# 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.

**Speaker:** Philippe Biane (Paris)

**Title:** Gog and Magog triangles and the Schutzenberger involution

**Date:** Wednesday, 7th March 2018

**Time:** 2pm

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

**Abstract:** I will present Gog and Magog triangles which are particular cases of Gelfand-Tsetlin triangles and which appear in many models of statistical mechanics. An open problem is to find a bijection between these classes of objects. I will explain an approach to this problem based on the Schutzenberger involution.

All are welcome.

**Title:** The two dimensional Yang-Mills measure and its large N limit**Speaker: ** Antoine Dahlqvist (UCD)**Date:** Wednesday, 14th Mar 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:** Large deviations for non-interacting trapped fermions

**Speaker:** Gregory Schehr (Orsay)

**Date:** Wednesday, 21st March 2018

**Time:** 2pm

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

**Abstract:** I will consider the (quantum) spatial fluctuations of N non-interacting fermions in an isotropic d-dimensional trapping potential at zero temperature, with a special focus on hard potentials. I will study the maximal radial distance, $r_{\max}$, of the fermions from the trap center and focus on the large deviations of $r_{\max}$ away from its typical position, both to the right (right tail) and to the left (left tail). In $d=1$, this question can be studied, in several cases, thanks to an exact mapping to random matrix models, where such large deviations regimes have been well studied in the recent past. I will show that in $d>1$ the large deviation regime to the left exhibits a quite unusual, and rather universal, intermediate regime. This intermediate regime can be studied in detail using the tools of determinantal point processes.

**Speakers:** Pierre Le Doussal and Alexandre Krajenbrink (Paris)

**Title:** Large deviation tails for the KPZ equation

**Date:** Wednesday, 18th April 2018

**Time:** 1.30pm - 3pm *** Note slightly earlier than usual time ***

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

**Abstract:** We present informally recent results on the large deviations for the distribution of the height of a growing interface described by the Kardar-Parisi-Zhang equation. The large time large deviation rate function is calculated using Coulomb gas methods. The short time rate functions are also obtained for several initial conditions. The numerical determination of the rate function is in good agreement with the predictions.

**Speakers:** Paolo Facchi (Bari)**Title:** Statistical mechanics of multipartite entanglement**Date:** Tuesday, 15th May 2018**Time:** 2pm - 3pm **Location:** UCD, 125 Science North (JK Lab)**Abstract:** Entanglement is a characteristic trait of quantum mechanics. However, beyond a few simple examples mostly referring to pure bipartite states, little is known about it. The complexity of the subject is manifested e.g. in the difficulties one encounters when attempting to quantify entanglement.

In this talk I will review some statistical aspects of the entanglement of random states in high-dimensional quantum systems. First I will show the emergence of a concentration-of-measure phenomenon in the entanglement spectrum of a large bipartite system, and unveil the presence of phase transitions. Then I will consider multipartite systems and look at a characterisation of multipartite entanglement in terms of the statistics of bipartite entanglement over all balanced bipartitions.

**Speaker:** James Martin (Oxford)

**Title:** Recursive structures in the multi-type ASEP

**Date:** Wednesday, 30th May 2018

**Time:** 2pm

**Location:** UCD, AG 1.01

**Abstract:** Consider a system of N particles with labels 1,2,..., N jumping on a ring of size N. Each of the N sites contains a single particle. Whenever particle i is immediately to the right of particle j with i<j, the two particles exchange places at rate 1. I'll review various descriptions of the stationary distribution of such "multi-type totally asymmetric simple exclusion processes", for example in terms of outputs of systems of Markovian queues in series, or in terms of traces of matrix products.

Suppose we now add jumps in the reverse direction (i.e. when particle i is to the left of particle j for i<j) with some rate q in (0,1). I'll describe how the constructions above generalise to this "partially asymmetric" case. A new queueing description has various nice consequences, both algebraic (e.g. a common denominator for the stationary probabilities of the various configurations as rational functions of q) and probabilistic (descriptions of the formation of "convoys" of nearby particles with similar labels for large N).

If time permits I’ll mention some other variants (zero-range processes, Busemann functions) and some open problems.

**Speaker:** Satya Majumdar (Orsay)

**Title:** Random Convex Hulls and Extreme Value Statistics: Applications to Ecology and Animal Epidemics

**Date:** Wednesday, 6th June 2018

**Time:** 2pm - 3pm

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

**Abstract:** Convex hull of a set of points in two dimensions roughly describes the shape of the set. In this talk, I will discuss the statistical properties of the convex hull for two stochastic processes in two dimensions: (i) a set of n independent planar Brownian paths (ii) a branching Brownian motion with death. We show how to compute exactly the mean perimeter and the mean area of the convex hull in these two problems. The first problem has application in estimating the home range of an animal population of size n, while the second will be used to estimate the spatial extent of the outbreak of animal epidemics. Our result also makes an interesting connection between random geometry and extreme value statistics.

**Speaker:**Fabio Deelan Cunden (UCD)

**Title:**Universality of the weak pushed-to-pulled transition in systems with repulsive interactions

**Date:**Wednesday, 6th June 2018

**Time:**3.30pm - 4.30pm

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

**Abstract:**We consider a d-dimensional gas in canonical equilibrium under pairwise screened Coulomb repulsion and external confinement, and subject to a volume constraint (hard walls). We show that its excess free energy displays a third-order singularity separating the pushed and pulled phases, irrespective of range of the pairwise interaction, dimension and details of the confining potential. The explicit expression of the excess free energy is universal and interpolates between the Coulomb (long-range) and the delta (zero-range) interaction. The order parameter of the transition---the `electrostatic pressure' generated by the surface excess charge---is determined by invoking a fundamental energy conservation argument.

This is joint work with Paolo Facchi, Marilena Ligabò and Pierpaolo Vivo.

**Speaker:**Mylène Maïda (Lille)

**Title:**The Sine beta process: a statistical physics approach

**Date:**Wednesday, 13th June 2018

**Time:**11am

**Location:**UCD, SCN 1.25 (JK Lab)

**Abstract:**The Sine process (corresponding to $\beta =2$) is a well known determinantal point process. It appears as the bulk limit of some particle systems in various contexts (random matrix ensembles, zeros of L-funtions, growth models etc.) Its universality properties are fascinating. More recently, a family of point processes for $\beta >0$ have been introduced as the bulk limit of Gaussian $\beta$ ensembles by Valko and Virag. Much less is known as soon as $\beta$ is different from 2. In an ongoing work with David Dereudre, Adrien Hardy and Thomas Leblé, we are using classical tools of statistical mechanics to better understand the Sine beta process.

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