{E})$, shown in
Figure 1(a). Each vertex in $\mathcal{V}$ represents a potential residence or
site. Two adjacent vertices are neighboring sites, and each site has six
nearest neighbors on $G_{\triangle}$. Some vertices
$\mathcal{U}\subseteq\mathcal{V}$ are designated urban sites. We denote the
set of agents as $\mathcal{A}$ and the poor agents as $\mathcal{P\subseteq
A}.$ Figure 1(a) shows an example of centralized placement of urban sites,
whereas Figure 1(b) shows the distributed placement, with urban sites depicted
as yellow hexagons.

We assume each agent $i$ is assigned a race $r(i)\in\\{\text{blue, red}\\}$
and wealth $w(i)\in\\{\text{rich, poor}\\}.$ Each site in $\mathcal{V}$ can
accommodate at most one agent. For simplicity of analysis, we assume that $n$
agents fully occupy all the sites on $G_{\triangle}$, where $|\mathcal{V}|=n.$
The size of the urban sites is assumed to be of a constant fraction of all the
sites, i.e., $|\mathcal{U}|=c\cdot n,$ where $c\in[0,1].$ As shown in Figure
1(c), we represent the race of an agent by color and the wealth of an agent by
the shade of each color; poor blue agents are referred to as cyan, poor red
agents are pink and blue and red are reserved for the rich members of each
color class.

Among the $n$ agents, $\mathcal{P}$ are the subset that are poor and the
fraction that are poor is denoted by $p$, so $|\mathcal{P}|=p\cdot n.$
Similarly, the fraction of the red $\mathcal{R}$ is $r$, and
$|\mathcal{R}|=r\cdot n.$ Among the red agents, we further denote the fraction
of the poor red $\mathcal{R}_{\rm p}$ as $r_{\rm p}$, and the fraction of rich
red $\mathcal{R}_{\rm r}$ as $r_{\rm r}$, so that $|\mathcal{R}_{\rm
p}|=r_{\rm p}n,$ and $|\mathcal{R}_{\rm r}|=r_{\rm r}n.$ Similarly, we define
the fraction of the blue $\mathcal{B}$ as $b$, and the fraction of poor blue
$\mathcal{B}_{\rm p}$ as $b_{\rm p}$, and the fraction of rich blue as
$\mathcal{B}_{\rm r}$ as $b_{\rm r}$.

A configuration (or a state) $\sigma$ is the set of vertices of
$G_{\triangle}$ occupied by an arrangement of $n$ agents with the urban sites
assignment information embedded, along with the race and wealth of each agent.
The state space (or configuration space) $\Omega$ is the set of all possible
configurations.

For a configuration $\sigma,$ we denote agent $i$ resides at a site
$\ell_{\sigma}(i)\in\mathcal{V}$. Agents living at neighboring sites are
neighbors and each agent can at most have six neighbors. Each agent $i$ is
assigned a race $r(i)$, wealth $w(i)$, and occupies a site $\ell_{\sigma}(i)$,
which it can recognize as an urban site or not. We define an indicator
function that takes agent $i$ as input and outputs true when $i$ is poor and
currently on the urban sites as the following:

$u_{\sigma}(i)\triangleq\left\\{\begin{matrix}1,&\text{if\,\,}i\in\mathcal{P},\ell_{\sigma}(i)\in\mathcal{U}\\\
0,&\text{otherwise}\end{matrix}\right.$

For a configuration $\sigma,$ the number of agents that are poor and on the
urban sites is defined to be
$p(\sigma)\triangleq\sum_{i\in\mathcal{A}}u_{\sigma}(i).$

Each agent is able to identify each neighbor‘s race, and thus can determine
the number of neighbors that share its color, $N_{\sigma}(i).$ An edge in a
configuration $\sigma$ with vertices occupied by agent $i$ and $j$ is racially
homogeneous if they are the same color (i.e., $r(i)=r(j)$) and racially
heterogeneous otherwise. We define the total number of racially heterogeneous
edges of a configuration $\sigma$ as $h(\sigma),$ and the total number of
racially homogeneous edges as $e(\sigma).$

(a)

(b)

(c)

Figure 1: A city lattice region $G_{\triangle}$ (a) with centralized urban
sites, (b) with distributed urban sites, and (c) fully occupied by agents.

The Markov chain $\mathcal{M}$ is defined so that it will converge to
$\pi(\sigma)=\lambda^{-h(\sigma)}\gamma^{p(\sigma)}/Z$, which generalizes the
Schelling probabilities to reflect the additional effect of urban sites. Each
agent $i$ is able to swap locations with any agent $j\in\mathcal{A},j\neq i$
in the city $G_{\triangle}$, and we denote it a swap move $s_{ij}.$ Beginning
at any configuration $\sigma_{0}\in\Omega,$ at each time step, given the
current configuration $\sigma$, the algorithm randomly picks two agents $i$
and $j$ at sites $\ell_{\sigma}(i)\in\mathcal{V}$ and
$\ell_{\sigma}(j)\in\mathcal{V}$ and tries to swap their positions with the
appropriate Metropolis probabilities (so agents are more likely to move if
they have fewer racially homogeneous neighbors, and with a dampening factor
$\frac{1}{\gamma}<1$ if the agent is poor and currently at the urban site.
Mathematically,

$P(\sigma:i\to j)=\frac{\lambda^{-N_{\sigma}(i)}}{\gamma^{u_{\sigma}(i)}},$

where $\lambda>1$, and $\gamma>1$. The probability of agents $i$ and agent $j$
swapping positions satisfies

$\displaystyle
P(\sigma:s_{ij})=\frac{1}{n^{2}}\lambda^{-N_{\sigma}(i)-N_{\sigma}(j)}\gamma^{-\sum_{k\in\\{i,j\\}}u_{\sigma}(k)}.$
(1)

Algorithm 1  Markov Chain $\mathcal{M}$.

1:Beginning at any configuration $\sigma_{0}$ with $n$ agents, repeat:

2:Choose two agents $i$ and $j$ uniformly at random in the current
configuration $\sigma$.

3:Choose $q\in(0,1)$ uniformly at random.

4:if
$q<\lambda^{-N_{\sigma}(i)-N_{\sigma}(j)}\gamma^{-\sum_{k\in\\{i,j\\}}u_{\sigma}(k)}$,
then then agents $i$ and $j$ swap positions.

5:else agents $i$ and $j$ keep their current locations.

It is easy to see that the Markov chain $\mathcal{M}$ is ergodic on the state
space $\Omega,$ since swap moves of $\mathcal{M}$ suffice to transform any
configuration to any other configuration (irreducible) and there is a non-zero
self-loop probability for $\lambda>1$ and $\gamma>1$ (aperiodic). Using
detailed balance it is easy to confirm that the Markov chain converges to

$\displaystyle\pi(\sigma)=\lambda^{-h(\sigma)}\gamma^{p(\sigma)}/Z,$ (2)

where $h(\sigma)$ is the number of racially heterogeneous edges in $\sigma$,
$p(\sigma)$ is the number of the poor on the urban sites in $\sigma$,
$Z=\sum_{\sigma\in\Omega}\lambda^{-h(\sigma)}\gamma^{p(\sigma)}$ is the
partition function, and $\Omega$ is the configuration space.

##  3 Urbanization of Poverty

We start by showing the urbanization of poverty, defined as the urban sites
being occupied by nearly the maximal number of the poor agents, with an
$\epsilon-$fraction of error tolerance, whether the urban sites are
centralized or distributed. We prove in Theorem 3 that under centralized
placement, for any $\lambda$ and $\epsilon$, as long as $\gamma$ is bigger
than a threshold depending only on $\epsilon$, we will observe the
urbanization of the poor at stationarity with high probability. See Figure
2(a) for simulations. We prove in Corollary 4 that when urban sites are
distributed throughout the lattice and the contribution of the wealth bias
$\gamma$ exceeds the contribution of the racial bias $\lambda$, we will also
see the urbanization of poverty (simulations in Figure 2(b)). Finally, we
prove that when urban sites are centralized, and for any $\lambda>1$, if
$\gamma>1$ is sufficiently small, then we are very unlikely to observe the
urbanization of poverty and both urban and nonurban sites will have a
nontrivial mix of rich and poor agents.

######  Definition 1.

For any $\epsilon\in(0,\frac{1}{2})$, a city is said to have
$\epsilon-$urbanization of poverty if the number of the poor on the urban
sites is at least $\min\\{c,p\\}n-\epsilon n.$

Here, $\epsilon$ expresses the tolerance for having rich agents on the urban
sites: smaller $\epsilon$ requires stricter urbanization of poverty, whereas
larger $\epsilon$ allows for smaller density of the poor on the urban sites.
$\min\\{c,p\\}n$ refers to the maximal possible number of the poor on the
urban sites. $\epsilon-$urbanization of poverty can be understood as the urban
sites will be accommodated by the maximal possible number of the poor minus a
small error tolerance of no more than $\epsilon n$.

######  Remark 2.

When the size of the urban sites can roughly accommodate all the poor, where
$p-m\epsilon<c<p+(m+1)\epsilon,$ a configuration satisfying the above
definition of $\epsilon-$urbanization of poverty can also be viewed as having
$(\beta,(m+1)\epsilon)-$wealth segregation: there are at most $\beta\sqrt{n}$
heterogeneous edges in terms of wealth, where $\beta>4\sqrt{p};$ and the
number of poor in the urban area $\mathcal{U}$ is at least $pn-(m+1)\epsilon
n$.

We prove that with centralized urban sites or distributed urban sites, for
large enough $\gamma$, we will most likely observe the urbanization of poverty
at stationarity in Theorem 3 and Corollary 4.

######  Theorem 3 (Centralized Urbanization of Poverty).

If $\gamma>16^{\frac{3(3\epsilon+2)}{2\epsilon^{2}}}$ and $\lambda>1$, with
the centralized urban sites, when $n$ is sufficiently large, then for
$\mathcal{M}$, configurations drawn from distribution $\pi$ have
$\epsilon-$urbanization of poverty with probability at least $1-\xi_{1}^{n}$,
where $0<\xi_{1}<1$.

First, we define $\Omega_{\neg\text{urb}}$ to be the set of the configurations
that do not have $\epsilon-$urbanization of poverty. To show Theorem 3, it
suffices to show $\pi(\Omega_{\neg\text{urb}})\leq\xi_{1}^{n}$. Then we use
Peierls argument (see Appendix A.1 for a short tutorial): mapping from non-
urbanized configurations to urbanized configurations with the bridging
technique shows that the map has an exponential gain in probability weight.
Thus the non-urbanized configurations are exponentially unlikely compared with
urbanized ones, even though the number of non-urbanized configurations is much
more than the urbanized ones, and some of those configurations have large
probability weights in terms of $\lambda^{-h(\sigma)}$.

Compared with [23, 6], the state space is enlarged by adding the wealth
dimension, which requires modifying the bridge system to encode both race and
wealth information of each agent, defined as a $\delta-$color-and-richness
bridge system specified in Appendix A.2. Moreover, compared with [23, 6], due
to the new term about $p(\sigma)$ in the stationary distribution (2), besides
the more complex energy-entropy tradeoffs between
$\frac{\pi(\sigma)}{\pi(\nu)}$ and $|f^{-1}(\nu)|$, now there is another
tradeoff inside the energy term (energy-energy tradeoff): because a mapped
configuration with big probability weight of $\gamma^{p(\sigma)-p(\nu)}$ can
correspond to small probability weight of $\lambda^{-(h(\sigma)-h(\nu))}$. We
design more careful mapping rules to balance and bound
$\frac{\pi(\sigma)}{\pi(\nu)}$ and $|f^{-1}(\nu)|$, and see Appendix A.3, A.4
and A.5 for the mapping technique summaries.

