##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics Seminar

## Subhabrata Sen

#### Harvard University

## Large deviations for dense random graphs: beyond mean-field

##### Abstract:

In a seminal paper, Chatterjee and Varadhan derived an LDP for the dense Erd\H{o}s-R\'{e}nyi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is \emph{inhomogeneous} or \emph{constrained}. \\ \\ In this talk, we will explore large deviations for dense random graphs, beyond the ``mean-field'' setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erd\H{o}s-R\'{e}nyi random graphs. \\ \\ Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.

Host: Benson Au

### April 15, 2021

### 11:00 AM

### For zoom ID and password email: bau@ucsd.edu

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