Allan Sly, Lingfu Zhang
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 $\mathbb{Z}^d$.
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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…
This paper is concerned with factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We study the mutual information of the values on two given vertices. If the vertices are neighbors (i.e., their distance is $1$), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$, of order $(d-1)^{-k/2}$. Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process the rate …
We study factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most $k(d-1) / (\sqrt{d-1})^k$, where $k$ denotes the distance of the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.
The noisy voter model with general initial conditions
本文研究一般图上的噪声选民模型，证明其截止时间由初始状态的自相关函数显式决定，并据此证实交替初始条件在1D Ising模型中对所有噪声值渐近最快。
Factor-of-iid Schreier decorations of lattices in Euclidean spaces
研究各种传递图上是否存在作为 iid 的 factor 的 Schreier 装饰，证明部分格具有此类装饰而某些图则不可能，并探讨对称平面格的 factor of iid 平衡定向及其在拟等距下的不变性。
Recently, quantitative versions of the Gibbard-Satterthwaite theorem were proven for $k=3$ alternatives by Friedgut, Kalai, Keller and Nisan and for neutral functions on $k \geq 4$ alternatives by Isaksson, Kindler and Mossel. We prove a quantitative version of the Gibbard-Satterthwaite theorem for general social choice functions for any number $k \geq 3$ of alternatives. In particular we show that for a social choice function $f$ on $k \geq 3$ alternatives and $n$ voters, which is $ε$-far from the family of nonmanipulable functions, a uniformly chosen voter profile is manipulable with probabi…
We want to find the convex combination $S$ of iid Bernoulli random variables that maximizes $\textbf{P}(S\geq t)$ for a given threshold~$t$. Csóka conjectured that such an $S$ is an average if $t\geq p$, where $p$ is the success probability of the Bernoulli random variables. We prove this conjecture for a range of $p$ and $t$.
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 …
In this paper we study intermittency for the parabolic Anderson equation $\partial u/\partial t=κΔu+γξu$ with $u:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$, where $κ\in[0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, $γ\in(0,\infty)$ is the coupling constant, and $ξ:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$ is a space--time random medium. The solution of this equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $ξ$. We focus on the case where $ξ$ is the voter model with opinions 0 and 1 that are updated according to a random walk tr…
Independence of the indicator functions of record values for Multivariate independent data
本文证明了在多维独立数据中，记录值指示函数的独立性，且提供了不同于一维中现有证明的详细证明方法。
Given integers $d \geq 2, n \geq 1$, we consider affine random walks on torii $(\mathbb{Z} / n \mathbb{Z})^{d}$ defined as $X_{t+1} = A X_{t} + B_{t} \mod n$, where $A \in \mathrm{GL}_{d}(\mathbb{Z})$ is an invertible matrix with integer entries and $(B_{t})_{t \geq 0}$ is a sequence of iid random increments on $\mathbb{Z}^{d}$. We show that when $A$ has no eigenvalues of modulus $1$, this random walk mixes in $O(\log n \log \log n)$ steps as $n \rightarrow \infty$, and mixes actually in $O(\log n)$ steps only for almost all $n$. These results generalize those on the so-called Chung-Diaconis-G…
We show that a slight modification of a theorem of Ruzmaikina and Aizenman on competing particle systems on the real line leads to a characterization of Poisson-Dirichlet distributions $PD(a,0)$. Precisely, let $s$ be a proper random mass-partition i.e. a random sequence $(s_i, i\in\N)$ such that $s_1\geq s_2\geq ...$ and $\sum_i s_i=1$ a.s. Consider ${W_i}_{i\in\N}$, an iid sequence of positive random variables with a density and such that $E[W^λ]$ is finite for all $λ\in\R$. It is shown that if the law of $s$ is invariant under a random multiplicative shift $s_i W_i$ of the atoms followed by…
Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space $\mathbb{R}^d$, $d\geq 3$, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in $\mathbb{R}^d$ as an isometry-invariant random partition of $\mathbb{R}^d$ to bounded polyhedra, and also as an isometry-invariant random partition of $\mathbb{R}^d$ to indistinguishable pieces. A new technique is develop…
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