We consider the discrete-time voter model on complete bipartite graphs and study the quasi-stationary distribution (QSD) for the model as the size of one of the partitions tends to infinity while the other partition remains fixed. We show that the QSDs converge weakly to a nontrivial limit which features a consensus with the exception of a random number of dissenting vertices in the "large" partition. Moreover, we explicitly calculate the law of the number of dissenters and show that it follows the heavy-tailed Sibuya distribution with parameter depending on the size of the "small" partition. Our results rely on a discrete-time analogue of the well-known duality between the continuous-time voter model and coalescing random walks which we develop in the paper.
The voter model on $\mathbb{Z}^d$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When $d \geq 3$, the set of (extremal) stationary distributions is a family of measures $μ_α$, for $α$ between 0 and 1. A configuration sampled from $μ_α$ is a strongly correlated field of 0's and 1's on $\mathbb{Z}^d$ in which the density of 1's is $α$. We consider such a configuration as a site percolation model on $\mathbb{Z}^d$. We prove that if $d \geq 5$, the probability of existence of a…
For the Voter Model on $\mathbb{Z}^d$, $d\geq 3$, we show that the (extremal) stationary distributions are isomorphic to Bernoulli shifts, and answer an open question asked by Steif and Tykesson. The proof gives explicit constructions of the stationary distributions as factors of IID processes on $\mathbb{Z}^d$.
Percolation on the stationary distributions of the voter model with stirring
Bernoulli Institute, University of Groningen（格罗宁根大学伯努利研究所）；Groningen Cognitive System and Materials Center, University of Groningen（格罗宁根大学认知系统与材料中心）；Institute of Mathematics, Statistics and Computer Science, University of São Paulo, Brazil（圣保罗大学数学、统计与计算机科学研究所）；CNRS and Sorbonne Université, Paris, France（法国国家科学研究中心与索邦大学）；Department of Statistics, University of Warwick（华威大学统计学系）
We consider two independent stationary random walks on large random regular graphs of degree $k\geq 3$ with $N$ vertices. On these graphs, the exponential approximations of the meeting times are known to follow from existing methods and form a basis for the voter model's diffusion approximations. The main result of this note improves the exponential approximations to an explicit form such that the first moments are asymptotically equivalent to $N(k-1)/[2(k-2)]$.
In the latent voter model, which models the spread of a technology through a social network, individuals who have just changed their choice have a latent period, which is exponential with rate $λ$, during which they will not buy a new device. We study site and edge versions of this model on random graphs generated by a configuration model in which the degrees $d(x)$ have $3 \le d(x) \le M$. We show that if the number of vertices $n \to\infty$ and $\log n \ll λ_n \ll n$ then the latent voter model has a quasi-stationary state in which each opinion has probability $\approx 1/2$ and persists in t…
The Invasion Model on the complete bibartitle graph was introduced and studied by physicists as a rudimentary model for opinion dynamics on complex networks. We identify the limit of the Quasistationary distribution for the model as one partition size tends to infinity. The limit is a highly dispersed measure. A distinctive feature of the model is that of two time scales with non-trivial interaction. The work and the results complement and are in sharp contrast to the analogous results on the closely related Voter Model.