###### Proof of Theorem 3.

For any $\sigma\in\Omega_{\neg\text{urb}}$, we first construct a
$\delta-$color-and-richness bridge system (see Appendix A.2 for definition and
Figure 5(a) for illustration) and define the mapping
$f(\sigma)=(f_{5}{\circ}f_{4}{\circ}f_{3}{\circ}f_{2}{\circ}f_{1})(\sigma)$,
where $\psi=f_{1}(\sigma)$ is the richness inversion mapping (defined in
Appendix A.3 and see Figure 5(b) for illustration), and $\tau=f_{2}(\psi)$ is
the color inversion mapping (defined in Appendix A.3 and see Figure 5(c) for
illustration). $\tau=(f_{2}{\circ}f_{1})(\sigma)$ eliminates the bridged
racially heterogeneous edges and the bridged poor agents. For
$(f_{5}{\circ}f_{4}{\circ}f_{3})(\tau)$ (defined in Appendix A.4), we first
assume the urban sites are centralized, under which we recover the same ratios
of each color and richness as in $\sigma$ in the centralized way defined in
Appendix A.5 (also see Figure 6 for illustrations). Then the upper bounds of
$h(\nu)-h(\sigma)\leq 3\alpha\sqrt{n}-z_{\rm c}$ and
$p(\sigma)-p(\nu)\leq-\delta n$ and $|f^{-1}(\nu)|\leq(z_{\rm
c}+1)9^{\alpha\sqrt{n}}4^{(\frac{3\delta+1}{4\delta})(z_{\rm c}+3n)}$ can be
obtained from Claim 26 and 27 which are shown in Appendix B. The color contour
length $z_{\rm c}$ is defined in the bridge system (Appendix A.2), which is
the sum of length of the contours separating the red (or pink) from the blue
(or cyan) in $\sigma$ with a no more than $\delta-$fraction omission. Finally,
substituting (2) and the bounds into the Peierls argument yields

$\displaystyle\pi(\Omega_{\neg\text{urb}})$
$\displaystyle=\sum_{\sigma\in\Omega_{\neg\text{urb}}}\pi(\sigma)\leq\sum_{\nu\in\Omega}\sum_{\sigma\in
f^{-1}(\nu)}\pi(\sigma)=\sum_{\nu\in\Omega}\pi(\nu)\frac{\sum_{\sigma\in
f^{-1}(\nu)}\pi(\sigma)}{\pi(\nu)}$
$\displaystyle\leq\sum_{\nu\in\Omega}\pi(\nu)\sum_{\sigma\in
f^{-1}(\nu)}\lambda^{h(\nu)-h(\sigma)}\gamma^{p(\sigma)-p(\nu)}$
$\displaystyle\leq\sum_{\nu\in\Omega}\pi(\nu)\sum_{z_{\rm c}=\sqrt{r\cdot
n}}^{3n}n^{6}(3n+1)(z_{\rm
c}+1)9^{\alpha\sqrt{n}}(\frac{4^{\frac{3\delta+1}{4\delta}}}{\lambda})^{z_{\rm
c}}(\frac{64^{\frac{3\delta+1}{4\delta}}}{\gamma^{\delta}})^{n},$ (3)

where $z_{\rm c}\leq 3n$ is because the color contour length is upper bounded
by the sum of all edges of $G_{\triangle}$, and $z_{\rm c}\geq\sqrt{r\cdot n}$
is due to the triangular lattice geometry, which is proved in Lemma 2.1 in
[7].

If $\lambda\geq 4^{\frac{3\delta+1}{4\delta}}$, as long as
$\gamma^{\delta/3}>4^{\frac{3\delta+1}{4\delta}}$, the sum will be
exponentially small for sufficiently large $n$. Or if
$1\leq\lambda<4^{\frac{3\delta+1}{4\delta}}$, the sum further yields
$\pi(\Omega_{\neg\text{urb}})\leq n^{6}\cdot(3n+1)\cdot
9^{\alpha\sqrt{n}}\cdot
3n\cdot(\frac{16^{\frac{3\delta+1}{4\delta}}}{\lambda\gamma^{\delta/3}})^{3n}.$
As long as $\gamma^{\delta/3}>16^{\frac{3\delta+1}{4\delta}}\geq
16^{\frac{3\delta+1}{4\delta}}/\lambda$, the sum will still be exponentially
small for sufficiently large $n$. Combining the two cases, we can see that as
long as $\gamma^{\delta/3}>16^{\frac{3\delta+1}{4\delta}}$ and $\lambda>1$,
$\pi(\Omega_{\neg\text{urb}})\leq\xi_{1}^{n},$ for $\xi_{1}\in(0,1).$
Substituting $\delta=\epsilon/2$ into
$\gamma^{\delta/3}>16^{\frac{3\delta+1}{4\delta}}$ yields Theorem 3. ∎

In the above theorem, under the centralized urban sites placement, to realize
the urbanization of poverty, we find that for $\lambda$ in certain threshold,
it suffices to have
$\lambda\cdot\gamma^{\delta/3}>16^{\frac{3\delta+1}{4\delta}}$, where the
wealth bias and the racial bias both help the urbanization of poverty.
Instead, in the following corollary where urban sites are distributed, we find
the competing relationship between $\gamma$ and $\lambda$, where when $\gamma$
is larger than $\lambda$ to a certain extent, the urbanization of poverty is
achieved.

######  Corollary 4 (Distributed Urbanization of Poverty).

If
$\gamma>4^{\frac{3(3\epsilon+2)}{2\epsilon^{2}}}\cdot\max\\{\lambda^{6/\epsilon},4^{\frac{3(3\epsilon+2)}{2\epsilon^{2}}}\\}$
and $\lambda>1$, with the distributed urban sites, when $n$ is sufficiently
large, then for $\mathcal{M}$, configurations drawn from distribution $\pi$
have $\epsilon-$urbanization of poverty with probability at least
$1-\xi_{1}^{n}$, where $0<\xi_{1}<1$.

###### Proof of Corollary 4.

The bridge system and mapping are the same as the proof of Theorem 3 except
that: for $(f_{5}{\circ}f_{4}{\circ}f_{3})(\tau)$, we recover the same ratios
of each color and richness as in $\sigma$ in the distributed way defined in
section A.5. Claim 27 still holds true. For Claim 26, it now follows from
Lemma 24 and 19 that $h(\nu)-h(\sigma)\leq 2cn-z_{\rm c}\leq 3n-z_{\rm c}$,
and $p(\sigma)-p(\nu)\leq-\delta n$ holds true, where $\delta=\epsilon/2.$
Substituting the bounds into the Peierls argument yields

$\displaystyle\pi(\Omega_{\neg\text{urb}})$
$\displaystyle\leq\sum_{\nu\in\Omega}\pi(\nu)\sum_{z_{\rm c}=\sqrt{r\cdot
n}}^{3n}n^{6}(3n+1)(z_{\rm
c}+1)9^{\alpha\sqrt{n}}(\frac{4^{\frac{3\delta+1}{4\delta}}}{\lambda})^{z_{\rm
c}}(\frac{\lambda^{3}64^{\frac{3\delta+1}{4\delta}}}{\gamma^{\delta}})^{n}.$
(4)

Similarly, when $\lambda\geq 4^{\frac{3\delta+1}{4\delta}}$, if
$\gamma^{\delta/3}>\lambda\cdot 4^{\frac{3\delta+1}{4\delta}}$, the sum will
be exponentially small for sufficiently large $n$. When
$1\leq\lambda<4^{\frac{3\delta+1}{4\delta}}$, the sum further yields
$\pi(\Omega_{\neg\text{urb}})\leq n^{6}\cdot(3n+1)\cdot
9^{\alpha\sqrt{n}}\cdot 3n\cdot(\frac{\lambda\cdot
16^{\frac{3\delta+1}{4\delta}}}{\lambda\gamma^{\delta/3}})^{3n}.$ As long as
$\gamma^{\delta/3}>16^{\frac{3\delta+1}{4\delta}}$, the sum will still be
exponentially small for sufficiently large $n$. Combining the two cases, we
can see that as long as
$\gamma^{\delta/3}>4^{\frac{3\delta+1}{4\delta}}\max\\{\lambda,4^{\frac{3\delta+1}{4\delta}}\\}$
and $\lambda>1$, $\pi(\Omega_{\neg\text{urb}})\leq\xi_{1}^{n},$ for
$\xi_{1}\in(0,1).$ Substituting $\delta=\frac{\epsilon}{2}$ the lower bound of
$\gamma$ yields Corollary 4. ∎

To complement Theorem 3, we further prove that for any $\lambda>1$, if
$\gamma>1$ but is smaller than a threshold, it is exponentially unlikely we
will observe urbanization of poverty at stationary under certain demographic
parameter choices. See the proof details in Appendix 5.

######  Theorem 5 (Dispersion of Poverty).

Given the centralized urban sites, $p<c<p+\epsilon$, $r_{\rm p}<r_{\rm
r}-\delta$ and $b_{\rm p}<b_{\rm r}-\delta$, for any $\lambda>1$, if
$1<\gamma<(\frac{r-\delta}{r_{\rm p}})^{\frac{r_{\rm
p}}{p}}(\frac{b-\delta}{b_{\rm p}})^{\frac{b_{\rm p}}{p}}/2,$ when $n$ is
sufficiently large, then for $\mathcal{M}$, configurations drawn from
distribution $\pi$ have $\epsilon-$urbanization of poverty with probability at
most $\xi_{2}^{n}$ for some constant $0<\xi_{2}<1$ and
$\delta=\frac{\epsilon}{2}.$

##  4 Urbanized Racial Segregation

In this section, we prove urbanized racial segregation, where major racial
clusters will be formed, and the race that is predominantly poor will
predominantly occupy the urban area. We begin by formally define
$(\beta,\delta)-$racial segregation as follows

######  Definition 6.

For $\beta>4\sqrt{r}$ and $\delta\in(0,\frac{1}{2})$, a city configuration
$\sigma$ is said to be $(\beta,\delta)-$racial segregated if there is a subset
of agents $R$ such that:

  * •

there are at most $\beta\sqrt{n}$ racially heterogeneous edges of $\sigma$
with exactly one endpoint in $R$;

  * •

the number of red agents in $R$ is at least $rn-\delta n$.

The parameter $\delta$ is the tolerance for having agents of the wrong color
within the red region $R$, with smaller $\delta$ corresponding to a stricter
segregation class. If one color class has fewer than $\delta n$ agents, then
the entire configuration space will be $(\beta,\delta)-$segregated, with
$R=\varnothing,$ or $\bar{R}=\varnothing.$ We require that each color class
has more than $\delta n$ agents and, accordingly, we need $\delta<1/2.$ The
parameter $\beta$ in the definition controls how small the boundary is between
the red region $R$ and the rest of the configuration, and the minimal value
$4\sqrt{r}$ corresponds to the case where the red forms a homogeneous hexagon
cluster.

Then urbanized segregation is defined as the union of Definition 6 and
Definition 1 as follows.

######  Definition 7.

For centralized urban sites, $\beta>4\sqrt{r},$ and
$\epsilon\in(0,\frac{1}{2})$, a city has $(\beta,\epsilon)-$urbanized racial
segregation if it is $(\beta,\epsilon)-$racially segregated and have
$\epsilon-$urbanization of poverty.

Like Definition 6, $\beta$ here controls how small the boundary is between the
red region $R$ and the rest of the city. $\epsilon$ expresses the tolerance
for having agents of the wrong color within the monochromatic color regions
and having rich agents on the urban sites. In the following theorem, we show
that for large enough $\lambda$ and $\gamma$, then with high probability,
$\mathcal{M}$ leads to urbanized segregation. See Figure 2(c) for simulated
visualizations.

