Alain-Sol Sznitman
We consider continuous-time random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph. This identity is closely linked to the generalized second Ray-Knight theorem, and uniquely determines the law of occupation times of random interlacements at level u.
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We define renormalized intersection local times for random interlacements of Lévy processes in R^{d} and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials.
We consider continuous time interlacements on Z^d, with d bigger or equal to 3, and investigate the scaling limit of their occupation times. In a suitable regime, referred to as the constant intensity regime, this brings Brownian interlacements on R^d into play, whereas in the high intensity regime the Gaussian free field shows up instead. We also investigate the scaling limit of the isomorphism theorem of arXiv:1111.4818. As a by-product, when d=3, we obtain an isomorphism theorem for Brownian interlacements.
We prove a shape theorem for the internal (graph) distance on the interlacement set $\mathcal{I}^u$ of the random interlacement model on $\mathbb Z^d$, $d\ge 3$. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.
We base ourselves on the construction of the two-dimensional random interlacements [12] to define the one-dimensional version of the process. For this constructions we consider simple random walks conditioned on never hitting the origin, which makes them transient. We also compare this process to the conditional random walk on the ring graph. Our results are the convergence of the vacant set on the ring graph to the vacant set of one-dimensional random interlacements, a central limit theorem for the interlacements' local time for sites far from the origin and the convergence in law of the loca…
It was shown by Le Jan that the occupation field of a Poisson ensemble of Markov loops ("loop soup") of parameter one-half associated to a transient symmetric Markov jump process on a network is half the square of the Gaussian free field on this network. We construct a coupling between these loops and the free field such that an additional constraint holds: the sign of the free field is constant on each cluster of loops. As a consequence of our coupling we deduce that the loop clusters of parameter one-half do not percolate on periodic lattices. We also construct a coupling between the random …
In this paper, we study geometric properties of the unique infinite cluster $Γ$ in a sufficiently supercritical Finitary Random Interlacements $\mathcal{FI}^{u,T}$ in $\mathbb{Z}^d, \ d\ge 3$. We prove that the chemical distance in $Γ$ is, with stretched exponentially high probability, of the same order as the Euclidean distance in $\mathbb{Z}^d$. This also implies a shape theorem parallel to those for Bernoulli percolation and random interlacements. We also prove local uniqueness of $\mathcal{FI}^{u,T}$, which says any two large clusters in $\mathcal{FI}^{u,T}$ "close to each other" will with…
In this paper we establish some properties of percolation for the vacant set of random interlacements, for d at least 5 and small intensity u. The model of random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560. It is known that, for small u, almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u, see arXiv:0808.3344 and arXiv:0805.4106. We estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin, given that it is finite. This comes as a by-…
We consider continuous-time random interlacements on Z^d, d greater or equal to 3, and investigate the percolation model where a site x of Z^d is occupied if the total amount of time spent at x by all the trajectories of the interlacement at level u > 0 exceeds some given non-negative parameter, and empty otherwise. Thus, the set of occupied sites forms a subset of the interlacement at level u. We also investigate percolation properties of empty sites. A recent isomorphism theorem arXiv:1111.4818 of Sznitman enables us to "translate" some of the relevant questions into the language of level-se…
We define two families of Poissonian soups of bidirectional trajectories on $\mathbb{Z}^2$, which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus $(\mathbb{Z}/N \mathbb{Z})^2$, started from the uniform distribution, run up to a time of order $(N\log N)^2$ and forced to avoid a fixed point. The local limit of the latter was recently established in arXiv:1502.03470. Our construction proceeds by considering, somewhat in the spirit of statistical mechanics, a sequence of finite-volume approximations, consisting of random walks av…
Percolation for the Gaussian free field on the cable system: counterexamples
该研究针对一般瞬态图上高斯自由场渗流的临界参数条件，构造反例证明现有两个弱条件非必要，揭示有质量图的典型临界值特征并建立相关场与随机交错的同构关系。
We investigate level sets of the Gaussian free field on continuous transient metric graphs $\tilde{\mathcal G}$ and study the capacity of its level set clusters. We prove, without any further assumption on the base graph $\mathcal{G}$, that the capacity of sign clusters on $\tilde{\mathcal G}$ is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on $\tilde{\mathcal G}$ are bounded or not. It also elucidates why the critical parameter for percolation of level sets on $\tilde{\mathcal G}$ vanishes in most instances in the m…
We show that supersymmetric (susy) hyperbolic isomorphism theorems that relate Vertex Reinforced Jump Processes and $H^{2|2}$ field, introduced in [2] and [3], are annealed version of isomorphism theorems relating Markov processes and Gaussian free field, with the help of a Bayes formula that relates susy hyperbolic field to susy free field. On the other hand, we also prove a BFS-Dynkin's isomorphism theorem for reinforced loop soup. Moreover, we provide yet another proof of BFS-Dynkin's isomorphism for VRJP a la Feynman-Kac.
We construct loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.
