相关论文: Local limit theorems via Landau-Kolmogorov inequal…
In this paper we prove a quantiative local limit theorem for the distribution of the number of triangles in the Erd\H{o}s-Renyi random graph $G(n,p)$, for a fixed $p\in (0,1)$. This proof is an extension of the previous work of Gilmer and…
We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both…
We define a multi-group version of the mean-field spin model, also called Curie-Weiss model. It is known that, in the high temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group…
Consider a `dense' Erd\H{o}s--R\'enyi random graph model $G=G_{n,M}$ with $n$ vertices and $M$ edges, where we assume the edge density $M/\binom{n}{2}$ is bounded away from 0 and 1. Fix $k=k(n)$ with $k/n$ bounded away from 0 and~1, and let…
Ferromagnetic exponential random graph models (ERGMs) are nonlinear exponential tilts of Erd\H{o}s-R\'enyi models, under which the presence of certain subgraphs such as triangles may be emphasized. These models are mixtures of metastable…
Consider a set of $n$ vertices, where each vertex has a location in $\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\mathbb{R}^d$, and a weight associated to it. Construct a random graph by placing edges independently for…
We analyze Ising/Curie-Weiss models on the (directed) Erd\H{o}s-R\'enyi random graph on $N$ vertices in which every edge is present with probability $p$. These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993]. We prove a…
In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures $\mu$ and $\nu$ such that $\nu$ is log-concave with respect to $\mu$.…
We study a Curie-Weiss model with a random external field generated by a dynamical system. Probabilistic limit theorems (weak law of large numbers, central limit theorems) are proven for the corresponding magnetization.
We discuss a Curie-Weiss model with two groups with different coupling constants within and between groups. For the total magnetisations in each group, we show bivariate laws of large numbers and a central limit theorem which is valid in…
We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all…
Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…
We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in…
This paper investigates the limiting behaviour of degree-degree correlation metrics for sequences of random graphs under a general assumption of local convergence in probability. We establish convergence results for Pearson's correlation…
We study the thermodynamic properties of the generalized non-convex multispecies Curie-Weiss model, where interactions among different types of particles (forming the species) are encoded in a generic matrix. For spins with a generic prior…
We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the "cavity" prediction for…
We prove a central limit theorem for a certain class of functions on sparse rank-one inhomogeneous random graphs endowed with additional i.i.d. edge and vertex weights. Our proof of the central limit theorem uses a perturbative form of…
The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the $d$-dimensional integer lattice, $\mathbb{Z}^d$. Under certain mild assumptions on the distribution, the…
Stein's method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the…
We introduce a general framework for studying anticoncentration and local limit theorems for random variables, including graph statistics. Our methods involve an interplay between Fourier analysis, decoupling, hypercontractivity of Boolean…