######  Theorem 8 (Urbanized Racial Segregation).

With centralized urban sites,
$\lambda>3^{\frac{\alpha}{\beta}}4^{\frac{3\delta+1}{4\delta}}$,
$\gamma^{\delta/3}>4^{\frac{3\delta+1}{4\delta}}$, and $n$ sufficiently large,
configurations from $\mathcal{M}$ drawn from distribution $\pi$ have
$(\beta,\epsilon)-$urbanized segregation with probability at least
$1-\xi^{\sqrt{n}}$ for some constant $0<\xi<1,$ and
$\delta=\frac{\epsilon}{2}.$

###### Proof of Theorem 8.

First, we define $U_{\beta,\epsilon}\subset\Omega$ to be the configurations
that have $(\beta,\epsilon)-$urbanized segregation. To prove Theorem 8, it
suffices to prove $\pi(\Omega\setminus U_{\beta,\epsilon})\leq\xi^{\sqrt{n}},$
where $\xi\in(0,1).$ We can further divide $\Omega\setminus
U_{\beta,\epsilon}$ into two parts: $\Omega_{\neg\text{urb}}$ that do not have
$\epsilon-$urbanization of poverty, and
$\Omega_{\text{urb}\wedge\neg\text{seg}}$ that have $\epsilon-$urbanization of
poverty and do not have $(\beta,\epsilon)-$segregation. Thus it suffices to
prove $\pi(\Omega_{\neg\text{urb}})\leq\xi_{1}^{n}$ and
$\pi(\Omega_{\text{urb}\wedge\neg\text{seg}})\leq\xi_{0}^{\sqrt{n}}$, for
$0<\xi_{1},\xi_{0}<1$. It follows from the proof of Theorem 3 that if
$\lambda\geq 4^{\frac{3\delta+1}{4\delta}}$, as long as
$\gamma^{\delta/3}>4^{\frac{3\delta+1}{4\delta}},$
$\pi(\Omega_{\neg\text{urb}})\leq\xi_{1}^{n}$. It is proved in Claim 31 in
Appendix D that If
$\lambda>3^{\frac{\alpha}{\beta}}4^{\frac{3\delta+1}{4\delta}}$,
$\pi(\Omega_{\text{urb}\wedge\neg\text{seg}})\leq\xi_{0}^{\sqrt{n}}$ for some
$\xi_{0}\in(0,1).$ Combining the two parts, to have $\pi(\Omega\setminus
S_{\beta,\epsilon})$ to be exponentially small for large $n$, it suffices to
have $\lambda>3^{\frac{\alpha}{\beta}}4^{\frac{3\delta+1}{4\delta}}$ and
$\gamma^{\delta/3}>4^{\frac{3\delta+1}{4\delta}}$. ∎

To complement Theorem 8, in the following theorem, we prove that for large
enough $\gamma$ but $\lambda>1$ smaller than a threshold, we will most likely
observe urbanization of poverty but racial integration outside the urban area
under certain demographic parameter choices. The proof technique is very
similar to the proof of Theorem 5. See Appendix E for proof details. A special
case of Theorem 9 is shown in Remark 10, where segregation of the poor red is
observed in the urban area with racial integration outside. See Figure 2(d)
for simulated visualizations.

######  Theorem 9 (Coexistence of Urbanization and Racial Integration).

With the centralized urban sites, for the demographics choices such that
$(\frac{p-\delta}{r_{\rm p}})^{r_{\rm p}}(\frac{1-p-\delta}{r_{\rm
r}})^{r_{\rm r}}>2^{r}$, if $1<\lambda^{3}<(\frac{p-\delta}{b_{\rm
p}})^{b_{\rm p}}(\frac{1-p-\delta}{b_{\rm r}})^{b_{\rm r}}/2^{r},$ and
$\gamma^{\delta/3}>16^{\frac{3\delta+1}{4\delta}},$ when $n$ is sufficiently
large, then for $\mathcal{M}$, configurations drawn from distribution $\pi$
have $\epsilon-$urbanization of poverty, and $(\beta,\delta)-$racial
integration outside the urban area, with probability at least $1-\xi_{3}^{n}$
for some constant $0<\xi_{3}<1$ and $\delta=\frac{\epsilon}{2}.$

######  Remark 10.

As a special case when the size of the urban sites can roughly accommodate all
the poor, where $p<c<p+\epsilon,$ if the demographics satisfies Theorem 9 with
$b_{\rm p}\leq m\epsilon$, then under the same bias parameter choices as
Theorem 9, the stationary configuration will have high probability of having
urbanized segregation of the poor red, where the density of the poor red on
the urban sites is at least $1-(m+1)\epsilon,$ and racial integration among
the rich outside the urban area.

##  5 Integration for Distributed Urban Sites

With the centralized urban sites, the wealth and homophily biases align to
cause segregation, as shown in Theorem 8. However, when the placement of urban
sites is distributed, the racial and wealth biases work against each other,
and we get integration if the wealth bias is strong enough to overcome the
homophily bias. (See Remark 38 in Appendix F for additional explanation.)

######  Theorem 11.

We assume we have a bank of urban sites with size $|\mathcal{U}|=cn$ evenly
partitioning the city like we find with bus routes. If the number of the poor
blue is small enough, $b_{\rm p}<\hat{b}_{\rm p}$, no matter how large
$\lambda$ is, as long as $\gamma^{\hat{b}_{\rm p}-b_{\rm
p}}>\lambda^{2c}64^{\frac{3\delta+1}{4\delta}}$, and $n$ is sufficiently
large, then configurations drawn from distribution $\pi$ will be
$(\beta,\delta)-$segregated with exponentially small probability
$\xi_{4}^{n}$, for some constant $0<\xi_{4}<1.$

Integration occurs because no matter how large the homophily bias weight
$\lambda^{-(h(\sigma)-h(\nu))}$ is, as long as the energy term arising from
the wealth bias, $\gamma^{p(\sigma)-p(\nu)}$, is larger, then the stationary
distribution will be very unlikely to be segregated. See Appendix F for proof
details.

##  6 Simulations

We supplement the theorems with simulations of $\mathcal{M}$, shown in Figure
2, for a city with income inequality starting from random initial locations of
agents. Figure 2 compares configurations after running $\mathcal{M}$ for the
same number of iterations, varying only the values of $\lambda$, $\gamma$, and
the placement of urban sites. Note that the parameter values of $\lambda$ and
$\gamma$ in the simulations are better than in our theorems, confirming that
our proven bounds are likely not tight.

Figure 2(a) verifies Theorem 3 and 9, showing the coexistence of urbanization
of poverty and racial integration outside the urban area under strong wealth
bias but slight racial bias. Specially, since the chosen urban area can
accommodate all the poor in a city and the city has severe income inequality,
Figure 2(a) can also be viewed as a verification of Remark 10, showing
segregation of the poor red in the urban area and integration outside.
Individuals in Figure 2(a) have small racial bias, so the wealth biases can
also drive racial segregation under centralized placement of urban sites.
Figure 2(b) verifies Corollary 4, showing the urbanization of poverty with
distributed urban sites. Compared with Figure 2(a), the pink cluster gets
dispersed via the distributed urban sites. Figure 2(c) verifies Theorem 8,
showing the urbanized segregation. Due to income inequality where most of the
poor are red, we can see the pink predominantly occupies the urban area. To
contrast, in Figure 3, where the demographic is without income equality, we
can see urbanized segregation and roughly the same amount of the poor red and
poor blue occupying the urban sites. Figure 2(d) verifies Theorem 11, showing
the mitigation of segregation via distributing the urban sites in the
existence of agents‘ strong racial bias, which should lead to Figure 2(c) if
not distributing the urban sites. Compared with Figure 2(b), whose segregation
level is even smaller, the difference is that agents in Figure 2(b) have
little racial bias, whereas in Figure 2(d) each agent has strong racial bias.
Figure 2(e) and 2(f) provide baselines of the main work. Figure 2(e) shows
segregation under strong racial bias without wealth bias, which is proved in
[6, 21]. Figure 2(f) shows integration under little racial and wealth bias,
which is proved in Theorem 5.

(a) mixed urbanization and racial integration outside the centralized urban
sites, with $\gamma=200,\lambda=1.01$.

(b) urbanization of poverty and integration with distributed urban sites with
$\gamma=200,\lambda=1.01$.

(c) urbanized segregation with $\gamma=200,\lambda=2$
.

(d) mitigation of segregation via distributing the urban sites, with
$\gamma=200,\lambda=2$.

(e) segregation no wealth bias when $\gamma=1,\lambda=2$.

(f) integration with $\gamma=1.01,\lambda=1.01$.

Figure 2: Simulations of $\mathcal{M}$ after five million iterations with
40$\%$ poor red(pink), 10$\%$ rich red (red), 40$\%$ rich blue (blue), and
10$\%$ poor blue (cyan) and a $46\%$ fraction of the urban sites.  Figure 3:
Urbanized segregation without income inequality under strong wealth and racial
biases ($\gamma=200,\lambda=1.01$). The racial and wealth distribution is
25$\%$ poor red, 25$\%$ rich red, 25$\%$ rich blue, and 25$\%$ poor blue.

## References

  * [1] George Barmpalias, Richard Elwes, and Andy Lewis-Pye.  Digital morphogenesis via schelling segregation.  In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 156–165. IEEE, 2014.
  * [2] Patrick Bayer and Robert McMillan.  Tiebout sorting and neighborhood stratification.  Journal of Public Economics, 96(11-12):1129–1143, 2012.
  * [3] Prateek Bhakta, Sarah Miracle, and Dana Randall.  Clustering and mixing times for segregation models on $\mathbb{Z}^{2}$.  In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 327–340. SIAM, 2014.
  * [4] Christian Borgs, Jennifer T Chayes, Alan Frieze, Jeong Han Kim, Prasad Tetali, Eric Vigoda, et al.  Torpid mixing of some monte carlo markov chain algorithms in statistical physics.  In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039), pages 218–229. IEEE, 1999.
  * [5] Christina Brandt, Nicole Immorlica, Gautam Kamath, and Robert Kleinberg.  An analysis of one-dimensional schelling segregation.  In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 789–804, 2012.
  * [6] Sarah Cannon, Joshua J. Daymude, Cem Gokmen, Dana Randall, and Andréa W. Richa.  A local stochastic algorithm for separation in heterogeneous self-organizing particle systems.  Proceedings of RANDOM, 2019.
  * [7] Sarah Cannon, Joshua J. Daymude, Dana Randall, and Andréa W. Richa.  A markov chain algorithm for compression in self-organizing particle systems.  In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, pages 279–288, 2016.
  * [8] William A.V. Clark.  Residential preferences and neighborhood racial segregation: A test of the schelling segregation model.  Demography, 28(1):1–19, 1991.
  * [9] William A.V. Clark and Mark Fossett.  Understanding the social context of the schelling segregation model.  Proceedings of the National Academy of Sciences, 105(11):4109–4114, 2008.
  * [10] Marie Cottrell, Madalina Olteanu, Julien Randon-Furling, and Aurelien Hazan.  Multidimensional urban segregation: an exploratory case study.  In 2017 12th International Workshop on Self-Organizing Maps and Learning Vector Quantization, Clustering and Data Visualization (WSOM), pages 1–7. IEEE, 2017.
  * [11] Mark Fossett.  Generative models of segregation: Investigating model-generated patterns of residential segregation by ethnicity and socioeconomic status.  The Journal of mathematical sociology, 35(1-3):114–145, 2011.
  * [12] Stefan Gerhold, Lev Glebsky, Carsten Schneider, Howard Weiss, and Burkhard Zimmermann.  Computing the complexity for schelling segregation models.  Communications in Nonlinear Science and Numerical Simulation, 13(10):2236–2245, 2008.
  * [13] Edward L. Glaeser, Matthew E. Kahn, and Jordan Rappaport.  Why do the poor live in cities? the role of public transportation.  Journal of urban Economics, 63(1):1–24, 2008.
  * [14] Robert B. Griffiths.  Peierls proof of spontaneous magnetization in a two-dimensional ising ferromagnet.  Physical Review, 136(2A):A437, 1964.
  * [15] Robert B. Griffiths.  The peierls argument for the existence of phase transitions.  In Mathematical Aspects of Statistical Mechanics, volume 5, pages 13–26. Amer. Math. Soc Providence, 1972.
  * [16] Erez Hatna and Itzhak Benenson.  Combining segregation and integration: Schelling model dynamics for heterogeneous population.  arXiv preprint arXiv:1406.5215, 2014.
  * [17] John Iceland, Daniel H. Weinberg, and Erika Steinmetz.  Racial and ethnic residential segregation in the United States 1980-2000, volume 8.  Bureau of Census, 2002.
  * [18] Katherine Kortum, Rajesh Paleti, Chandra R. Bhat, and Ram M. Pendyala.  Joint model of residential relocation choice and underlying causal factors.  Transportation research record, 2303(1):28–37, 2012.
  * [19] Alexander J. Laurie and Narendra K. Jaggi.  Role of‘vision‘in neighbourhood racial segregation: a variant of the schelling segregation model.  Urban Studies, 40(13):2687–2704, 2003.
  * [20] Sonia Lehman-Frisch.  Segregation, spatial (in) justice, and the city.  Berkeley Planning Journal, 24(1), 2011.
  * [21] Shengkai Li, Bahnisikha Dutta, Sarah Cannon, Joshua J. Daymude, Ram Avinery, Enes Aydin, Andréa W. Richa, Daniel I. Goldman, and Dana Randall.  Programming active cohesive granular matter with mechanically induced phase changes.  Science Advances, 7(17):eabe8494, 2021.
  * [22] Douglas S. Massey and Nancy A. Denton.  The dimensions of residential segregation.  Social forces, 67(2):281–315, 1988.
  * [23] Sarah Miracle, Dana Randall, and Amanda Pascoe Streib.  Clustering in interfering binary mixtures.  In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 652–663. Springer, 2011.
  * [24] Rocco Paolillo and Jan Lorenz.  How different homophily preferences mitigate and spur ethnic and value segregation: Schelling’s model extended.  Advances in Complex Systems, 21(06n07):1850026, 2018.
  * [25] Liliana Perez, Suzana Dragicevic, and Jonathan Gaudreau.  A geospatial agent-based model of the spatial urban dynamics of immigrant population: A study of the island of montreal, canada.  Plos one, 14(7):e0219188, 2019.
  * [26] Mark Pollicott and Howard Weiss.  The dynamics of schelling-type segregation models and a nonlinear graph laplacian variational problem.  Advances in Applied Mathematics, 27(1):17–40, 2001.
  * [27] Dana Randall.  Slow mixing of glauber dynamics via topological obstructions.  In Symposium on Discrete Algorithms: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, volume 22, pages 870–879, 2006.
  * [28] Anand Sahasranaman and Henrik Jeldtoft Jensen.  Ethnicity and wealth: The dynamics of dual segregation.  PloS one, 13(10):e0204307, 2018.
  * [29] Thomas C. Schelling.  Dynamic models of segregation.  Journal of mathematical sociology, 1(2):143–186, 1971.
  * [30] Abhinav Singh, Dmitri Vainchtein, and Howard Weiss.  Schelling’s segregation model: Parameters, scaling, and aggregation.  Demographic Research, 21:341–366, 2009.
  * [31] Dietrich Stauffer.  Social applications of two-dimensional ising models.  American Journal of Physics, 76(4):470–473, 2008.
  * [32] Dietrich Stauffer and Sorin Solomon.  Ising, schelling and self-organising segregation.  The European Physical Journal B, 57(4):473–479, 2007.
  * [33] Tiit Tammaru, Szymon Marcińczak, Raivo Aunap, Maarten van Ham, and Heleen Janssen.  Relationship between income inequality and residential segregation of socioeconomic groups.  Regional Studies, 54(4):450–461, 2020.

