In a recent paper by two of the authors, the concepts of upwards and downwards $ε$-movability were introduced, mainly as a technical tool for studying dynamical percolation of interacting particle systems. In this paper, we further explore these concepts which can be seen as refinements or quantifications of stochastic domination, and we relate them to previously studied concepts such as uniform insertion tolerance and extractability.
We prove the stochastic domination for determinantal processes associated with finite rank projection kernels. The result was first proved by Lyons in discrete setting. We avoid the machinery of matroids in order to obtain a proof that works in a general setting. We prove another result on the stochastic domination of two determinantal processes where the kernels are represented with respect to different measures. Combining this result with Lyons' theorem on the Stochastic domination we obtain a result on the stochastic domination for the last passage time in a directed last passage percolatio…
A competition model on $\mathbb{Z}_+^{2}$ governed by directed last passage percolation is considered. A stochastic domination argument between subtrees of the last passage percolation is put forward.
There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z^{d-1} joined by a perpendicular copy) into the open set of site percolation on Z^d, whenever the parameter p is close enough to 1 or the Lipschitz constant is sufficiently large. This is proved using several new results and techniques involving stochastic domination, in contexts that include a process of independent overlapping intervals on Z, and first-passage percolation on general graphs.
There is a mathematical error in the first version of this paper. A new corrected version will be posted when the error is fixed, possibly with a modified title.
Local criteria for global connectivity comparisons: beyond stochastic domination
University of California, Los Angeles（加州大学洛杉矶分校）；Technische Universität Braunschweig（布伦瑞克工业大学）；Weierstrass Institute for Applied Analysis and Stochastics, Berlin（柏林魏尔斯特拉斯应用分析与随机性研究所）；University of Sheffield（谢菲尔德大学）；California Institute of Technology（加州理工学院）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究网络阿尔弗雷德·雷尼数学研究所）
Stochastic dominations for FK percolation and sharp thinning thresholds for the Ising energy field
Université Claude Bernard Lyon 1（里昂第一大学）；CNRS UMR 5208, Institut Camille Jordan（法国国家科学研究中心联合研究实验室5208，卡米尔·若当研究所）；Universidad de Chile（智利大学）；Centro de Modelamiento Matemático (AFB170001)（数学建模中心）；UMI-CNRS 2807（中法科研联合单位2807）
Consider a sequence of positive integers $\{k_n,n\ge1\}$, and an array of nonnegative real numbers $\{a_{n,i},1\le i\le k_n,n\ge1\}$ satisfying $\sup_{n\ge 1}\sum_{i=1}^{k_n}a_{n,i}=C_0\in (0,\infty).$ This paper introduces the concept of $\{a_{n,i}\}$-stochastic domination. We develop some techniques concerning this concept and apply them to remove an assumption in a strong law of large numbers of Chandra and Ghosal [Acta. Math. Hungarica, 1996]. As a by-product, a considerable extension of a recent result of Boukhari [J. Theoret. Probab., 2021] is established and proved by a different method…
In this note we establish several inequalities and monotonicity properties for the free energy of directed polymers under certain stochastic orders: the usual stochastic order, the Laplace transform order and the convex order. For the latter our results cover also many classical disordered systems.
Let G=(V,E) be a locally finite graph. Let \vec{p}\in[0,1]^V. We show that Shearer's measure, introduced in the context of the Lovasz Local Lemma, with marginal distribution determined by \vec{p} exists on G iff every Bernoulli random field with the same marginals and dependency graph G dominates stochastically a non-trivial Bernoulli product field. Additionaly we derive a lower non-trivial uniform bound for the parameter vector of the dominated Bernoulli product field. This generalizes previous results by Liggett, Schonmann & Stacey in the homogeneous case, in particular on the k-fuzz of Z. U…
We prove for the contact process on $Z^d$, and many other graphs, that the upper invariant measure dominates a homogeneous product measure with large density if the infection rate $λ$ is sufficiently large. As a consequence, this measure percolates if the corresponding product measure percolates. We raise the question of whether domination holds in the symmetric case for all infinite graphs of bounded degree. We study some asymmetric examples which we feel shed some light on this question. We next obtain necessary and sufficient conditions for domination of a product measure for ``downward'' F…
We study how iterated convolutions of probability measures compare under stochastic domination. We give necessary and sufficient conditions for the existence of an integer $n$ such that $μ^{*n}$ is stochastically dominated by $ν^{*n}$ for two given probability measures $μ$ and $ν$. As a consequence we obtain a similar theorem on the majorization order for vectors in $\R^d$. In particular we prove results about catalysis in quantum information theory.
