4 Statistical physics approach to asymptotic enumeration & large deviations in random graphs-Perkins

Title: Statistical physics approach to asymptotic enumeration and large deviations in random graphs - part 4
Presented to IAS-PCMI by Will Perkins, Georgia Tech

Abstract: This course will introduce some basics of statistical physics (Gibbs measures, partition functions, phase transitions) and some tools from statistical physics and algorithms (cluster expansion, coupling, Markov chain mixing) and apply these tools to study two related combinatorial problems: asymptotic enumeration of combinatorial structures (for example, counting the number of triangle-free graphs with a given edge density) and large deviations in random graphs (for example, the lower-tail large deviation problem for triangles in G(n,p)).
Prerequisites: some probability theory (expectation, variance, central limit theorems, Markov chains, concentration inequalities). No previous knowledge of statistical physics assumed or needed.
Optional background reading: Statistical Mechanics of Lattice Systems, Friedli & Velenik. Markov Chains and Mixing Times, Levin & Peres. The Method of Hypergraph Containers, Balogh, Morris, & Samotij.
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Lecture notes & problem sets
https://www.ias.edu/pcmi/pcmi-2025-gs...
PCMI 2025 GSS Lecture Notes and Problem Sets - IAS/Park City Mathematics Institute
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PCMI 2025 Research Topic: Probabilistic and Extremal Combinatorics
Organized by Julia Böttcher (LSE), Jacob Fox (Stanford University), Penny Haxell (University of Waterloo), Robert Morris (IMPA), and Wojciech Samotij (Tel Aviv University).

Extremal and probabilistic combinatorics are two central branches of contemporary discrete mathematics. The first of these two branches studies how large (or how small) a discrete structure can be given that it satisfies a certain set of restrictions; the second investigates random combinatorial objects using a blend of combinatorial methods and tools of probability theory. These two fields have been growing at a stunning rate over the last few decades and are nowadays considered to be an important part of mainstream mathematical research.
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The aim of the planned summer graduate program at PCMI is to provide in-depth introduction to several preeminent themes and methods in extremal and probabilistic combinatorics, with particular emphasis on strong connections of these fields with other areas of mathematics such as analysis, geometry, number theory, statistical physics, and theoretical computer science. The core of the program will be nine graduate mini-courses taught by a diverse group of leading researchers in the field renowned for their clear and engaging lecturing styles. In parallel, we plan thematic workshops aimed at more senior researchers as well as activities for undergraduate students.