##  Appendix A : Technical Summary

###  A.1 Peierls arguments

The goal of a Peierls argument is to show that the probability that a sample
drawn from the stationary distribution $\pi$ of a Markov chain falls into a
target set is exponentially small in $n$, which yields
$\pi(\Omega_{t})\leq\xi^{n}$ for some constant $\xi\in(0,1).$ It maps from
configurations in the target set $\Omega_{\rm t}$ to configurations in the
configuration space $\Omega$ and shows that the map has an exponential gain in
probability. Thus the targeted configurations are exponentially unlikely in
$\Omega$. Mathematically, the mapping $\nu=f(\sigma)$ is defined from
$\sigma\in\Omega_{\rm t}$ to $\nu\in\Omega$, which yields

$\displaystyle\pi(\Omega_{\rm t})$ $\displaystyle=\sum_{\sigma\in\Omega_{\rm
t}}\pi(\sigma)\leq\sum_{\nu\in\Omega}\sum_{\sigma\in
f^{-1}(\nu)}\pi(\sigma)=\sum_{\nu\in\Omega}\pi(\nu)\frac{\sum_{\sigma\in
f^{-1}(\nu)}\pi(\sigma)}{\pi(\nu)}.$ (5)

In order to show $\pi(\Omega_{\rm t})\leq\xi^{n}$, the mapping needs to be
carefully defined to get the upper bounds of the probability ratio
$\frac{\pi(\sigma)}{\pi(\nu)}$ and the number of the preimages $|f^{-1}(\nu)|$
for any given $\nu$. Usually a mapped configuration $\nu$ with large
probability ratio $\frac{\pi(\sigma)}{\pi(\nu)}$ will also have many possible
preimages, which leads to the tradeoffs between $\frac{\pi(\sigma)}{\pi(\nu)}$
and $|f^{-1}(\nu)|$, known as an energy/entropy tradeoff.

To facilitate the mapping operations $f(\cdot)$ and the counting of
$|f^{-1}(\nu)|$, certain bridge systems have been defined in [23, 6] to
systematically encode the agents‘ colors to facilitate inverting the map (to
bound the number of preimages). Compared to [23, 6], here the configuration
space is enlarged by adding the wealth dimension, which requires extending the
bridge system to encode the multi-dimensional information simultaneously.
Moreover, because of the additional wealth term is reflected in the stationary
distribution, more careful mapping rules are required to account for both the
tradeoffs between $\frac{\pi(\sigma)}{\pi(\nu)}$ and $|f^{-1}(\nu)|$, and the
tradeoffs between the wealth bias and the homophily bias in the probability
measure.

###  A.2 Bridge Systems

Throughout the proofs, we use red and blue to represent the races, and
richness of the color to represent the wealth of each agent (rich red is red;
poor red is pink; rich blue is blue; poor blue is cyan). We first need to
extend the bridging technique to expand the encoded information dimension from
color only to both color and richness over the methods in [23, 6], within our
context. The following shows our adapted bridging technique.

Lattice Duality. The hexaagonal dual to the triangular lattice $G_{\triangle}$
is obtained by creating a vertex at the centroid of each unit triangle in
$G_{\triangle}$ and connecting two of these vertices if their corresponding
unit triangles have a common edge (as shown in Figure 4(a), the obtained
hexagonal lattice is denoted by $G_{\rm{hex}}$). Each edge $e^{{}^{\prime}}\in
G_{\rm{hex}}$ crosses a unique edge $e\in G_{\triangle}$ and separates two
adjacent agents living at the of $e$. There is a bijection between edges of
$G_{\triangle}$ and edges of $G_{\rm{hex}}$, associating an edge of
$G_{\triangle}$ with the unique edge of $G_{\rm{hex}},$ and vice versa.

Color Contours and Color Bridges. If an edge $e^{{}^{\prime}}\in G_{\rm{hex}}$
separates two agents heterogeneous in race, we call it a color edge. We define
a color contour to be made up of color edges and is a self-avoiding polygon in
$G_{\rm hex}$ that never visits the same vertex twice except to start and end
at the same place. The color contour is denoted in green as shown in Figure
4(b). The color bridges are shown in dashed green. They are self-avoiding
walks on $G_{\rm{hex}}$ that connect color contours to the boundary.

Richness Contours and Richness Bridges. If an edge $e^{{}^{\prime}}\in
G_{\rm{hex}}$ separates two agents heterogeneous in wealth, we call it a
richness edge. We define a richness contour to be made up of richness edges
and is a self-avoiding polygon, which is denoted in orange as shown in Figure
4(c). The richness bridges are shown in dashed orange, which connects richness
contours to the boundary.

(a)

(b)

(c)

Figure 4: (a) The hexagon lattice $G_{\triangle}$ dual to the city lattice
$G_{\triangle}$. (b) Demonstration of the color contours and color bridges.
(c) Demonstration of the richness contours and richness bridges.

$\delta-$Color-and-Richness Bridge System. We first define the $\delta-$color
bridge system similarly as [23, 6] as the following. For any color component
$F$ 111A color component is a maximal simply connected subset $F$ of agents
where all agents in $F$ adjacent to a location not in $F$ have the same race,
which we call the color of F., let $C_{\rm c}$ be a collection of color
contours of $F$. The color bridges collection $B_{\rm c}$ connects each color
contour in $C_{\rm c}$ to the boundary of $F$.

An agent $P$ is bridged in terms of color in $F$ if there exists a path
through agents of the same race as $P$ to the boundary of $F$ or a bridged
color contour in $C_{\rm c}$. An agent is unbridged in terms of color if such
a path does not exist. Then we define that $(B_{\rm c},C_{\rm c})$ is a
$\delta-$color bridge system for $F$ if

  * •

$|B_{\rm c}|\leq|C_{\rm c}|(1-\delta)/2\delta$, where $|B_{\rm c}|$ is the
total number of edges in $B_{\rm c}$ and $|C_{\rm c}|$ is the total number of
edges in $C_{\rm c}$;

  * •

the number of unbridged agents in terms of color in $F$ is $\leq\delta|F|$,
where $|F|$ is the number of agents in $F$.

Note $\delta\in[0,1]$ controls how much color information is omitted by the
$\delta-$color bridge system, and see proof for Lemma 7.2 in [6] for the
construction way of a $\delta-$color bridge system for any $F$.

######  Lemma 12 ([6]).

For any color component $F$, there exists a $\delta-$color bridge system for
$F$.

Similarly, we can also define the $\delta-$richness bridge system, and the
richness component. The following lemma holds similarly.

######  Lemma 13 ([6]).

For any richness component $F$, there exists a $\delta-$richness bridge system
for $F$.

We call the joint $\delta-$color and $\delta-$richness bridge system a
$\delta-$color-and-richness bridge system. For any configuration
$\sigma\in\Omega_{\rm t},$ we can construct a $\delta-$color-and-richness
bridge system. See Figure 5(a) for illustrations. where at most $\delta n$
agents are not bridged in terms of color, and at most $\delta n$ agents are
not bridged in terms of richness. Combining Lemma 12 and Lemma 13, the
following lemma holds.

######  Lemma 14.

For any finite region $F$, there exists a $\delta-$color-and-richness bridge
system for $F$.

Crossing Contours and Non-Crossing Contours. The contour that touches the
boundary of the defined domain is called a crossing contour. The contour that
does not touch the boundary is called a non-crossing contour. For example,
Figure 4(b) has one crossing color contour and three non-crossing color
contours.

The sum of the number of edges of a contour is called the length of the
contour. We denote the length of the bridged non-crossing contours as $y$, and
length of the crossing contours as $x$. The length of the bridged non-crossing
color contours is denoted by $y_{\rm c}$. The length of the crossing color
contours is $x_{\rm c}$. We can also define non-crossing richness contours
$y_{\rm r}$ and crossing richness contours $x_{\rm r}$ in similar way. It
follows that $y=y_{\rm c}+y_{\rm r}$ and $x=x_{\rm c}+x_{\rm r}$. We call
$z_{\rm c}\triangleq x_{\rm c}+y_{\rm c}$ the bridged color contour length,
and $z_{\rm r}\triangleq x_{\rm r}+y_{\rm r}$ the bridged richness contour
length.

For a given bridged color or richness contour length, the following lemmas
bound the number of possible bridge systems, which can be counted in a depth-
first way (see proof details in [6]).

######  Lemma 15 (Lemma 7.6 in [6]).

For a given $z_{\rm r}=y_{\rm r}+x_{\rm r}$, there are at most $(z_{\rm
r}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm r}}$ ways of
constructing a $\delta-$richness bridge system.

######  Lemma 16 (Lemma 7.6 in [6]).

For a given $z_{\rm c}=y_{\rm c}+x_{\rm c}$, there are at most $(z_{\rm
c}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm c}}$ ways of
constructing a $\delta-$color bridge system.

Combining Lemma 15 and 16 yields the following lemma.

######  Lemma 17.

For a given $z_{\rm c}=y_{\rm c}+x_{\rm c}$, there are at most $(\max\\{z_{\rm
r}\\}+1)(z_{\rm c}+1)9^{\alpha\sqrt{n}}4^{(\frac{3\delta+1}{4\delta})(z_{\rm
c}+\max\\{z_{\rm r}\\})}$ ways of constructing a $\delta-$color-and-richness
bridge system.

(a)

(b)

(c)

Figure 5: (a) A configuration $\sigma$ with a $\delta-$color-and-richness
bridge system. (b) Richness inversion in $\psi=f_{1}(\sigma)$. (c) Color
inversion in $\tau=f_{2}(\psi)$.

###  A.3 The Inversion Mappings

For any configuration $\sigma\in\Omega_{\rm t}$, after bridging, in order to
get the upper bounds of $h(\nu)-h(\sigma)$ and $p(\sigma)-p(\nu)$, we define
the mappings to first eliminate the bridged racially heterogeneous edges
$h(\sigma)$ and bridged poor agents, and then define mappings to recover the
racially heterogeneous edges and the poor agents on the urban sites in various
designed ways. Different proof targets will lead to different recovery
mappings, shown in the theorems in the following sections. For this part, we
define the richness inversion function $f_{1}(\cdot)$ and color inversion
function $f_{2}(\cdot)$ which eliminate most of the racially heterogeneous
edges and poor agents respectively.