We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. We begin by considering the Ising model on the homogeneous tree of degree $d$, $\Td$. For given interaction parameters $J_1$, $J_2>0$ and external field $h_1\in\RR$, we compute the smallest external field $\tilde{h}$ such that the plus measure with parameters $J_2$ and $h$ dominates the plus measure with parameters $J_1$ and $h_1$ for all $h\geq\tilde{h}$. Moreover, we discuss continuity of $\tilde{h}$ with respect to the three parameters $J_1$, $J_2$, $h$ and also how the plus measures a…
The ordinary contact process is used to model the spread of a disease in a population. In this model, each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy. In this paper, we introduce and study the contact process in a randomly evolving environment. Here we associate to every individual an independent two-state, $\{0,1\},$ background process. Given $δ_0<δ_1,$ if the background process is in state $0,$ the individual (if infected) becomes healthy at rate $δ_0,$ while if the background process is in state $1,$ it becomes healthy at rate $δ_1.$…
Liggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d.\ Bernoulli product measure. In particular, they proved this for $\mathbb{Z}^d$ and (for infection rate sufficiently large) $d$-ary homogeneous trees $T_d$. In this paper we prove some space-time versions of their results. We do this by combining their methods with specific properties of the contact process and general correlation inequalities. One of our main results concerns the contact process on $T_d$ with $d\geq2$. We show that, for l…
For n>=1 let X_n be a vector of n independent Bernoulli random variables. We assume that X_n consists of M "blocks" such that the Bernoulli random variables in block i have success probability p_i. Here M does not depend on n and the size of each block is essentially linear in n. Let X'_n be a random vector having the conditional distribution of X_n, conditioned on the total number of successes being at least k_n, where k_n is also essentially linear in n. Define Y'_n similarly, but with success probabilities q_i>=p_i. We prove that the law of X'_n converges weakly to a distribution that we ca…
This note develops Rio's proof [C. R. Math. Acad. Sci. Paris, 1995] of the rate of convergence in the Marcinkiewicz--Zygmund strong law of large numbers to the case of sums of dependent random variables with regularly varying normalizing constants. It allows us to obtain a complete convergence result for dependent sequences under uniformly bounded moment conditions. This result is new even when the underlying random variables are independent. The main theorems are applied to three different dependence structures: (i) $m$-pairwise negatively dependent random variables, (ii) $m$-extended negativ…
We prove an exponential concentration bound for cover times of general graphs in terms of the Gaussian free field, extending the work of Ding-Lee-Peres and Ding. The estimate is asymptotically sharp as the ratio of hitting time to cover time goes to zero. The bounds are obtained by showing a stochastic domination in the generalized second Ray-Knight theorem, which was shown to imply exponential concentration of cover times by Ding. This stochastic domination result appeared earlier in a preprint of Lupu, but the connection to cover times was not mentioned.
We study Pippenger's model of Boolean networks with unreliable gates. In this model, the conditional probability that a particular gate fails, given the failure status of any subset of gates preceding it in the network, is bounded from above by some $ε$. We show that if we pick a Boolean network with $n$ gates at random according to the Barak-Erdős model of a random acyclic digraph, such that the expected edge density is $c n^{-1}\log n$, and if $ε$ is equal to a certain function of the size of the largest reflexive, transitive closure of a vertex (with respect to a particular realization of t…
In this note we discuss vacant set level set percolation on a transient weighted graph. It interpolates between the percolation of the vacant set of random interlacements and the level set percolation of the Gaussian free field. We employ coupling and derive a stochastic domination from which we deduce in a rather general set-up a certain monotonicity property of the percolation function. In the case of regular trees this stochastic domination leads to a strict inequality between some eigenvalues related to Ornstein-Uhlenbeck semi-groups for which we have no direct analytical proof. It underpi…
