Limiting behavior of relative Rényi entropy in a non-regular location shift family

具有正则变化机制的非线性 CSBP 在爆炸附近的行为

Behaviors near explosion of nonlinear CSBPs with regularly varying mechanisms

本文研究非线性连续状态分支过程的爆炸问题，给出显式积分判别法，并依据正则变化指数划分三种爆炸情形，刻画了爆炸前的重整化极限分布。

Limiting behavior for a general class of voter models with confidence threshold

This article is concerned with a general class of stochastic spatial models for the dynamics of opinions. Like in the voter model, individuals are located on the vertex set of a connected graph and update their opinion at a constant rate based on the opinion of their neighbors. However, unlike in the voter model, the set of opinions is represented by the set of vertices of another connected graph that we call the opinion graph: when an individual interacts with a neighbor, she imitates this neighbor if and only if the distance between their opinions, defined as the graph distance induced by th…

Limiting spectral distribution of Gram matrices associated with functionals of $β$-mixing processes

We give asymptotic spectral results for Gram matrices of the form $ n^{-1}\mathcal{X}_n \mathcal{X}_n^T$ where the entries of $\mathcal{X}_n$ are dependent across both rows and columns. More precisely, they consist of short or long range dependent random variables having moments of second order and that are functionals of an absolutely regular sequence. We also give a concentration inequality of the Stieltjes transform and we prove that, under an arithmetical decay condition on the $β$-mixing coefficients, it is almost surely concentrated around its expectation. Applications to examples of pos…

局部正则树状扩张图上极限临界Potts测度的刻画

Characterizing the limiting critical Potts measures on locally regular-tree-like expander graphs

研究收敛到正则树的扩张图上Potts模型的临界极限，证明其局部弱极限为自由与连线Gibbs测度的混合，并推广了现有零场及偶数度数结果。

Random Networks with Heterogeneous Reciprocity

Users of social networks display diversified behavior and online habits. For instance, a user's tendency to reply to a post can depend on the user and the person posting. For convenience, we group users into aggregated behavioral patterns, focusing here on the tendency to reply to or reciprocate messages. The reciprocity feature in social networks reflects the information exchange among users. We study the properties of a preferential attachment model with heterogeneous reciprocity levels, give the growth rate of model edge counts, and prove convergence of empirical degree frequencies to a lim…

Cokernels of adjacency matrices of random $r$-regular graphs

We study the distribution of the cokernels of adjacency matrices (the Smith groups) of certain models of random $r$-regular graphs and directed graphs, using recent mixing results of Mészáros. We explain how convergence of such distributions to a limiting probability distribution implies asymptotic nonsingularity of the matrices, giving another perspective on recent results of Huang and Mészáros on asymptotic nonsingularity of adjacency matrices of random regular directed and undirected graphs, respectively. We also remark on the new distributions on finite abelian groups that arise, in partic…

Limiting behavior of the distance of a random walk

This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $n/2$. Here, we study three new examples. It is trivial that the distance for random walk on the hypercube is smooth and is given by one simple formula. In the case of random adjacent transpositions, we find that there is no phase transition even though the distance has different scalings in three different regimes. In the case of a random 3-regular graph, there is a ph…

Exploratory LQG Mean Field Games with Entropy Regularization

We study a general class of entropy-regularized multi-variate LQG mean field games (MFGs) in continuous time with $K$ distinct sub-population of agents. We extend the notion of actions to action distributions (exploratory actions), and explicitly derive the optimal action distributions for individual agents in the limiting MFG. We demonstrate that the optimal set of action distributions yields an $ε$-Nash equilibrium for the finite-population entropy-regularized MFG. Furthermore, we compare the resulting solutions with those of classical LQG MFGs and establish the equivalence of their existenc…

高阶马尔可夫链的边际平稳分布与收敛速率

Marginal Stationary Distributions and Convergence Rates of Higher Order Markov Chains

研究正则高阶马尔可夫链，解释其极限概率分布为边际平稳分布，证明其性质并建立收敛速率结果，还引入边际混合时间，扩展了一阶马尔可夫链混合时间概念到高阶链。

Two-scale multitype contact process: coexistence in spatially explicit metapopulations

It is known that the limiting behavior of the contact process strongly depends upon the geometry of the graph on which particles evolve: while the contact process on the regular lattice exhibits only two phases, the process on homogeneous trees exhibits an intermediate phase of weak survival. Similarly, we prove that the geometry of the graph can drastically affect the limiting behavior of multitype versions of the contact process. Namely, while it is strongly believed (and partly proved) that the coexistence region of the multitype contact process on the regular lattice reduces to a subset of…

Random non-Abelian G-circulant matrices. Spectrum of random convolution operators on large finite groups

We analyse the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by M. Meckes for the Abelian case. We show that for regular sequences of groups the limiting distribution of eigenvalues (resp. singular values) is a mixture of eigenvalue (resp. singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular for the sequence of symmetric groups, the limiting distri…

The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric ($β$ = 1), Hermitian ($β$ = 2), and Hermitian self-dual ($β$ = 4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the one-band condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the …