Richness Inversion. First, we represent each agent with two bits, with poor
red denoted $00$, poor blue denoted $01$, rich red denoted $10$ and rich blue
denoted $11.$ We define the richness inversion function $f_{1}(\cdot)$ as
follows: for any agent $ij$, where $i$ is the richness bit, and $j$ is the
color bit, the richness bit is flipped to ${(i+b)(\text{mod}2)}$ for agent
$ij$ that is surrounded by $b$ bridged richness contours or unbridged crossing
richness contours (see the left corner‘s contour as an example of richness
contours in Figure 5(b)). The color bit remains unchanged. See Figure 5(b) for
illustrations.

######  Lemma 18 (Lemma 7.5 in [6]).

For any configuration $\psi=f_{1}(\sigma)$, there are at most $\delta n$ poor
agents, and they are unbridged; no additional color edges are introduced; and
for any mapped configuration $\psi$, there is only one preimage for a given
$\delta-$richness bridge system.

Color Inversion. We define the function color inversion $f_{2}(\cdot)$ as: for
any agent $ij$, where $i$ is the richness bit, and $j$ is the color bit, the
color bit is flipped to ${(j+b)(\text{mod}2)}$ for the agent $ij$ that is
surrounded by $b$ bridged color contours or unbridged crossing color contours
(like the red agent on the right boundary in Figure 5(b)). The richness bit
remains unchanged. See Figure 5(c) for illustrations.

######  Lemma 19 (Lemma 7.8 in [6]).

For any $\tau=f_{2}(\psi)$, $(x_{\rm c}+y_{\rm c})$ of the original racially
heterogeneous edges in $\psi$ are eliminated: $h(\tau)-h(\psi)\leq-(x_{\rm
c}+y_{\rm c})$; no additional poor agents are introduced; and for any mapped
configuration $\tau$ with a given bridge system, there is only one preimage
that can be mapped to it.

###  A.4 The Color and Richness Recovery Mappings

After eliminating the bridged poor agents and the bridged racially
heterogeneous edges in $f_{1}(\cdot)$ and $f_{2}(\cdot)$, we need to recover
the same ratio of each color and richness as in $\sigma$, which is defined in
$f_{3}(\cdot)$, $f_{4}(\cdot)$ and $f_{5}(\cdot)$ as the following.

Pink Recovery. For any $\tau=(f_{2}{\circ}f_{1})(\sigma)$, we define the pink
recovery function $f_{3}(\tau)$ as to flip the agents‘ colors to pink starting
from a fixed place in a given order except when encountering the following
unbridged agents: we flip the unbridged pink to cyan, cyan to red, and red to
blue. The flipping process stops once reaching the correct number of the pink
agents as in $\sigma.$

######  Lemma 20.

For any mapped configuration $\zeta$, if the starting location of $f_{3}$ and
the flipping order are specified, there are at most $n$ preimages that can be
mapped to it: $|f_{3}^{-1}(\zeta)|\leq n$.

###### Proof.

Given $\zeta$, it suffices to recover its preimage if we are given the
stopping place and there are at most $n$ possible stopping places. ∎

Cyan Recovery. For any $\zeta=(f_{3}{\circ}f_{2}{\circ}f_{1})(\sigma)$, we
define the cyan recovery function $f_{4}(\zeta)$ as to flip the agents
starting from the stopping place of $f_{3}(\cdot)$ in a given order to cyan
except when encountering the following unbridged agents: we remain the
unbridged pink to pink, red to red, and flip cyan to blue. The flipping will
be stopped after reaching the right number of the cyan agents as in $\sigma.$
The proof of Lemma 21 is similar to the proof of Lemma 20. Given any mapped
configuration $\phi$, it suffices to recover its preimage $f_{4}^{-1}(\phi)$
if we are given the starting location and the stopping place of
$f_{4}(\cdot)$, which is bounded by $n^{2}.$

######  Lemma 21.

For any mapped configuration $\phi$, if the flipping order is specified, there
are at most $n^{2}$ preimages that can be mapped to it:
$|f_{4}^{-1}(\phi)|\leq n^{2}$.

Red Recovery. For any
$\phi=(f_{4}{\circ}f_{3}{\circ}f_{2}{\circ}f_{1})(\sigma)$, we define the red
recovery function $f_{5}(\cdot)$ as to flip agents starting from the stopping
place of $f_{4}(\cdot)$ in a given order to red except when encountering the
following unbridged agents: we flip the unbridged red to blue, cyan to pink,
and pink to cyan. To guarantee the right number of the cyan and pink in the
mapped configuration $\nu$, whenever we flip an unbridged cyan to pink during
this phase, we flip one pink back to cyan starting from the stop location of
$f_{3}(\cdot)$. If we encounter non-pink agents, we first recover its colors
before $f_{3}(\cdot)$ and use the rule of flipping the unbridged agents for
$f_{4}(\cdot)$. Whenever we flip an unbridged pink to cyan, we flip one cyan
to pink starting from the starting location of $f_{4}(\cdot)$, and if we
encounter agents that are not cyan, we first recover its colors before
$f_{4}(\cdot)$ and use the rule of $f_{3}(\cdot)$. We stop such operations
after reaching the right number of the red for $\nu$.

The proof of Lemma 22 is similar as Lemma 20. To show the upper bound of the
number of preimages $|f_{5}^{-1}(\nu)|$ for a given $\nu$: we first need to
find the stop location of $f_{4}(\cdot)$, which has at most $n$ possibilities.
Then we complement the colors of the first $k$ elements of $\nu$, where
$k\in\\{0,1,...,n-1\\}$, and possibly we also need to find the stop location
of $f_{3}$ (same location as the starting location of $f_{4}$), which has at
most $n$ possibilities. Altogether there are at most $n^{3}$ different
preimages.

######  Lemma 22.

For any mapped configuration $\nu$, if the flipping order is specified, there
are at most $n^{3}$ preimages that can be mapped to it: $|f_{5}^{-1}(\nu)|\leq
n^{3}$.

###  A.5 The Centralized Recovery and Distributed Recovery

(a)

(b)

(c)

Figure 6: (a) Demonstration of $\zeta=f_{3}(\tau)$. (b) Demonstration of
$\phi=f_{4}(\zeta)$. (c) Demonstration of $\nu=f_{5}(\phi)$.

Centralized Recovery. If the urban sites are centralized, like shown in Figure
1(a), for the pink, cyan, and red recoveries defined in $f_{3},f_{4}$ and
$f_{5}$, the starting location of $f_{3}$ and the flipping order of each
function can be specified in the centralized way: the starting location of
$f_{3}$ is specified to be the center of the urban area, and the flipping
order for $f_{3},f_{4},f_{5}$ are specified as in clockwise direction and loop
to the immediate outer layer when completing flipping one clockwise cycle like
shown in Figure 6(a), 6(b), and 6(c). In such a way, the following upper bound
can be obtained.

######  Lemma 23.

If the recoveries are specified in the centralized way, for any
$\nu=(f_{5}{\circ}f_{4}{\circ}f_{3})(\tau),$ there are at most
$3\alpha\sqrt{n}$ racially heterogeneous edges introduced by the recovery
operations: $h(\nu)-h(\tau)\leq 3\alpha\sqrt{n}$.

###### Proof.

In the defined way of centralized recovery, racially heterogeneous edges are
created along the hexagon boundaries between the pink and cyan regions, the
cyan and red regions, and the red and the rest region. Each boundary can be
upper bounded by the perimeter of the fundamental domain, which is
$\alpha\sqrt{n}$, and in total $3\alpha\sqrt{n}$.

Inside each pink, cyan, and red region, the unbridged agents will not create
additional racially heterogeneous edges: for any
$\tau=(f_{2}{\circ}f_{1})(\sigma)$, It follows from the definition of $f_{2}$
and Lemma 19 that the bridged agents in $\tau$ are either cyan or blue. The
flipping rules for the unbridged agents in $f_{3},f_{4}$ and $f_{5}$ thus can
be verified to not introduce additional racially heterogeneous edges. Hence we
get $h(\nu)-h(\tau)\leq 3\alpha\sqrt{n}.$ ∎

Distributed Recovery. If the urban sites evenly partition the city(diamond-
shaped city) like shown in Figure 1(b), for the pink, cyan, and red recoveries
defined in $f_{3},f_{4}$ and $f_{5}$, the starting location of $f_{3}$ and the
flipping order of each function can be specified in the distributed way: the
starting location of $f_{3}$ is specified to be the most top left corner of
the urban site, and the flipping order for $f_{3},f_{4},f_{5}$ are specified
as to only flip the agents on the urban sites row after row, and then flip
agents on the non-urban sites row after row after finishing flipping all the
agents on the urban sites. In such a way, the following upper bound can be
obtained.

######  Lemma 24.

If the recoveries are specified in the distributed way, for any
$\nu=(f_{5}{\circ}f_{4}{\circ}f_{3})(\tau),$ there are at most $2cn$ racially
heterogeneous edges introduced by the recovery operations: $h(\nu)-h(\tau)\leq
2cn$.

###### Proof.

In the defined way of distributed recovery, compared with $\tau$, racially
heterogeneous edges are possibly created along the diamond boundaries between
the urban and non-urban sites (see Figure 1(b) for demonstration), which can
be upper bounded by the sum of perimeters of all the small diamonds.

The perimeter of a diamond can be upper bounded by $4\cdot k$: since the urban
sites, with total size $c\cdot n$, evenly partitions the finite lattice, the
number of row urban sites is $\frac{cn}{2}.$ Each row of urban sites has
$\sqrt{n}$ sites, so there are $\frac{c\sqrt{n}}{2}$ rows of urban sites.
Since each column of urban sites also has $\sqrt{n}$ sites, we can get the
side length of a diamond to be
$k=\frac{\sqrt{n}}{\frac{c\sqrt{n}}{2}-1}\geq\frac{2}{c}$. Since the side
length of one diamond is $k,$ the total number of the diamonds can be upper
bounded by $(\frac{\sqrt{n}}{k})^{2}$. Hence the sum of perimeters of all the
diamonds can be upper bounded by $\frac{4n}{k}\leq 2cn,$ which yields
$h(\nu)-h(\tau)\leq 2cn.$ ∎

######  Lemma 25.

For any $\nu=(f_{5}{\circ}f_{4}{\circ}f_{3}{\circ}f_{2}{\circ}f_{1})(\sigma),$
if the recovery is specified in the centralized way or distributed way, the
number of the poor agents on the urban sites is lower bounded by
$p(\nu)\geq(\min\\{c,p\\}-\delta)n.$

###### Proof.

It follows from Lemma 18 that the number of poor agents is at most $\delta n$
in the mapped $\psi=f_{1}(\sigma)$ and they are unbridged. It follows from
Lemma 19 that no additional poor agents will be introduced for any
$\tau=f_{2}(\psi).$ In the worst case scenario, these $\delta n$ poor agents
will not be recovered on the urban sites in
$\nu=(f_{5}{\circ}f_{4}{\circ}f_{3})(\tau)$. For the rest of the poor agents,
because of the recovery way in the order of pink followed by cyan then red, it
yields that $p(\nu)\geq(\min\\{c,p\\}-\delta)n$. ∎

##  Appendix B : Proof of Theorem 3 (Urbanization of Poverty)

The claims in the proof of Theorem 3 are shown and proved as follows.

######  Claim 26.

For any $\sigma\in\Omega_{\neg\text{urb}}$ with a given bridged color contour
length $z_{\rm c}$, for the defined mapping $\nu=f(\sigma),$
$h(\nu)-h(\sigma)\leq 3\alpha\sqrt{n}-z_{\rm c}$ and
$p(\sigma)-p(\nu)\leq-\delta n$, where $\delta=\epsilon/2.$

###### Proof of Claim 26.

It follows from Lemma 19 that for any $\sigma\in\Omega_{\neg\text{urb}}$,
$h(\tau)-h(\sigma)\leq-z_{\rm c}$. It follows from Lemma 23 that
$h(\nu)-h(\tau)\leq 3\alpha\sqrt{n}.$ Combining the two inequalities, we get
$h(\nu)-h(\sigma)\leq 3\alpha\sqrt{n}-z_{\rm c}$.

