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We prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our techniques rely on a…
Building on earlier work introducing the notion of "mod-Gaussian" convergence of sequences of random variables, which arises naturally in Random Matrix Theory and number theory, we discuss the analogue notion of "mod-Poisson" convergence.…
We establish a central limit theorem for the sum of $\epsilon$-independent random variables, extending both the classical and free probability setting. Central to our approach is the use of graphon limits to characterize the limiting…
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random…
We introduce the concept of Randomly Modulated Gaussian Processes as a unifying framework for modeling, analyzing and classifying anomalous diffusion models in heterogeneous media. This formulation incorporates correlations in the…
We consider large uniform labeled random graphs in different classes with prescribed decorations in their modular decomposition. Our main result is the estimation of the number of copies of every graph as an induced subgraph. As a…
In this work, we obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the "spectrum" of partitions of a large integer n (under the Plancherel measure). More specifically, we show that,…
Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting non-universality phenomenon. Though they have bounded variance, their fluctuations are asymptotically non-Gaussian…
We study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity. We prove that for a rather wide class of test functions the fluctuations of linear eigenvalue…
The purpose of this article is to present a general method to find limiting laws for some renormalized statistics on random permutations. The model considered here is Ewens sampling model, which generalizes uniform random permutations. We…
In this paper, we relate the framework of mod-$\phi$ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is that of Fourier analysis in the Wiener algebra, allowing…
We consider the adjacency matrix $A$ of a large random graph and study fluctuations of the function $f_n(z,u)=\frac{1}{n}\sum_{k=1}^n\exp\{-uG_{kk}(z)\}$ with $G(z)=(z-iA)^{-1}$. We prove that the moments of fluctuations normalized by…
In this paper, we study modulus of continuity and rate of convergence of series of conditionally sub-Gaussian random fields. This framework includes both classical series representations of Gaussian fields and LePage series representations…
The study of the normalized sum of random variables and its asymptotic behaviour has been and continues to be a central chapter in probability and statistical mechanics. When those variables are independent the central limit theorem ensures…
In this paper we complete our understanding of the role played by the limiting (or residue) function in the context of mod-Gaussian convergence. The question about the probabilistic interpretation of such functions was initially raised by…
We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix…
In this article we will examine a "generalized topological sigma model." This so-called "generalized topological sigma model" is the M-Theoretic analog of the standard topological sigma model of string theory. We find that the observables…
We study the fluctuations in the discrete spectrum of the hyperbolic Laplacian for the modular domain using smooth counting functions. We show that in a certain regime, these have Gaussian fluctuations.
Standard Gaussian graphical models (GGMs) implicitly assume that the conditional independence among variables is common to all observations in the sample. However, in practice, observations are usually collected form heterogeneous…
We study the number of occurrences of any fixed vincular permutation pattern. We show that this statistics on uniform random permutations is asymptotically normal and describe the speed of convergence. To prove this central limit theorem,…