Large deviations for empirical entropies of Gibbsian sources

Large deviation principles for empirical measures of colored random graphs

For any finite colored graph we define the empirical neighborhood measure, which counts the number of vertices of a given color connected to a given number of vertices of each color, and the empirical pair measure, which counts the number of edges connecting each pair of colors. For a class of models of sparse colored random graphs, we prove large deviation principles for these empirical measures in the weak topology. The rate functions governing our large deviation principles can be expressed explicitly in terms of relative entropies. We derive a large deviation principle for the degree distr…

Averaged vs. quenched large deviations and entropy for random walk in a dynamic random environment

We consider random walk with bounded jumps on a hypercubic lattice of arbitrary dimension in a dynamic random environment. The environment is temporally independent and spatially translation invariant. We study the rate functions of the level-3 averaged and quenched large deviation principles from the point of view of the particle. In the averaged case the rate function is a specific relative entropy, while in the quenched case it is a Donsker-Varadhan type relative entropy for Markov processes. We relate these entropies to each other and seek to identify the minimizers of the level-3 to level…

Large deviation results for Critical Multitype Galton-Watson trees

In this article, we prove a joint large deviation principle in $n$ for the \emph{empirical pair measure} and \emph{ empirical offspring measure} of critical multitype Galton-Watson trees conditioned to have exactly $n$ vertices in the weak topology. From this result we extend the large deviation principle for the empirical pair measures of Markov chains on simply generated trees to cover offspring laws which are not treated by \cite[Theorem~2.1]{DMS03}. For the case where the offspring law of the tree is a geometric distribution with parameter \sfrac{1}{2}, we get an exact rate function. All o…

Large Deviations for a Non-Centered Wishart Matrix

We investigate an additive perturbation of a complex Wishart random matrix and prove that a large deviation principle holds for the spectral measures. The rate function is associated to a vector equilibrium problem coming from logarithmic potential theory, which in our case is a quadratic map involving the logarithmic energies, or Voiculescu's entropies, of two measures in the presence of an external field and an upper constraint. The proof is based on a two type particles Coulomb gas representation for the eigenvalue distribution, which gives a new insight on why such variational problems sho…

On the Renyi Divergence, Joint Range of Relative Entropies, and a Channel Coding Theorem

This paper starts by considering the minimization of the Renyi divergence subject to a constraint on the total variation distance. Based on the solution of this optimization problem, the exact locus of the points $\bigl( D(Q\|P_1), D(Q\|P_2) \bigr)$ is determined when $P_1, P_2, Q$ are arbitrary probability measures which are mutually absolutely continuous, and the total variation distance between $P_1$ and $P_2$ is not below a given value. It is further shown that all the points of this convex region are attained by probability measures which are defined on a binary alphabet. This characteriz…

Hierarchical Dirichlet Process and Relative Entropy

The Hierarchical Dirichlet process is a discrete random measure serving as an important prior in Bayesian non-parametrics. It is motivated with the study of groups of clustered data. Each group is modelled through a level two Dirichlet process and all groups share the same base distribution which itself is a drawn from a level one Dirichlet process. It has two concentration parameters with one at each level. The main results of the paper are the law of large numbers and large deviations for the hierarchical Dirichlet process and its mass when both concentration parameters converge to infinity.…

Large deviations for many Brownian bridges with symmetrised initial-terminal condition

Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some non-degenerate initial measure on some fixed time interval $[0,β]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $σ(i)$-th motion, where $σ$ is a uniformly distributed random permutation of $1,...,N$. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature $1/β$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Di…

带凝聚的空间粒子过程：Gibbs测度方法、凝胶化与Smoluchowski方程

Spatial particle processes with coagulation: Gibbs-measure approach, gelation and Smoluchowski equation

本文用Gibbs测度方法研究空间Marcus–Lushnikov凝聚过程，建立大偏差原理并利用速率函数极小化子构造Smoluchowski方程解，给出凝胶化相变判据。

Concentration of information content for convex measures

We establish sharp exponential deviation estimates of the information content as well as a sharp bound on the varentropy for the class of convex measures on Euclidean spaces. This generalizes a similar development for log-concave measures in the recent work of Fradelizi, Madiman and Wang (2016). In particular, our results imply that convex measures in high dimensions are concentrated in an annulus between two convex sets (as in the log-concave case) despite their possibly having much heavier tails. Various tools and consequences are developed, including a sharp comparison result for Rényi entr…

从熵约束到强化过程：多尺度测度的概率起源

From entropic constraints to reinforced processes: a probabilistic origin of multiscale measures

本文提出强化多项式过程，通过大偏差原理揭示了多尺度测度的概率起源，为多尺度Gibbs结构提供了严谨的随机基础。

末次通过渗流中的网格熵——一种超加性临界指数方法

Grid entropy in last passage percolation -- a superadditive critical exponent approach

该研究在独立同分布末次通过渗流框架下提出网格熵概念，重构了吉布斯自由能变分公式，改进了熵的已知界，验证了有限网格熵测度集与经验测度极限点集的一致性，还部分解答了有向聚合物情形的Hoffman问题。