Phase transitions in edge-weighted exponential random graphs

Dimension reduction in vertex-weighted exponential random graphs

We investigate the behavior of vertex-weighted exponential random graphs. We show that vertex-weighted exponential random graphs with edge weights induced by products of independent vertex weights are approximate mixtures of graphs whose vertex weight vector is a near fixed point of a certain vector equation. For graphs with Hamiltonians counting cliques, it is demonstrated that, under appropriate conditions, every solution to this equation is close to a block vector with a small number of communities. We prove that for the cases of positive weights and small weights in the Hamiltonian in part…

线性抛物型方程指数稳定性的加权半群方法

A weighted semigroup approach to exponential stability in linear parabolic equations

本文采用加权半群方法，证明了带一般漂移项和零阶系数的线性抛物型方程初边值问题解的指数L²稳定性，该结果对漂移项具有鲁棒性。

关于随机向量加权和的短 Edgeworth 展开

On short Edgeworth expansions for weighted sums of random vectors

本文研究 k 维独立同分布随机向量加权和的渐近行为，证明在五阶绝对矩有限时，Edgeworth 修正的收敛阶为 O(1/n^{3/2})，推广了 Bobkov 的一维结果。

Berry-Esseen and Edgeworth approximations for the tail of an infinite sum of weighted gamma random variables

Consider the sum $Z = \sum_{n=1}^\infty λ_n (η_n - \mathbb{E}η_n)$, where $η_n$ are i.i.d.~gamma random variables with shape parameter $r > 0$, and the $λ_n$'s are predetermined weights. We study the asymptotic behavior of the tail $\sum_{n=M}^\infty λ_n (η_n - \mathbb{E}η_n)$ which is asymptotically normal under certain conditions. We derive a Berry-Essen bound and Edgeworth expansions for its distribution function. We illustrate the effectiveness of these expansions on an infinite sum of weighted chi-squared distributions.

Large Deviations for Weighted Sums of Stretched Exponential Random Variables

We consider the probability that a weighted sum of $n$ i.i.d. random variables $X_j$, $j = 1, . . ., n$, with stretched exponential tails is larger than its expectation and determine the rate of its decay, under suitable conditions on the weights. We show that the decay is subexponential, and identify the rate function in terms of the tails of $X_j$ and the weights. Our result generalizes the large deviation principle given by Kiesel and Stadtmüller [8] as well as the tail asymptotics for sums of i.i.d. random variables provided by Nagaev [10, 11]. As an application of our result, motivated by…

A Riemann-Roch theorem on a weighted infinite graph

A Riemann-Roch theorem on graph was initiated by M. Baker and S. Norine. In their article [2], a Riemann-Roch theorem on a finite graph with uniform vertex-weight and uniform edge-weight was established and it was suggested a Riemann-Roch theorem on an infinite graph was feasible. In this article, we take an edge-weighted infinite graph and focus on the importance of the spectral gaps of the Laplace operators defined on its finite subgraphs naturally given by Q-valued positive weights on the edges. We build a potential theoretic scheme for proof of a Riemann-Roch theorem on the edge-weighted i…

随机图加权染色的线性与局部正确性

Linearity and Local Correctness in Weighted Colourings of Random Graphs

本文研究二项随机图的加权色数推广，证明其随最大顶点度数的线性增长依赖于边权重矩的有界性，并得出平衡子图局部正确加权染色的尖锐阈值结果。

Weighted cogrowth formula for free groups

We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ endowed with variable edge lengths, by an arbitrary subgroup $G$ of $F_n$. Our main result, which generalizes Grigorchuk's cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on $G \backslash {\rm Cay}(F_n)$ to the Poincaré exponent of $G$. Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.

Edgeworth expansion for Bernoulli weighted mean

In this work, we derive an Edgeworth expansion for the Bernoulli weighted mean $\hatμ = \frac{\sum_{i=1}^n Y_i T_i}{\sum_{i=1}^n T_i}$ in the case where $Y_1, \dots, Y_n$ are i.i.d. non semi-lattice random variables and $T_1, \dots, T_n$ are Bernoulli distributed random variables with parameter $p$. We also define the notion of a semi-lattice distribution, which gives a more geometrical equivalence to the classical Cramér's condition in dimensions bigger than 1. Our result provides a first step into the generalization of classical Edgeworth expansion theorems for random vectors that contain bo…

The hard-to-soft edge transition: exponential moments, central limit theorems and rigidity

The local eigenvalue statistics of large random matrices near a hard edge transitioning into a soft edge are described by the Bessel process associated with a large parameter $α$. For this point process, we obtain 1) exponential moment asymptotics, up to and including the constant term, 2) asymptotics for the expectation and variance of the counting function, 3) several central limit theorems and 4) a global rigidity upper bound.