## Student Probability Seminar, Fall 2020

### Wednesday 2-3 PM, 939 Evans

This is the UC Berkeley Student Probability Seminar, run by students in both the Mathematics and Statistics Departments at UC Berkeley. We meet on Wednesdays on Zoom from 2-3pm, right before the probability seminar.

This semester we will be studying statistical mechanics models on the lattice. We will mainly rely on Hugo Duminil-Copin's notes, which are freely accessible here. We may also occasionally use Freidli-Velenik's book on the same topic and Grimmett's book on the random cluster model.

Previous Seminars: Spring 2020, Fall 2019, Spring 2019 , Fall 2018 , Fall 2017 , Summer 2017
 DATE SPEAKER TITLE (click for abstract below) August 26 Adam Jaffe Introductory talk September 2 Ella Hiesmayr Phase transitions and graphical representations September 9 Jake Calvert Correspondence between the random cluster and Potts models September 16 Jake Calvert Correspondence between the random cluster and Potts models (cont'd) September 23 Jorge Garza Vargas The critical value of the random cluster model September 30 Yujin Kim The loop representation of the random-cluster model and the continuity of the phase transition October 7 Yujin Kim (cont'd) October 14 Ian Francis Continuity of the phase transition for $$q\leq 4$$ October 21 Mehdi Ouaki The random current representation of the Ising model October 28 November 4 November 11 November 18

### Introductory talk

In this introduction to statistical-mechanics on lattices, we'll provide some general motivation from physics, introduce a few canonical models from the theory, and discuss the important notion of phase transitions. At the end we'll explore the idea of "graphical representation" which, when possible, allows one to study a model via percolation theory. The talk will contain some precise definitions, but we'll mostly focus on heuristics, intuition, and developing a common vocabulary for discussing the topics.

### Phase transitions and graphical representations

Ella Hiesmayr

Today we will first look at the definition and possible regimes of the transition between the ordered and the disordered phase in lattice spin models. We will then introduce graphical representations of such models and review some key properties of Bernoulli percolation, which will be used as a tool later on.

### Correspondence between the random cluster and Potts models

Jake Calvert

We will show that the critical inverse temperature of the Potts model can be phrased in terms of the critical value of a corresponding random cluster model.

### The critical value of the random cluster model

Jorge Garza Vargas

The main goal of this talk is to compute the critical value of the random-cluster model in the plane. To this end, we will study duality and review Zhang's argument carefully. Time permitting, we will start discussing the next topic: loop representation of the random-cluster model.

### The loop representation of the random-cluster model and the continuity of the phase transition

Yujin Kim

We analyze the nature of the RCM at the critical point through the "loop" representation, which provides new, geometric insight into the RCM that is particularly valuable at the critical point.

### Continuity of the phase transition for $$q\leq 4$$

Ian Francis

We will develop tools necessary to prove the sub-exponential decay of correlations for free boundary conditions, which implies continuity of phase transitions for $$q \leq 4$$. In particular, we will discuss windings, parafermionic observables, and discrete contour integrals.

### The random current representation of the Ising model.

Mehdi Ouaki

We present a new graphical representation of the ferromagnetic Ising model. We introduce the notion of currents and sources on a graph and give the corresponding percolation configuration. Some correlation inequalities are then derived and used to give a proof of the sharpness of the phase transition for Ising models.