For any $\sigma\in\Omega_{\neg\text{urb}}$,
$p(\sigma)\leq(\min\\{c,p\\}-\epsilon)n$ is satisfied. It follows from Lemma
25 that $p(\nu)\geq(\min\\{c,p\\}-\delta)n$. Combining the inequalities,
setting $\epsilon=2\delta$, it follows that $p(\sigma)-p(\nu)\leq-\epsilon
n+\delta n=-\delta n$. ∎

######  Claim 27.

For a given color contour length $z_{\rm c}$, for any $\nu=f(\sigma)$, the
number of configurations in $\Omega_{\neg\text{urb}}$ that can map to $\nu$ is
upper bounded by: $|f^{-1}(\nu)|\leq(z_{\rm
c}+1)9^{\alpha\sqrt{n}}4^{(\frac{3\delta+1}{4\delta})(z_{\rm c}+3n)}.$

###### Proof of Claim 27.

We denote $\nu=f_{5}(\phi)$, $\phi=f_{4}(\zeta)$, $\zeta=f_{3}(\tau)$,
$\tau=f_{2}(\psi)$, and $\psi=f_{1}(\sigma)$. Since
$f(\sigma)=(f_{5}{\circ}f_{4}{\circ}f_{3}{\circ}f_{2}{\circ}f_{1})(\sigma)$,
for any $\nu$, it follows that
$|f^{-1}(\nu)|\leq|f_{5}^{-1}(\nu)|\cdot|f_{4}^{-1}(\phi)|\cdot|f_{3}^{-1}(\zeta)|\cdot|f_{2}^{-1}(\tau)|\cdot|f_{1}^{-1}(\psi)|$.
It follows from Lemma 22 that $|f_{5}^{-1}(\nu)|\leq n^{3}$. It follows from
Lemma 21 that $|f_{4}^{-1}(\phi)|\leq n^{2}$. It follows from Lemma 20 that
$|f_{3}^{-1}(\zeta)|\leq n$.

For a given color contour length $z_{\rm c}$, the number of all possible
bridge systems $b$ is upper bounded by $b\leq(\max\\{z_{\rm r}\\}+1)(z_{\rm
c}+1)9^{\alpha\sqrt{n}}4^{(\frac{3\delta+1}{4\delta})(z_{\rm c}+\max\\{z_{\rm
r}\\})}$ in Lemma 17. For the triangular lattice with $n$ vertices, the sum of
all edges is $3n$. Because of the lattice duality, we conclude $z_{\rm r}\leq
3n.$ Thus $b\leq(3n+1)(z_{\rm
c}+1)9^{\alpha\sqrt{n}}4^{(\frac{3\delta+1}{4\delta})(z_{\rm c}+3n)}$. It
follows from Lemma 18 and 19 that for a given bridge system and a given image,
the number of the corresponding preimages of both $f_{2}$ and $f_{1}$ is one.

Hence, we conclude that for a given $z_{\rm c}$,
$|f^{-1}(\nu)|\leq|f_{5}^{-1}(\nu)|\cdot|f_{4}^{-1}(\phi)|\cdot|f_{3}^{-1}(\zeta)|\cdot
b\leq n^{6}(3n+1)(z_{\rm
c}+1)9^{\alpha\sqrt{n}}4^{(\frac{3\delta+1}{4\delta})(z_{\rm c}+3n)}$. ∎

##  Appendix C : Proof of Theorem 5 (Dispersion of Poverty)

###### Proof of Theorem 5.

Let $\Omega_{\rm{urb}}\in\Omega$ be the set of configurations that have
$\epsilon-$urbanization of poverty. To prove Theorem 5, it suffices to prove
$\pi(\Omega_{\rm{urb}})\leq\xi_{2}^{n}$ for some $\xi_{2}\in(0,1)$ and large
enough $n$.

For each $\sigma\in\Omega_{\rm{urb}},$ we construct a $\delta-$richness bridge
system and define the mapping $\nu=d(\sigma)=(d_{2}\circ f_{1})(\sigma)$ as
the following: we first do the richness inversion and obtain
$\tau=f_{1}(\sigma)$; next for $\tau$, we randomly flip the blue to cyan until
the right number of the cyan, and we randomly flip the red to pink until the
right number of the pink and obtain $\nu=d_{2}(\tau).$

######  Claim 28.

For any $\sigma\in\Omega_{\rm{urb}}$, for the defined mapping $\nu=d(\sigma),$
$h(\nu)-h(\sigma)\leq 0$ and $p(\sigma)-p(\nu)\leq cn.$

###### Proof of Claim 28.

Because the color information remains unchanged for $f_{1}$ and $d_{2}$,
$h(\nu)-h(\sigma)\leq 0$. For any $\sigma$, $p(\sigma)\leq\min\\{c,p\\}n$. The
scenario under discussion follows that $p<c$, thus $p(\sigma)\leq pn.$ Since
for any $\nu$, $p(\nu)\geq 0,$ hence $p(\sigma)-p(\nu)\leq pn.$ ∎

######  Claim 29.

For a given richness contour length $z_{\rm r}$, for any $\nu=d(\sigma)$, the
number of configurations in $\Omega_{\rm{urb}}$ that can map to $\nu$ is upper
bounded by:
$|d^{-1}(\nu)|\leq(\beta\sqrt{n}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}\beta\sqrt{n}}2^{pn}.$

###### Proof of Claim 29.

We denote $\tau=f_{1}(\sigma)$ and $\nu=d_{2}(\tau)$. Since
$d(\sigma)=(d_{2}{\circ}f_{1})(\sigma)$, for any $\nu,$ it follows that
$|d^{-1}(\nu)|\leq|f_{1}^{-1}(\tau)|\cdot|d_{2}^{-1}(\nu)|$. For any
$\nu\in\Omega,$ the number of preimages $|d_{2}^{-1}(\nu)|$ can be upper
bounded by $2^{pn}$ by recording whether each poor agent is flipped or not.

Because every configuration $\sigma\in\Omega_{\rm{urb}}$ satisfies
$\epsilon-$urbanization, assuming we also have $p<c<p+\epsilon,$ then $\sigma$
also satisfies $(\beta,\epsilon)-$wealth segregation (see definition in Remark
2). Hence $z_{\rm r}$ can be upper bounded by $z_{\rm r}\leq\beta\sqrt{n}$
(See Lemma 7.4 in [6] for details). Thus it follows from Lemma 15 that for any
given $z_{\rm r}$, the number of $\delta-$richness bridge systems can be upper
bounded by
$(\beta\sqrt{n}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}\beta\sqrt{n}}$.
It follows from Lemma 18 that for any $\tau$, for a given $\delta-$richness
bridge system, the number of preimages is one. Hence we conclude for any
$\tau$ with a given $z_{\rm r}$, the number of preimages is upper bounded by:
$|f_{1}^{-1}(\tau)|\leq(\beta\sqrt{n}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}\beta\sqrt{n}}$.
Combining the two inequalities, we conclude
$|d^{-1}(\nu)|\leq(\beta\sqrt{n}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}\beta\sqrt{n}}2^{pn}$.
∎

######  Claim 30.

For any given $\sigma\in\Omega_{\rm{urb}},$ we denote $D(\sigma)$ to be the
set of all possible images $\nu=d(\sigma)$ mapped from $\sigma$. It follows
that $|D(\sigma)|\geq(\frac{r-\delta}{r_{\rm p}})^{(r_{\rm
p}-\delta)n}(\frac{b-\delta}{b_{\rm p}})^{(b_{\rm p}-\delta)n}$.

###### Proof of Claim 30.

It follows from the definition of $f_{1}$ that for any given $\sigma,$ there
is only one configuration can be obtained from $f_{1}(\sigma)$. For any given
$\tau$, we define $D_{2}(\tau)$ to be the set of all possible configurations
obtained from $d_{2}(\tau)$ by flipping the pink and cyan back; then
$|D_{2}(\tau)|=\begin{pmatrix}rn-a_{\tau}\\\ r_{\rm
p}n-a_{\tau}^{{}^{\prime}}\end{pmatrix}\cdot\begin{pmatrix}bn-a_{\tau}\\\
b_{\rm p}n-a_{\tau}^{{}^{\prime\prime}}\end{pmatrix}$, where $a_{\tau}$ is the
number of unbridged agents in $\tau$, $a_{\tau}^{{}^{\prime}}$ is the number
of unbridged pink, and $a_{\tau}^{{}^{\prime\prime}}$ is the number of
unbridged cyan. Thus $|D_{2}(\tau)|$ can be further lower bounded by

$\displaystyle|D_{2}(\tau)|\geq(\frac{r-a_{\tau}}{r_{\rm
p}-a_{\tau}^{{}^{\prime}}})^{(r_{\rm
p}-a_{\tau}^{{}^{\prime}})n}(\frac{b-a_{\tau}}{b_{\rm
p}-a_{\tau}^{{}^{\prime\prime}}})^{(b_{\rm
p}-a_{\tau}^{{}^{\prime\prime}})n}\geq(\frac{r-\delta}{r_{\rm p}})^{(r_{\rm
p}-\delta)n}(\frac{b-\delta}{b_{\rm p}})^{(b_{\rm p}-\delta)n}.$ (6)

Hence $|D(\sigma)|$ can be lower bounded by $|D(\sigma)|\geq
1\cdot|D_{2}(\tau)|\geq(\frac{r-\delta}{r_{\rm p}})^{(r_{\rm
p}-\delta)n}(\frac{b-\delta}{b_{\rm p}})^{(b_{\rm p}-\delta)n}.$ ∎

Finally, we define a weighted bipartite graph $G(\Omega_{\rm{urb}},\Omega,E)$
with an edge of weight $\pi(\sigma)$ between $\sigma\in\Omega_{\rm{urb}}$ and
$\nu\in\Omega$. The total weight of edges is

$\displaystyle\sum_{\sigma\in\Omega_{\rm{urb}}}\pi(\sigma)\cdot|D(\sigma)|\geq\pi(\Omega_{\rm{urb}})(\frac{r-\delta}{r_{\rm
p}})^{(r_{\rm p}-\delta)n}(\frac{b-\delta}{b_{\rm p}})^{(b_{\rm p}-\delta)n}.$
(7)

On the other hand, the weight of the edges is at most

$\displaystyle\sum_{\nu\in\Omega}\sum_{\sigma\in
d^{-1}(\nu)}\max_{\sigma\in\Omega_{\rm{urb}}}{\pi(\sigma)}=\sum_{\nu\in\Omega}\pi(\nu)\sum_{\sigma\in
d^{-1}(\nu)}\frac{\max_{\sigma\in\Omega_{\rm{urb}}}{\pi(\sigma)}}{\pi(\nu)}|d^{-1}(\nu)|$
$\displaystyle\leq\sum_{\nu\in\Omega}\pi(\nu)\sum_{z_{\rm
r=\sqrt{pn}}}^{\beta\sqrt{n}}\lambda^{\max(h(\nu)-h(\sigma))}\gamma^{\max(p(\sigma)-p(\nu))}(\beta\sqrt{n}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}\beta\sqrt{n}}2^{pn}$
$\displaystyle\leq\gamma^{pn}\beta\sqrt{n}(\beta\sqrt{n}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}\beta\sqrt{n}}2^{pn},$
(8)

where the inequalities in Claim 28 and 29 has been substituted in the above
derivation. Combining (7) and (C), we have

$\displaystyle\pi(\Omega_{\rm{urb}})(\frac{r-\delta}{r_{\rm p}})^{(r_{\rm
p}-\delta)n}(\frac{b-\delta}{b_{\rm p}})^{(b_{\rm
p}-\delta)n}\leq\beta\sqrt{n}(\beta\sqrt{n}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}\beta\sqrt{n}}\gamma^{cn}2^{pn}.$

For large enough $n$, to have $\pi(\Omega_{\rm{urb}})\leq\xi_{3}^{n}$ for some
$\xi_{3}\in(0,1)$, it suffices to have

$\gamma^{pn}2^{pn}<(\frac{r-\delta}{r_{\rm p}})^{(r_{\rm
p}-\delta)n}(\frac{b-\delta}{b_{\rm p}})^{(b_{\rm
p}-\delta)n}<(\frac{r-\delta}{r_{\rm p}})^{r_{\rm p}n}(\frac{b-\delta}{b_{\rm
p}})^{b_{\rm p}n},$

which can be rewritten as

$\gamma<(\frac{r-\delta}{r_{\rm p}})^{\frac{r_{\rm
p}}{p}}(\frac{b-\delta}{b_{\rm p}})^{\frac{b_{\rm p}}{p}}/2.$

Since $\gamma>1,$ to make the right hand side of the above inequality greater
than one, it suffices to have $\frac{r-\delta}{r_{\rm p}}>2$ and
$\frac{b-\delta}{b_{\rm p}}>2$, which can be rewritten as $r_{\rm p}<r_{\rm
r}-\delta$ and $b_{\rm p}<b_{\rm r}-\delta.$ ∎

##  Appendix D : Proof of Theorem 8 (Urbanized Racial Segregation)

Claim 31 in the proof of Theorem 8 are shown and proved as follows.

