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Xiaolin Zeng

Publications and source records attributed to Xiaolin Zeng.

24 records · Page 2Linked to original sources
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↗
arXiv2015

Random convex analysis (I): separation and Fenchel-Moreau duality in random locally convex modules

To provide a solid analytic foundation for the module approach to conditional risk measures, our purpose is to establish a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the $(\varepsilon,λ)$--topology and the locally $L^0$-- convex topology). This paper is focused on the part of separation and Fenchel-Moreau duality in random locally convex modules. The key point of this paper is to give the precise relation between random conjugate spaces of a random locally convex module under the two kinds of topologies, which enables us to not only give a thorough treatment of separation between a point and a closed $L^{0}$-convex subset but also establish the complete Fenchel-Moreau duality theorems in random locally convex modules under the two kinds of topologies.

math.FA↗
arXiv2015

Random convex analysis (II): continuity and subdifferentiability theorems in $L^{0}$--pre--barreled random locally convex modules

In this paper, we continue to study random convex analysis. First, we introduce the notion of an $L^0$--pre--barreled module. Then, we develop the theory of random duality under the framework of a random locally convex module endowed with the locally $L^0$--convex topology in order to establish a characterization for a random locally convex module to be $L^0$--pre--barreled, in particular we prove that the model space $L^{p}_{\mathcal{F}}(\mathcal{E})$ employed in the module approach to conditional risk measures is $L^0$--pre--barreled, which forms the most difficult part of this paper. Finally, we prove the continuity and subdifferentiability theorems for a proper lower semicontinuous $L^0$--convex function on an $L^{0}$--pre--barreled random locally convex module. So the principal results of this paper may be well suited to the study of continuity and subdifferentiability for $L^0$--convex conditional risk measures.

math.FA↗
arXiv2015

How Vertex reinforced jump process arises naturally

We prove that the only nearest neighbor jump process with local dependence on the occupation times satisfying the partial exchangeability property is the vertex reinforced jump process, under some technical conditions. This result gives a counterpart to the characterization of edge reinforced random walk given by Rolles.

math.PR↗
arXiv2013

A Russo Seymour Welsh Theorem for critical site percolation on \(\mathbb{Z}^2\)

The Russo-Seymour-Welsh Theorem for Z^2 bond or T (triangular lattice) site percolation states that at criticality, for all fixed real λ, the probability of the existence of a horizontal occupied crossing of each rectangle with size n*λn is not degenerated when n tends to infinity. Turning to site percolation on Z^2, where the self duality does not hold anymore, we prove that the analogue statement of the RSW Theorem will still be true in this case. The proof uses appropriate finite size criteria and a result of Kesten which allows us to extend existing crossings without losing too much probability. As a consequence, there is no infinite cluster at criticalty. Our object in this short paper is twofold. Our first goal is to give a proof of a RSW Theorem for Z 2 site percolation. Since the proof uses in an essential way a celebrated result by Kesten on the so called "box crossing", our second goal in this paper is to present a self-contained proof of this theorem in perhaps a more accessible language.

math.PR↗
arXiv2013

On random convex analysis -- the analytic foundation of the module approach to conditional risk measures

To provide a solid analytic foundation for the module approach to conditional risk measures, this paper establishes a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the $(\varepsilon,λ)$--topology and the locally $L^0$-- convex topology). Then, we make use of the advantage of the $(\varepsilon,λ)$--topology and grasp the local property of $L^0$--convex conditional risk measures to prove that every $L^{0}$--convex $L^{p}$--conditional risk measure ($1\leq p\leq+\infty$) can be uniquely extended to an $L^{0}$--convex $L^{p}_{\mathcal{F}}(\mathcal{E})$--conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter, which shows that the study of $L^p$--conditional risk measures can be incorporated into that of $L^{p}_{\mathcal{F}}(\mathcal{E})$--conditional risk measures. In particular, in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of $L^{0}$--convex conditional risk measures.

math.FA↗