Generating Galton-Watson trees using random walks and percolation for the Gaussian free field
本文通过有限随机游走轨迹探索构造Galton-Watson树，结合Dynkin型同构定理，将高斯自由场渗流临界参数$h_*$严格为正的结果推广到任意超临界后代分布和随机电导情形，并证明正相关促进渗流。
These notes originated in a series of lectures I gave in Marseille in May, 2013. I was invited to give an introduction to the isomorphism theorems, originating with Dynkin, which connect Markov local times and Gaussian processes. This is an area I had worked on some time ago, and even written a book about, but had then moved on to other things. However, isomorphism theorems have become of interest once again, both because of new applications to the study of cover times of graphs and Gaussian fields, and because of new isomorphism theorems for non-symmetric Markov processes and their connection…
Consider the stochastic heat equation $\partial_t u = \sL u + \dot{W}$, where $\sL$ is the generator of a [Borel right] Markov process in duality. We show that the solution is locally mutually absolutely continuous with respect to a smooth perturbation of the Gaussian process that is associated, via Dynkin's isomorphism theorem, to the local times of the replica-symmetric process that corresponds to $\sL$.In the case that $\sL$ is the generator of a Lévy process on $\R^d$, our result gives a probabilistic explanation of the recent findings of Foondun et al.
We consider Gaussian fields of real symmetric, complex Hermitian or quaternionic Hermitian matrices over an electrical network, and describe how the isomorphisms between these fields and random walks give rise to topological expansions encoded by ribbon graphs. We further consider matrix-valued Gaussian fields twisted by an orthogonal, unitary or symplectic connection. In this case the isomorphisms involve traces of holonomies of the connection along random walk loops parametrized by boundary cycles of ribbon graphs.
We investigate the percolation phase transition for level sets of the Gaussian free field on $\mathbb{Z}^d$, with $d\geqslant 3$, and prove that the corresponding critical parameter $h_*(d)$ is strictly positive for all $d\geqslant3$, thus settling an open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on $\mathbb{Z}^d$, for all $d\geqslant 3$. Among other things, our construction of an infinite cluster above small, but positive level $h$ involves random interlacements at level $u>0$, a random subset of $\mathbb{Z}^d$ with…
In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing primarily on models with long-range correlations. Our results are in the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We also prove a shape theorem for balls in the chemical distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the vacant set of random int…
Traces left by random walks on large random graphs: local limits
本文发展理论分析随机游走在随机参考顶点附近留下的迹，证明其弱极限定理，要求随机游走具适当混合性质且图模型满足更强的Benjamini-Schramm收敛变体，极限对象可显式给出。
In the simple case of a Bernoulli shift on two symbols, zero and one, by permuting the symbols, it is obvious that any two equal entropy shifts are isomorphic. We show that the isomorphism can be realized by a factor that maps a binary sequence to another that is coordinatewise smaller than or equal to the original sequence.
The geometry of random walk isomorphism theorems
本文利用自旋系统的连续对称性（平移、洛伦兹 boosts、旋转）将随机游走同构定理推广到双曲和球面几何，并给出若干应用，包括关联衰减、预解式及 Sabot--Tarrès 公式的新证明。
The Density Formula Approach for Non-reversible Isomorphism Theorems, with Applications
本文通过密度公式方法证明了非可逆同构定理，并将其应用于永久过程的比较不等式推导以及非可逆马尔可夫链覆盖时间的上界估计。
We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $\mathbb{H}^n$ or its supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin--Wagner theorem for these sigma models. This result is of intrinsic interest for the s…
Isomorphism Theorems between Models of Mixed Choice (Revised)
本文将混合非确定性与概率选择的 powercone 模型与 previsions 模型相关联，基于 Hahn-Banach 分离定理变体和 Schröder-Simpson 定理证明两者在拓扑假设下同构，并修复了原版中的引理证明。
Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism
该研究利用Satake同构与Hall-Littlewood多项式，刻画非阿基米德域上分裂约化群乘积的奇异数与角元，推广已有结果并证明其最小分离性，推导相关概率极限定理，建立代数分解与概率极限的统一框架。
We consider the translation dynamics on the $C^*$-algebra $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ of two sided infinite tensor product of $d$ dimensional matrices $\!M^{(n)}(\IC)=\!M_d(\IC)$ over the field of complex numbers $\IC$ and its restriction to the maximal abelian $C^*$ sub-algebra $\ID^e = \otimes_{n \in \IZ}\!D_e^{(n)}(\IC)$ of $\IM$, where each $\!D_e^{(n)}(\IC)=\!D_d(\IC)$ is the algebra of $d$ dimensional diagonal matrices with respect to an orthonormal basis $e=(e_i)$ of $\IC^d$. We prove that any two translation invariant Kolmogorov pure states of $\IM$ give unitarily equivale…
Reinforced Loop Soup via Wilson's Algorithm
本文通过改进的威尔逊算法构建了强化环汤，并建立了其与标准马尔可夫同构定理的联系。
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