######  Claim 31.

If $\lambda>3^{\frac{\alpha}{\beta}}4^{\frac{3\delta+1}{4\delta}}$,
$\pi(\Omega_{\text{urb}\wedge\neg\text{seg}})\leq\xi_{0}^{\sqrt{n}}$ for some
$\xi_{0}\in(0,1).$

###### Proof of Claim 31.

To prove $\pi(\Omega_{\text{urb}\wedge\neg\text{seg}})\leq\xi_{0}^{\sqrt{n}}$,
we define the bridge system and the mapping
$g(\cdot)=(g_{3}{\circ}g_{2}{\circ}f_{2})(\cdot)$ from the set
$\Omega_{\text{urb}\wedge\neg\text{seg}}$ to $\Omega$ as the following: for
any $\sigma\in\Omega_{\text{urb}\wedge\neg\text{seg}}$, we construct a
$\delta-$color bridge system for it. Then we do color inversion like defined
in $f_{2}(\cdot)$, and get $\psi=f_{2}(\sigma).$

$\tau=g_{2}(\psi)$ is defined as the following: starting from the center of
the urban area, we flip the color bit ($0$ to $1$ and $1$ to $0$) of each
agent layer by layer in a given order until the right number of pink and cyan
is reached.

$\nu=g_{3}(\tau)$ is then defined as the following: starting from outside the
urban area boundary, we flip the color bit ($0$ to $1$ and $1$ to $0$) of each
agent layer by layer in a given order until the right number of red and blue
is reached. During this phase, whenever we flip the color bit for an unbridged
pink, we go back to the stopping location of $g_{2}(\cdot)$ to flip one more
cyan to pink, and vise versa.

######  Claim 32.

For any $\sigma\in\Omega_{\text{urb}\wedge\neg\text{seg}}$ with a given
bridged color contour length $z_{\rm c}$, for the defined mapping
$\nu=g(\sigma),$ $h(\nu)-h(\sigma)\leq 2\alpha\sqrt{n}-z_{\rm c}$ and
$p(\sigma)-p(\nu)\leq 0.$

###### Proof of Claim 32.

It follows from Lemma 19 that $h(\psi)-h(\sigma)\leq-z_{\rm c}$ and the
richness information remains unchanged. For the color recoveries
$(g_{3}{\circ}g_{2})(\psi)$, similar as the proving strategy of Lemma 23, we
at most introduce $2\alpha\sqrt{n}$ racially heterogeneous edges compared with
$h(\psi)$ due to the hexagon boundaries created between pink and cyan, and
cyan and red. Each boundary can be upper bounded by the perimeter of the
fundamental domain $\alpha\sqrt{n}$. The richness information remains
unchanged. Thus $h(\nu)-h(\sigma)\leq h(\nu)-h(\psi)+h(\psi)-h(\sigma)\leq
2\alpha\sqrt{n}-z_{\rm c}$ and $p(\sigma)-p(\nu)\leq 0.$ ∎

######  Claim 33.

For a given color contour length $z_{\rm c}$, for any $\nu=g(\sigma)$, the
number of configurations in $\Omega_{\text{urb}\wedge\neg\text{seg}}$ that can
map to $\nu$ is upper bounded by: $|g^{-1}(\nu)|\leq n^{3}(z_{\rm
c}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm c}}.$

###### Proof of Claim 33.

For any given mapped $\tau$, the number of preimages $|g_{2}^{-1}(\tau)|$ is
at most $n$, since we only need to know the stopping locations of the flipping
operations, and there are at most $n$ possibilities of the stopping location.
For any given mapped $\nu$, the number of preimages $|g_{3}^{-1}(\tau)|$ is at
most $n^{2}$, since we need the stopping location information of
$g_{2}(\cdot),$ which is upper bounded by $n$ and the stopping location of
$g_{3}(\cdot)$ is also upper bounded by $n.$

To bound $|g^{-1}(\nu)|$ for a given $\nu$ and a given $z_{\rm c}=x_{\rm
c}+y_{\rm c}$, we can first bound the number of possible bridge systems for a
given $z_{\rm c}$, which yields $(z_{\rm
c}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm c}}$. See proof
details of this bound from Lemma 7.6 in [6]. For any configuration $\psi\in
f_{2}(\sigma)$ with a given bridge system, there is only one configuration
$\sigma$ that can be mapped to $\psi$ (Lemma 19). Thus combining with
$|g_{2}^{-1}(\tau)|$ and $|g_{3}^{-1}(\nu)|$, it yields $|g^{-1}(\nu)|\leq
n^{3}(z_{\rm c}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm c}}.$ ∎

Finally, substituting the bounds into Peierls Argument (5) yields

$\displaystyle\pi(\Omega_{\text{urb}\wedge\neg\text{seg}})$
$\displaystyle\leq\sum_{\nu\in\Omega}\pi(\nu)\sum_{z_{\rm
c}=\beta\sqrt{n}}^{3n}n^{3}(z_{\rm
c}+1)3^{\alpha\sqrt{n}}(\frac{4^{\frac{3\delta+1}{4\delta}}}{\lambda})^{z_{\rm
c}}$ $\displaystyle\leq\sum_{\nu\in\Omega}\pi(\nu)\sum_{z_{\rm
c}=\beta\sqrt{n}}^{3n}n^{3}(z_{\rm
c}+1)(\frac{3^{\frac{\alpha}{\beta}}4^{\frac{3\delta+1}{4\delta}}}{\lambda})^{z_{\rm
c}},$ (9)

where $z_{\rm c}\geq\beta\sqrt{n}$ is due to
$\sigma\in\Omega_{\text{urb}\wedge\neg\text{seg}}$ does not satisfy
$(\beta,\epsilon)-$segregation (see Lemma 7.4 in [6] for details), and $z_{\rm
c}\leq\beta\sqrt{n}$, which is the sum of all edges of $G_{\triangle}$. If
$\lambda>3^{\frac{\alpha}{\beta}}4^{\frac{3\delta+1}{4\delta}}$, the sum will
be exponentially small for sufficiently large $n$, which means
$\pi(\Omega_{\text{urb}\wedge\neg\text{seg}})\leq\xi_{0}^{\sqrt{n}}$ for some
$\xi_{0}\in(0,1).$ ∎

##  Appendix E : Proof of Theorem 9 (Urbanization and Integration)

###### Proof of Theorem 9.

First, we define $\Omega_{\rm{urb}\wedge\neg\text{seg}}\subset\Omega$ to be
the configurations that have $\epsilon-$urbanization of the poor and
$(\beta,\delta)-$integration outside the urban area. To prove Theorem 9, it
suffices to prove with all but exponentially small probability, a sample drawn
from (2) is not in $\Omega_{\rm{urb}\wedge\neg\text{seg}}$:
$\pi(\Omega\setminus\Omega_{\rm{urb}\wedge\neg\text{seg}})\leq\xi_{3}^{\sqrt{n}},$
where $\xi_{3}\in(0,1),$ and $n$ is sufficiently large.

We can further divide the configuration space
$\Omega\setminus\Omega_{\rm{urb}\wedge\neg\text{seg}}$ into two parts: the set
of configurations $\Omega_{\neg\text{urb}}$ that do not have
$\epsilon-$urbanization of poverty, and the set of configurations
$\Omega_{\text{urb}\wedge\text{seg}}$ that have $\epsilon-$urbanization of
poverty and $(\beta,\delta)-$segregation. Since
$\Omega\setminus\Omega_{\rm{urb}\wedge\neg\text{seg}}=\Omega_{\neg\text{urb}}+\Omega_{\text{urb}\wedge\text{seg}}$,
to prove
$\pi(\Omega\setminus\Omega_{\rm{urb}\wedge\neg\text{seg}})\leq\xi_{3}^{\sqrt{n}},$
it suffices to prove $\pi(\Omega_{\neg\text{urb}})\leq\xi_{1}^{n}$ and
$\pi(\Omega_{\text{urb}\wedge\text{seg}})\leq\xi_{0}^{\sqrt{n}}$, for some
constant $0<\xi_{1},\xi_{0}<1$, and sufficiently large $n$.

It follows from Theorem 3 that for
$\gamma^{\delta/3}>16^{\frac{3\delta+1}{4\delta}}$ and $\lambda>1$,
$\pi(\Omega_{\neg\text{urb}})\leq\xi_{1}^{n}$ for some $\xi_{1}\in(0,1).$ To
prove the second part
$\pi(\Omega_{\text{urb}\wedge\text{seg}})\leq\xi_{0}^{\sqrt{n}}$, for each
$\sigma\in\Omega_{\text{urb}\wedge\text{seg}},$ we construct a $\delta-$color
bridge system. Then we define the mapping $s=(s_{2}\circ f_{2})(\cdot)$: we do
the color inversion and obtain $\tau=f_{2}(\sigma);$ next for $\tau$, we
randomly flip the cyan to pink until the right number of the pink, and we
randomly flip the blue to red until the right number of the red and obtain
$\nu=s_{2}(\tau).$

######  Claim 34.

For any $\sigma\in\Omega_{\text{urb}\wedge\text{seg}}$ with bridged color
contour length $z_{\rm c}$, for the defined mapping $\nu=s(\sigma),$
$h(\nu)-h(\sigma)\leq-z_{\rm c}+3n$ and $p(\sigma)-p(\nu)\leq 0.$

###### Proof of 34.

Because the richness information remains unchanged for $f_{1}$ and $s_{2}$,
$p(\sigma)-h(\nu)\leq 0$. It follows from Lemma 19 that for a given $z_{\rm
c}$ and any $\tau=f_{2}(\sigma)$, $h(\tau)-h(\sigma)\leq-z_{\rm c}$. The
maximal number of racially heterogeneous edges created by $\nu=s_{2}(\tau)$
can be bounded by $3n$, which is sum of all the edges in $G_{\triangle}$.
Hence $h(\nu)-h(\sigma)=h(\nu)-h(\tau)+h(\tau)-h(\sigma)\leq 3n-z_{\rm c}.$ ∎

######  Claim 35.

For a given color contour length $z_{\rm c}$, for any $\nu=s(\sigma)$, the
number of configurations in $\Omega_{\text{urb}\wedge\text{seg}}$ that can map
to $\nu$ is upper bounded by: $|s^{-1}(\nu)|\leq(z_{\rm
c}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm c}}2^{rn}.$

###### Proof of Claim 35.

We denote $\tau=f_{2}(\sigma)$ and $\nu=s_{2}(\tau)$. Since
$s(\sigma)=(s_{2}{\circ}f_{2})(\sigma)$, for any $\nu,$ it follows that
$|s^{-1}(\nu)|\leq|f_{2}^{-1}(\tau)|\cdot|s_{2}^{-1}(\nu)|$. For any
$\nu\in\Omega,$ the number of preimages $|s_{2}^{-1}(\nu)|$ can be upper
bounded by $2^{rn}$ by recording whether each red agent is flipped or not.

