arXiv subjects

Xiaolin Zeng

Publications and source records attributed to Xiaolin Zeng.

At least 19 recordsLinked to original sources

arXiv2026

VRJP recurrence and fractional-moment decay for the $H^{2|2}$ model's effective field on the hierarchical lattice

We prove recurrence of the vertex-reinforced jump process on the hierarchical lattice for spectral dimension $d<2$ for every value of the conductance parameter $\overline{W}$, and at the critical spectral dimension $d=2$ for sufficiently strong reinforcement, i.e., sufficiently small $\overline{W}$. The key estimate is a fractional-moment bound for the Green's function of the associated random Schrödinger operator, expressed as geometric decay across hierarchical scales for the effective $H^{2|2}$ field. The proof combines the fractional-moment method with an exact hierarchical coarse-graining identity, which turns the path expansion into a recursion over block scales and controls the combinatorial growth created by long-range edges. Together with existing long-range-order results in the regime $d>2$, these estimates identify the recurrent side of the hierarchical VRJP phase diagram, leaving only the weak-reinforcement critical regime open.

math.PR↗

arXiv2026

The $H^{2|2}$ monotonicity theorem revisited

We use supersymmetric localization and integration by parts to derive variational and convex correlation inequalities in statistical physics. As a primary application, we give an alternative proof of the monotonicity theorem for the $H^{2|2}$ supersymmetric hyperbolic sigma model. This recovers a result of Poudevigne without relying on probabilistic couplings.

math.PR↗

arXiv2025

Reinforced Loop Soup via Wilson's Algorithm

The goal of this note is twofold: first, we explain the relation between the isomorphism theorems in the context of vertex reinforced jump process discovered in [BHS19, BHS21] and the standard Markovian isomorphism theorems for Markovian jump processes; second, we introduce the vertex reinforced counterpart of the standard Poissonian loop soup developed by Le Jan [LJ10]. To this end, we propose an algorithm that can be viewed as a variant of Wilson's algorithm with reinforcement. We establish the isomorphism theorems for the erased loops and the random walk from this algorithm, and in particular provide a concrete construction of the reinforced loop soup via a random process with a reinforcement mechanism.

math.PR↗

arXiv2025

Fine-Graining and Continuous Space Scaling Limit of the $H^{2|2}$ Model on the Hierarchical Lattice

We extend the exact coarse-graining result of Disertori, Merkl and Rolles~\cite{MR4517733} for the random field of $H^{2|2}$-model to the random Schrödinger operator representation of the $H^{2|2}$-model. We also introduce a fine-graining procedure as the reverse operation, and establish an associated exponential martingale property. Applying this fine-graining procedure to the $H^{2|2}$-model on the Dyson hierarchical lattice, we establish its continuous space scaling limit as a non-trivial random measure on $[0,1]$. This random measure is almost surely singular with respect to the Lebesgue measure if and only if the Vertex Reinforced Jump Process on the Dyson hierarchical lattice is recurrent. If the process is transient, the random measure almost surely has an absolutely continuous component. The density of this component is everywhere non-trivial and can be identified with the pointwise limit of an exponential martingale associated with the $H^{2|2}$-model on the Dyson hierarchical lattice.

math.PR↗

arXiv2025

Graphical Negative Multinomial and Multinomial Models with Dirichlet-type priors

Bayesian statistical graphical models are typically classified as either continuous and parametric (Gaussian, parameterized by the graph-dependent precision matrix with Wishart-type priors) or discrete and non-parametric (with graph-dependent structure of probabilities of cells and Dirichlet-type priors). We propose to break this dichotomy by introducing two discrete parametric graphical models on finite decomposable graphs: the graph negative multinomial and the graph multinomial distributions (the former related to the Cartier-Foata theorem for the graph genereted free quotient monoid). These models interpolate between the product of univariate negative binomial laws and the negative multinomial distribution, and between the product of binomial laws and the multinomial distribution, respectively. We derive their Markov decompositions and provide related probabilistic representations. We also introduce graphical versions of the Dirichlet and inverted Dirichlet distributions, which serve as conjugate priors for the two discrete graphical Markov models. We derive explicit normalizing constants for both graphical Dirichlet laws and establish their independence structure (a graphical version of neutrality), which yields a strong hyper Markov property for both Bayesian models. We also provide characterization theorems for graphical Dirichlet laws via respective graphical versions of neutrality, which extends previously known results.

math.PR↗

arXiv2025

Phase transition in the Integrated Density of States of the Anderson model arising from a supersymmetric sigma model