An exact spectral analysis of the Markov Propagator for the Voter model is presented for the complete graph, and extended to the complete bipartite graph and uncorrelated random networks. Using a well-defined Martingale approximation in diffusion-dominated regions of phase space, which is almost everywhere for the Voter model, this method is applied to compute analytically several key quantities such as exact expressions for the $m$ time step propagator of the Voter model, all moments of consensus times, and the local times for each macrostate. This spectral method is motivated by a related me…
We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \ge 3$. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are clo…
We propose an explicit construction of the stationary state of Extended Bipartite Matching (EBM) models, as defined in (Busic et. al., 2013). We use a Loynes-type backwards scheme similar in flavor to that in (Moyal et al., 2017), allowing to show the existence and uniqueness of a bi-infinite perfect matching under various conditions, for a large class of matching policies and of bipartite matching structures. The key algebraic element of our construction is the sub-additivity of a suitable stochastic recursive representation of the model, satisfied under most usual matching policies. By doing…
In the $R$-spread out, $d$-dimensional voter model, each site $x$ of $\mathbb{Z}^d$ has state (or 'opinion') 0 or 1 and, with rate 1, updates its opinion by copying that of some site $y$ chosen uniformly at random among all sites within distance $R$ from $x$. If $d \geq 3$, the set of (extremal) stationary measures of this model is given by a family $μ_{α, R}$, where $α\in [0,1]$. Configurations sampled from this measure are polynomially correlated fields of 0's and 1's in which the density of 1's is $α$ and the correlation weakens as $R$ becomes larger. We study these configurations from the …
Given a continuous time Markov Chain $\{q(x,y)\}$ on a finite set $S$, the associated noisy voter model is the continuous time Markov chain on $\{0,1\}^S$, which evolves in the following way: (1) for each two sites $x$ and $y$ in $S$, the state at site $x$ changes to the value of the state at site $y$ at rate $q(x,y)$; (2) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates $\{q(x,y)\}$ and the corresponding stationary distributions are almost uniform, then the mixing time has a sharp cutoff at time $\log|S|/2$ with a window of order 1. Lubetzky an…
We consider a discrete-time voter model process on a set of nodes, each being in one of two states, either 0 or 1. In each time step, each node adopts the state of a randomly sampled neighbor according to sampling probabilities, referred to as node interaction parameters. We study the maximum likelihood estimation of the node interaction parameters from observed node states for a given number of realizations of the voter model process. In contrast to previous work on parameter estimation of network autoregressive processes, whose long-run behavior is according to a stationary stochastic proces…
We consider a stochastic bipartite matching model consisting of multi-class customers and multi-class servers. Compatibility constraints between the customer and server classes are described by a bipartite graph. Each time slot, exactly one customer and one server arrive. The incoming customer (resp. server) is matched with the earliest arrived server (resp. customer) with a class that is compatible with its own class, if there is any, in which case the matched customer-server couple immediately leaves the system; otherwise, the incoming customer (resp. server) waits in the system until it is …
Consider the switch chain on the set of $d$-regular bipartite graphs on $n$ vertices with $3\leq d\leq n^{c}$, for a small universal constant $c>0$. We prove that the chain satisfies a Poincaré inequality with a constant of order $O(nd)$; moreover, when $d$ is fixed, we establish a log-Sobolev inequality for the chain with a constant of order $O_d(n\log n)$. We show that both results are optimal. The Poincaré inequality implies that in the regime $3\leq d\leq n^c$ the mixing time of the switch chain is at most $O\big((nd)^2 \log(nd)\big)$, improving on the previously known bound $O\big((nd)^{1…
In this paper, we initiate the study of "Generalized Divide and Color Models". A very special interesting case of this is the "Divide and Color Model" (which motivates the name we use) introduced and studied by Olle Häggström. In this generalized model, one starts with a finite or countable set $V$, a random partition of $V$ and a parameter $p\in [0,1]$. The corresponding Generalized Divide and Color Model is the $\{0,1\}$-valued process indexed by $V$ obtained by independently, for each partition element in the random partition chosen, with probability $p$, assigning all the elements of the p…
We continue our study of the parabolic Anderson equation $\partial u/\partial t = κΔu + γξu$ for the space-time field $u\colon\,\Z^d\times [0,\infty)\to\R$, where $κ\in [0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, $γ\in (0,\infty)$ is the coupling constant, and $ξ\colon\,\Z^d\times [0,\infty)\to\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $ξ$, both living on $\Z^d$. In earlier work we considered three choices for $ξ$: independent simple rand…
In this paper, we present a result similar to the shift-coupling result of Thorisson (1996) in the context of random graphs and networks. The result is that a given random rooted network can be obtained by changing the root of another given one if and only if the distributions of the two agree on the invariant sigma-field. Several applications of the result are presented for the case of unimodular networks. In particular, it is shown that the distribution of a unimodular network is uniquely determined by its restriction to the invariant sigma-filed. Also, the theorem is applied to the existenc…
If we consider the contact process with infection rate $λ$ on a random graph on $n$ vertices with power law degree distributions, mean field calculations suggest that the critical value $λ_c$ of the infection rate is positive if the power $α>3$. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399--1404]. Here, we show that the critical value $λ_c$ is zero for any value of $α>3$, and the contact process starting from all vertices infected, with a probability…
We consider local Markov chain Monte-Carlo algorithms for sampling from the weighted distribution of independent sets with activity $ł$, where the weight of an independent set $I$ is $ł^{|I|}$. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree $d$ and $ł<ł_c(d)$, where $ł_c(d)$ is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the $d$-regular infinite tree. We show that for $d \geq 3$, $ł$ just above $ł_c(…