It follows from Lemma 16 that for any given $z_{\rm c}$, the number of
$\delta-$color bridge systems can be upper bounded by $(z_{\rm
c}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm c}}$. It follows
from Lemma 19 that for any $\tau$, for a given $\delta-$color bridge system,
the number of preimages is one. Hence we conclude for any $\tau$ with a given
$z_{\rm c}$, the number of preimages is upper bounded by:
$|f_{2}^{-1}(\tau)|\leq(z_{\rm
c}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm c}}$. Combining the
two inequalities, we conclude $|s^{-1}(\nu)|\leq(z_{\rm
c}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm c}}2^{rn}$. ∎

######  Claim 36.

For any given $\sigma\in\Omega_{\text{urb}\wedge\text{seg}},$ we denote
$S(\sigma)$ to be the set of all possible images $\nu=s(\sigma)$ mapped from
$\sigma$. It follows that $|S(\sigma)|\geq(\frac{p-\delta}{r_{\rm
p}})^{(r_{\rm p}-\delta)n}(\frac{1-p-\delta}{r_{\rm r}})^{(r_{\rm
r}-\delta)n}.$

###### Proof of Claim 36.

It follows from the definition of $f_{2}$ that for any given $\sigma,$ there
is only one configuration can be obtained from $f_{2}(\sigma)$. For any given
$\tau$, we define $S_{2}(\tau)$ to be the set of all possible configurations
obtained from $s_{2}(\tau)$ by flipping the pink and red back; then
$|S_{2}(\tau)|=\begin{pmatrix}pn-a_{\tau}\\\ r_{\rm
p}n-a_{\tau}^{{}^{\prime}}\end{pmatrix}\cdot\begin{pmatrix}n-pn-a_{\tau}\\\
r_{\rm r}n-a_{\tau}^{{}^{\prime\prime}}\end{pmatrix}$, where $a_{\tau}$ is the
number of unbridged agents in $\tau$, $a_{\tau}^{{}^{\prime}}$ is the number
of unbridged pink, and $a_{\tau}^{{}^{\prime\prime}}$ is the number of
unbridged red. Thus $|S_{2}(\tau)|$ can be further lower bounded by

$\displaystyle|S(\tau)|\geq(\frac{p-a_{\tau}}{r_{\rm
p}-a_{\tau}^{{}^{\prime}}})^{(r_{\rm
p}-a_{\tau}^{{}^{\prime}})n}(\frac{1-p-a_{\tau}}{r_{\rm
r}-a_{\tau}^{{}^{\prime\prime}}})^{(r_{\rm
r}-a_{\tau}^{{}^{\prime\prime}})n}\geq(\frac{p-\delta}{r_{\rm p}})^{(r_{\rm
p}-\delta)n}(\frac{1-p-\delta}{r_{\rm r}})^{(r_{\rm r}-\delta)n}.$ (10)

Hence $|S(\sigma)|$ can be lower bounded by $|S(\sigma)|\geq
1\cdot|S_{2}(\tau)|\geq(\frac{p-\delta}{r_{\rm p}})^{(r_{\rm
p}-\delta)n}(\frac{1-p-\delta}{r_{\rm r}})^{(r_{\rm r}-\delta)n}.$ ∎

Finally, we define a weighted bipartite graph
$G(\Omega_{\text{urb}\wedge\text{seg}},\Omega,E)$ with an edge of weight
$\pi(\sigma)$ between $\sigma\in\Omega_{\text{urb}\wedge\text{seg}}$ and
$\nu\in\Omega$. The total weight of edges is

$\displaystyle\sum_{\sigma\in\Omega_{\text{urb}\wedge\text{seg}}}\pi(\sigma)\cdot|S(\sigma)|\geq\pi(\Omega_{\text{urb}\wedge\text{seg}})(\frac{p-\delta}{r_{\rm
p}})^{(r_{\rm p}-\delta)n}(\frac{1-p-\delta}{r_{\rm r}})^{(r_{\rm
r}-\delta)n}.$ (11)

On the other hand, the weight of the edges is at most

$\displaystyle\sum_{\nu\in\Omega}\sum_{\sigma\in
s^{-1}(\nu)}\max_{\sigma\in\Omega_{\text{urb}\wedge\text{seg}}}{\pi(\sigma)}=\sum_{\nu\in\Omega}\pi(\nu)\sum_{\sigma\in
s^{-1}(\nu)}\frac{\max_{\sigma\in\Omega_{\text{urb}\wedge\text{seg}}}{\pi(\sigma)}}{\pi(\nu)}|s^{-1}(\nu)|$
$\displaystyle\leq\sum_{\nu\in\Omega}\pi(\nu)\sum_{z_{\rm
c=\sqrt{rn}}}^{\beta\sqrt{n}}\lambda^{\max(h(\nu)-h(\sigma))}\gamma^{\max(p(\sigma)-p(\nu))}(z_{\rm
c}+1)3^{\alpha\sqrt{n}}4^{\frac{3\delta+1}{4\delta}z_{\rm c}}2^{rn}$
$\displaystyle\leq\sum_{z_{\rm c}=\beta_{\min}\sqrt{n}}^{\beta\sqrt{n}}(z_{\rm
c}+1)3^{\alpha\sqrt{n}}(\frac{4^{\frac{3\delta+1}{4\delta}}}{\lambda})^{z_{\rm
c}}\lambda^{3n}2^{rn}.$ (12)

where the inequalities in Claim 35 and 36 has been substituted in the above
derivation. Combining (11) and (E), we have

$\displaystyle\pi(\Omega_{\text{urb}\wedge\text{seg}})(\frac{p-\delta}{r_{\rm
p}})^{(r_{\rm p}-\delta)n}(\frac{1-p-\delta}{r_{\rm r}})^{(r_{\rm
r}-\delta)n}\leq\sum_{z_{\rm c}=\beta_{\min}\sqrt{n}}^{\beta\sqrt{n}}(z_{\rm
c}+1)3^{\alpha\sqrt{n}}(\frac{4^{\frac{3\delta+1}{4\delta}}}{\lambda})^{z_{\rm
c}}\lambda^{3n}2^{rn}.$ (13)

For large enough $n$, to have
$\pi(\Omega_{\text{urb}\wedge\text{seg}})\leq\xi_{3}^{n}$ for some
$\xi_{3}\in(0,1)$, it suffices to have

$\lambda^{3n}2^{rn}<(\frac{p-\delta}{r_{\rm p}})^{(r_{\rm
p}-\delta)n}(\frac{1-p-\delta}{r_{\rm r}})^{(r_{\rm
r}-\delta)n}<(\frac{p-\delta}{r_{\rm p}})^{r_{\rm
p}n}(\frac{1-p-\delta}{r_{\rm r}})^{r_{\rm r}n},$

which can be rewritten as

$\lambda^{3}<(\frac{p-\delta}{r_{\rm p}})^{r_{\rm p}}(\frac{1-p-\delta}{r_{\rm
r}})^{r_{\rm r}}/2^{r}.$

Since $\lambda>1,$ to make the right hand side of the above inequality greater
than one, it suffices to have $(\frac{p-\delta}{r_{\rm p}})^{r_{\rm
p}}(\frac{1-p-\delta}{r_{\rm r}})^{r_{\rm r}}>2^{r}.$ Combing the above
parameter choices with Theorem 3, which requires
$\gamma^{\delta/3}>16^{\frac{3\delta+1}{4\delta}}$ and $\lambda>1$, we
conclude Theorem 9. ∎

##  Appendix F : Proof of Theorem 11 (Integration for Distributed
$\mathcal{U}$)

###### Proof of Theorem 11.

First we define the configuration space $S_{\beta,\delta}$ to be the set of
configurations that are $(\beta,\delta)-$segregated. To prove Theorem 11, it
suffices to prove $\pi(S_{\beta,\delta})\leq\xi_{4}^{n}$, where
$\xi_{4}\in(0,1).$ The bridging and the mapping
$\nu=f(\sigma)=(f_{5}{\circ}f_{4}{\circ}f_{3}{\circ}f_{3}{\circ}f_{1})(\sigma)$
are defined as the following: we first construct a $\delta-$color-and-richness
bridge system for $\sigma\in S_{\beta,\delta}$ (see Appendix A.2 for details).
Then we do richness inversion and color inversion like defined in
$f_{1}(\cdot)$ and $f_{2}(\cdot)$. Then we do the color and richness recovery
$\nu=(f_{5}{\circ}f_{4}{\circ}f_{3}{\circ})(\tau)$ in the distributed way as
specified in section A.5. The following claim can be proved after the mapping.

######  Claim 37.

For any $\sigma\in S_{\beta,\delta}$ with a given bridged color contour length
$z_{\rm c}$, for the defined mapping $\nu=f(\sigma),$ $h(\nu)-h(\sigma)\leq
2cn-z_{\rm c}$ and $p(\sigma)-p(\nu)\leq b_{\rm p}n-\hat{b}_{\rm p}n$, where
$\hat{b}_{\rm p}\triangleq\min\\{c,p\\}-(r+\delta)c-2\delta.$

###### Proof of Claim 37.

For any configuration $\sigma\in S_{\beta,\delta}$, the poor agents on the
urban sites are either in the red region $R$ or outside $R$. It follows from
the definition of $(\beta,\delta)-$segregation that the size of $R$ is at most
$(r+\delta)n$. Since the urban sites are evenly distributed with the total
size $c\cdot n$, the maximal number of the urban sites in $R$ is
$(r+\delta)cn.$ Hence the maximal number of the poor agents on the urban sites
in $R$ is $(r+\delta)cn.$ Outside $R$, the poor agents on the urban sites
could be unbridged agents whose number is upper bounded by $\delta\cdot n$, or
the cyan agents whose number is $b_{\rm p}n$. Hence the total number of the
poor on the urban sites for any $\sigma$ follows that
$p(\sigma)\leq(r+\delta)cn+\delta n+b_{\rm p}n.$

After the richness and color inversions, it follows from Lemma 18 and 19 that
for any $\tau=(f_{2}\circ f_{1})(\sigma),$ $h(\tau)-h(\sigma)\leq-z_{\rm c}$
and the number of the poor in $\tau$ is less than $\delta n$ and they are
unbridged. After the color and richness recovery, it follows from Lemma 25
that $p(\nu)\geq\min\\{c,p\\}n-\delta n$. Hence $p(\sigma)-p(\nu)\leq b_{\rm
p}n-\hat{b}_{\rm p}n$, where $\hat{b}_{\rm
p}=\min\\{c,p\\}-(r+\delta)c-2\delta$.

It also follows from Lemma 24 that for any
$\nu=(f_{5}{\circ}f_{4}{\circ}f_{3}{\circ})(\tau)$, $h(\nu)-h(\tau)\leq 2cn.$
Combining with $h(\tau)-h(\sigma)\leq-z_{\rm c}$, we get $h(\nu)-h(\sigma)\leq
2cn-z_{\rm c}.$ ∎

For a given color contour length $z_{\rm c},$ for any $\nu=f(\sigma),$ the
number of preimages follows from Claim 27. Similarly, we use Peierls Argument
(5), substituting the related bounds into which yields

$\displaystyle\pi(S_{\beta,\delta})$
$\displaystyle\leq\sum_{\nu\in\Omega}\pi(\nu)\sum_{z_{\rm c}=\sqrt{r\cdot
n}}^{\beta\sqrt{n}}n^{6}(3n+1)(z_{\rm
c}+1)9^{\alpha\sqrt{n}}(\frac{4^{\frac{3\delta+1}{4\delta}}}{\lambda})^{z_{\rm
c}}(\frac{\lambda^{2c}64^{\frac{3\delta+1}{4\delta}}}{\gamma^{\hat{b}_{\rm
p}-b_{\rm p}}})^{n},$ (14)

where $z_{\rm c}\geq\sqrt{r\cdot n}$ is due to the triangular lattice
geometry, which is proved in Lemma 2.1 in [7], and $z_{\rm
c}\leq\beta\sqrt{n}$ is due to $\sigma\in S_{\beta,\delta}$ and the definition
of $(\beta,\delta)-$segregation. If $b_{\rm p}<\hat{b}_{\rm p}$ and
$\gamma^{\hat{b}_{\rm p}-b_{\rm
p}}>\lambda^{2c}64^{\frac{3\delta+1}{4\delta}}$, the sum will be exponentially
small given large enough $n$, which means
$\pi(S_{\beta,\delta})\leq\xi_{4}^{n}$ for some $\xi_{4}\in(0,1).$ ∎

######  Remark 38.

If the number of the poor blue satisfies $b_{\rm p}<\hat{b}_{\rm
p}\triangleq\min\\{c,p\\}-(r+\delta)c-2\delta,$ we can conclude the ratio
between the poor blue and the poor red is smaller than the ratio between the
blue and the red: $\frac{b_{\rm p}}{r_{\rm p}}<\frac{b}{r},$ which is
understood as income inequality.

###### Proof.

If $c\leq p$: it follows that $b_{\rm
p}<c-(r+\delta)c-2\delta=(b-\delta)c-2\delta<(b-\delta)c<(b-\delta)p<b\cdot
p$, which can be written as $\frac{b_{\rm p}}{r_{\rm p}}<\frac{b\cdot
p}{r_{\rm p}}=b+b\cdot\frac{b_{\rm p}}{r_{\rm p}}$. Hence we can get
$\frac{b_{\rm p}}{r_{\rm p}}<\frac{b}{r}.$ If $p\leq c$, it follows that
$b_{\rm p}<p-(r+\delta)c-2\delta<p-(r+\delta)p-2\delta<p(b-\delta)<b\cdot p$.
Hence the same conclusion $\frac{b_{\rm p}}{r_{\rm p}}<\frac{b}{r}$ follows. ∎