We study the Integrated Density of States (IDS) of the random Schrödinger operator appearing in the study of certain reinforced random processes in connection with a supersymmetric sigma-model. We rely on previous results on the supersymmetric sigma-model to obtain lower and upper bounds on the asymptotic behavior of the IDS near the bottom of the spectrum in all dimension. We show a phase transition for the IDS between weak and strong disorder regime in dimension larger or equal to three, that follows from a phase transition in the corresponding random process and supersymmetric sigma-model. In particular, we show that the IDS does not exhibit Lifshitz tails in the strong disorder regime, confirming a recent conjecture. This is in stark contrast with other disordered systems, like the Anderson model. A Wegner type estimate is also derived, giving an upper bound on the IDS and showing the regularity of the function.

math-ph↗

arXiv2023

Enumeration of maximum matchings of graphs

Counting maximum matchings in a graph is of great interest in statistical mechanics, solid-state chemistry, theoretical computer science, mathematics, among other disciplines. However, it is a challengeable problem to explicitly determine the number of maximum matchings of general graphs. In this paper, using Gallai-Edmonds structure theorem, we derive a computing formula for the number of maximum matching in a graph. According to the formula, we obtain an algorithm to enumerate maximum matchings of a graph. In particular, The formula implies that computing the number of maximum matchings of a graph is converted to compute the number of perfect matchings of some induced subgraphs of the graph. As an application, we calculate the number of maximum matchings of opt trees. The result extends a conclusion obtained by Heuberger and Wagner[C. Heuberger, S. Wagner, The number of maximum matchings in a tree, Discrete Math. 311 (2011) 2512--2542].

math.CO↗

arXiv2023

A multi-dimensional version of Lamperti's relation and the Matsumoto-Yor processes

This paper presents a multidimensional extension of the Matsumoto-Yor properties related to exponential functionals of drifted Brownian motion. The extension involves the interaction of geometric Brownian motions which are indexed by the vertices of a finite weighted graph, and the random potential associated with the Vertex Reinforced Jump process on this graph. We prove in this context a counterpart of Lamperti's transformation, of the Markov property of the Matsumoto-Yor process and of the intertwining relation.

math.PR↗

arXiv2022

On $L^0$-convex compactness in random locally convex modules

For the study of some typical problems in finance and economics, Žitković %[G. Žitković, Convex compactness and its applications, Math. Finan. Eco., 3(1)(2010) 1--12] introduced convex compactness and gave many remarkable applications. Recently, motivated by random convex optimization and random variational inequalities, Guo, et al introduced $L^0$-convex compactness, developed the related theory of $L^0$-convex compactness in random normed modules and further applied it to backward stochastic equations. %[T.X. Guo, et al, Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations, J. Math. Anal. Appl., 483(2020) 123644]. In this paper, we extensively study $L^0$-convexly compact sets in random locally convex modules so that a series of fundamental results are obtained. First, we show that every $L^0$-convexly compact set is complete (hence is also closed and has the countable concatenation property). Then, we prove that any $L^0$-convexly compact set is linearly homeomorphic to a weakly compact subset of some locally convex space, and simultaneously establish the equivalence between $L^0$-convex compactness and convex compactness for a closed $L^0$-convex set. Finally, we establish Tychonoff type, James type and Banach-Alaoglu type theorems for $L^0$-convex compactness, respectively.

math.FA↗

arXiv2020

A multi-dimensional version of Lamperti's relation and the Matsumoto-Yor opposite drift theorem

A classic result on the 1-dimensional Brownian motion shows that conditionally on its first hitting time of 0, it has the distribution of a 3-dimensional Bessel bridge. By applying a certain time-change to this result, Matsumoto and Yor showed a theorem giving a relation between Brownian motions with opposite drifts. The relevant time change is the one appearing in Lamperti's relation. Sabot and Zeng showed that a family of Brownian motions with interacting drifts, conditioned on the vector of hitting times of 0, also has the distribution of independent 3-dimensional Bessel bridges. Moreover, the distribution of these hitting times is related to a random potential that appears in the study of the vertex-reinforced jump process. The aim of this paper is to prove a multivariate version of the Matsumoto-Yor opposite drift theorem, by applying a Lamperti-type time change to the previous family of interacting Brownian motions. Difficulties arise since the time change progresses at different speeds on different coordinates.

math.PR↗

arXiv2020

On $H^{2|2}$ Isomorphism theorems and reinforced loop soup

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.

math.PR↗

arXiv2019

A note on recurrence of the Vertex reinforced jump process and fractional moments localization

We give a simple proof for recurrence of vertex reinforced jump process on $\mathbb{Z}^d$, under strong reinforcement. Moreover, we show how the previous result implies that linearly edge-reinforced random walk on \ $\mathbb{Z}^d$ is {recurrent} for strong reinforcement. Finally, we prove that the $H^{(2|2)}$ model on $\mathbb{Z}^d$ localizes at strong disorder. Even though these results are well-known, we propose a unified approach, {which also has the advantage to provide shorter proofs}, and relies on estimating fractional moments, introduced by Aizenman and Molchanov.

