# Probability seminars 2017/2018

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Title: Jacobi triple product via the exclusion process

Speaker: Marton Balazs (Bristol)

Date: Wednesday, 17th January 2018

Time: 2pm

Location: UCD, S3.56 Science South

Abstract: I will give a brief overview of very simple, hence maybe less investigated structures in interacting particle systems: reversible product blocking measures. These turn out to be more general than most people would think, in particular asymmetric simple exclusion and nearest-neighbour asymmetric zero range processes both enjoy them. But a careful look reveals that these two are really the same process. Exploitation of this fact will give rise to the Jacobi triple product formula - an identity previously known from number theory and combinatorics. I will derive it from pure probability this time, and I hope to surprise my audience as much as we got surprised when this identity first popped up in our notebooks.

Date: Wednesday, 24th January 2018

Time: 2pm

Location: UCD, 125 Science North (JK Lab)

Speaker: Theodoros Assiotis (Warwick)

Title: "Determinantal structures in (2+1)-dimensional growth and decay models."

**Abstract:** "I will talk about an inhomogeneous growth and decay model with a wall present in which the growth and decay rates on a single horizontal slice of the surface can be chosen essentially arbitrarily depending on the position. This model turns out to have a determinantal structure and most remarkably for a certain, the fully packed, initial condition the correlation kernel can be calculated explicitly in terms of one dimensional orthogonal polynomials on the positive half line and their orthogonality measures."

**Title: ** "The genealogical structure of Galton-Watson trees."

**Speaker:** Samuel Johnston (UCD)

**Date: ** Wednesday, 31st January 2018

**Time:** 2pm

**Location:** Room 1.25 O’Brien Centre for Science North**Abstract:**

Consider a continuous-time Galton-Watson branching process. If we condition the population to survive until a large fixed time T, and then choose k individuals at random from those alive at that time, what does the ancestral tree relating these k individuals look like?

**Title:** Tableaux combinatorics and the Abelian sandpile model on two classes of graphs**Speaker:** Mark Dukes (UCD)**Date:** Wednesday, 7th February 2018**Time:** 2pm**Location:** UCD, 125 Science North (JK Lab)**Abstract:**

The Abelian sandpile model is a model of discrete diffusion and can be considered as a process on any abstract graph. A state of the model is an assignment of grains of sand to vertices of the graph. If the number of grains at a vertex is less than its degree then that vertex is called stable, and a stable state is one in which every vertex is stable. However, should the number of grains at a vertex exceed its degree, then this vertex may topple and send a grain of sand to each of its neighbours. Recurrent states of this model are those stable states that appear in the long term limit. In this talk I will outline a collection of results concerning recurrent states of the sandpile model on both the complete bipartite graph and the Ferrers graph.

**Title:** Two-time distribution in last-passage percolation**Speaker:** Kurt Johansson (Stockholm)**Date:** Wednesday, 14th February 2018**Time:** 2pm**Location:** UCD, 125 Science North (JK Lab)**Abstract:**

I will discuss a new approach to computing the two-time distribution in last-passage

percolation with geometric weights. This can be interpreted as the correlations of the height

function at a spatial point at two different times in the equivalent interpretation as a discrete

polynuclear growth model. The new approach is rather close to standard random matrix theory

(or determinantal point process) computations. I will give some background and also present

some aspects of the computations involved.

**Title:** The two dimensional Yang-Mills measure and its large N limit**Speaker: ** Antoine Dahlqvist (UCD)**Date:** Wednesday, 14th March 2018**Time:** 2pm**Location:** UCD, 125 Science North (JK Lab)**Abstract:**

The Yang-Mills measure is a model of mathematical physics that stems from the physics of the standard model, describing the interactions between elementary particles. We shall explain how it gives raise to random matrix models in two dimensions that are very closely related to the Brownian motion on compact Lie groups. Given a surface, playing the role of space-time, and a compact Lie group, associated to the type of interaction, it is the data of a random mapping that sends any path to a matrix of the group in a multiplicative way. From the pioneering work of G. t’Hooft, it was conjectured in the physics literature that these models simplify when, the surface being kept fixed, the dimension of the group goes to infinity. We shall see that they display different behaviors in regards of the choice of surface and how they can be analysed thanks to differential equations involving deformations of loops, known as Makeenko-Migdal equations.

**Title:** 3D positive lattice walks and spherical triangles

**Speaker:** Kilian Raschel (Tours)

**Date:** Wednesday, 11th April 2018

**Time:** 2pm

**Location:** UCD, 125 Science North (JK Lab)

**Abstract:**

In this talk we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. We focus on the critical exponent, which admits a universal formula in terms of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. Our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard factorization, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with a 10^{-10} precision.

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