math.PR↗

arXiv2019

Some results on the Orlicz space generated from a random normed module

Noting the important role the abstract $L^p$ space has played in the development of random normed modules, in this paper we introduce and study the Orlicz space generated from a random normed module. First, we give a basic dual space representation theorem which identify the dual of the Orlicz heart of a random normed module with the Orlicz space generated from the random conjugate space. Then, we establish the respective equivalence relations of the strict convexity and uniform convexity of this abstract Orlicz space to the random strict convexity and random uniform convexity of the underlying random normed module. These results demonstrate that it is possible to use the Orlicz space theory in the further development of random nomed modules.

math.FA↗

arXiv2018

A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs

This paper concerns the Vertex reinforced jump process (VRJP), the Edge reinforced random walk (ERRW) and their link with a random Schrödinger operator. On infinite graphs, we define a 1-dependent random potential $β$ extending that defined in [20] on finite graphs, and consider its associated random Schrödinger operator $H_β$. We construct a random function $ψ$ as a limit of martingales, such that $ψ=0$ when the VRJP is recurrent, and $ψ$ is a positive generalized eigenfunction of the random Schrödinger operator with eigenvalue $0$, when the VRJP is transient. Then we prove a representation of the VRJP on infinite graphs as a mixture of Markov jump processes involving the function $ψ$, the Green function of the random Schrödinger operator and an independent Gamma random variable. On ${\mathbb Z}^d$, we deduce from this representation a zero-one law for recurrence or transience of the VRJP and the ERRW, and a functional central limit theorem for the VRJP and the ERRW at weak reinforcement in dimension $d\ge 3$, using estimates of [10,8]. Finally, we deduce recurrence of the ERRW in dimension $ d=2$ for any initial constant weights (using the estimates of Merkl and Rolles, [15,17]), thus giving a full answer to the old question of Diaconis. We also raise some questions on the links between recurrence/transience of the VRJP and localization/delocalization of the random Schrödinger operator $H_β$.

math.PR↗

arXiv2018

Hitting times of interacting drifted Brownian motions and the vertex reinforced jump process

Consider a negatively drifted one dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a 3- dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the Vertex Reinforced Jump Process ([17, 18]). These Brownian motions with interacting drifts have remarkable properties with respect to restriction and conditioning, showing hidden Markov properties. This family of processes are closely related to the martingale that plays a crucial role in the analysis of the vertex reinforced jump process and edge reinforced random walk ([18]) on infinite graphs.

math.PR↗

arXiv2017

On random convex analysis

Recently, based on the idea of randomizing space theory, random convex analysis has been being developed in order to deal with the corresponding problems in random environments such as analysis of conditional convex risk measures and the related variational problems and optimization problems. Random convex analysis is convex analysis over random locally convex modules. Since random locally convex modules have the more complicated topological and algebraic structures than ordinary locally convex spaces, establishing random convex analysis will encounter harder mathematical challenges than classical convex analysis so that there are still a lot of fundamentally important unsolved problems in random convex analysis. This paper is devoted to solving some important theoretic problems. First, we establish the inferior limit behavior of a proper lower semicontinuous $L^0$--convex function on a random locally convex module endowed with the locally $L^0$--convex topology, which makes perfect the Fenchel--Moreau duality theorem for such functions. Then, we investigate the relations among continuity, locally $L^0$--Lipschitzian continuity and almost surely sequent continuity of a proper $L^0$--convex function. And then, we establish the elegant relationships among subdifferentiability, Gâteaux--differentiability and Fréchét--differentiability for a proper $L^0$--convex function defined on random normed modules. At last, based on the Ekeland's variational principle for a proper lower semicontinuous $\bar{L}^0$--valued function, we show that $\varepsilon$--subdifferentials can be approximated by subdifferentials. We would like to emphasize that the success of this paper lies in simultaneously considering the $(\varepsilon, λ)$--topology and the locally $L^0$--convex topology for a random locally convex module.

math.FA↗

arXiv2016

Speed of Vertex reinforced jump process on Galton-Watson trees

We give an alternative proof of the fact that the vertex reinforced jump process on Galton- Watson tree has a phase transition between recurrence and transience as a function of c, the initial local time, see [3]. Further, applying the techniques in [1], we show a phase transition between positive speed and null speed for the discrete time process associated in the transient regime.

math.PR↗

arXiv2016

The Vertex Reinforced Jump Process and a Random Schrödinger operator on finite graphs

We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the Inverse Gaussian distribution. Considered as the potential of a random Schrödinger operator, this exponential family is related to the random field that gives the mixing measure of the Vertex Reinforced Jump Process (VRJP), and hence to the mixing measure of the Edge Reinforced Random Walk (ERRW), the so-called magic formula. In particular, it yields by direct computation the value of the normalizing constants of these mixing measures, which solves a question raised by Diaconis. The results of this paper are instrumental in [Sabot-Zeng,2015], where several properties of the VRJP and the ERRW are proved, in particular a functional central limit theorem in transient regimes, and recurrence of the 2-dimensional ERRW.

math.PR↗