# Probability seminars 2018/2019

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**Speaker:** Antoine Dahlqvist (UCD)

**Title:** Free probability and permutation invariant random matrices

**Date:** Wednesday, 10th October 2018

**Time:** 3pm

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

**Abstract:** Consider two independent sequences of Hermitian random matrices indexed by their size. Assume that both empirical measures of the eigenvalues converge weakly. What can be said asymptotically about the eigenvalues of their sum, of their product? When one of the sequence is invariant in law by unitary conjugation, an answer can be given thanks to the notion of free independence, as introduced by D. Voiculescu. In this talk, we shall consider what happens when this very assumption is dropped. We will review recent progresses addressing this question, focusing on ensembles where unitary invariance is weakened into the invariance by conjugation with permutation matrices. Surprisingly, we will see that free probability is still relevant for this problem.

**Speaker:** Elia Bisi (UCD)

**Title:** How long does it take to go through a sequence of N queues?

**Date:** Wednesday, 17th October 2018

**Time:** 3pm

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

**Abstract:** We present a model for customers queueing in a sequence of service stations, where the service times are independent exponential random variables. We reformulate this in terms of systems of interacting particles (totally asymmetric simple exclusion process), stochastic growth (corner growth model), and lattice paths (point-to-line last passage percolation). We then derive an exact formula for these equivalent models in terms of representation theoretic functions known as symplectic characters. Thanks to such a rich algebraic structure, in the large N limit we obtain fluctuations of order cube root of N and a limiting distribution from random matrix theory. This central limit theorem (very different from the classical Gaussian one!) permits setting our models in the framework of the KPZ universality class.

**Speaker:** Jon Warren (Warwick)**Title:** Random matrices and point to line last passage percolation**Date:** Wednesday, 14th November 2018**Time: ** 3pm**Location:** UCD, SCN 1.25 (JK Lab)**Abstract:** The all-time supremum of a Brownian motion with negative drift is exponentially distributed. A generalization of this classical fact to random matrices may be obtained by combining work of Nguyen and Remenik with work of Baik and Rains to show that the supremum of the largest eigenvalue of a Hermitian Brownian motion with drift is equal in distribution to a certain function of several independent exponentially distributed random variables. Moreover this function is a point to line last passge time. I will discuss a multidimensional extension of this identity in law which involves the invariant measure of a system of reflecting Brownian motions with a wall.

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**Title: **Wishart matrices and Hurwitz numbers

** Speakers: **Antoine Dahlqvist (UCD)

**Date**: Wednesday 21^{st} November 2018

** Time**: 3pm

** Location: **Room 1.25, Science Centre North

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**Abstract:**

From the 70’s onwards, it has been understood that random matrices built with Gaussian variables, such as the one of the Gaussian unitary ensemble, could be used to compute generating series of combinatorial objects such as discrete surfaces, indexed by their genus. Besides, random matrices have been used to model a variety of scattering phenomena in complex systems including heavy nuclei, disordered mesoscopic conductors, and chaotic quantum billiards. Wishart matrices (also well known as the Laguerre unitary ensemble distribution) is a model with many applications that belongs to both settings. I will explain how this relation was recently developed and used to solve an integrality conjecture in quantum chaotic transport, thanks to formulas in algebraic combinatorics. We shall see that it is based on a rich interplay between random matrices and the combinatorics of factorisations in the symmetric group, counted by the so-called monotone Hurwitz numbers.

Joint work with Fabio Deelan Cunden and Neil O’Connell.

**Title: ** Free fermions and α-determinantal processes

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

**Date:**Wednesday 28th November 2018

**Time:**3pm

**Location:**Room 1.25, Science Centre North

**Abstract:**

Determinantal point processes were introduced in the '70s as a consistent description of non-interacting fermions in quantum mechanics. Determinantal processes arise naturally in several other settings, including eigenvalues of random matrices and nonintersecting paths. Another, perhaps not so well-known class of point processes are the so-called α-determinantal processes. Using the Gaussian case (harmonic oscillator) as paradigmatic example, I will illustrate a new limit procedure to construct α-determinantal processes out of fermionic processes. Joint work with Satya N. Majumdar and Neil O’Connell.

**Speaker:** Nial Friel

**Title:** Informed sub-sampling MCMC: Approximate Bayesian inference for large datasets

**Date: ** Wednesday, 23rd January 2019

**Time: ** 3pm

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

**Abstract:** This talk introduces a framework for speeding up Bayesian inference for large datasets. We design a Markov chain whose transition kernel uses an (unknown) fraction of (fixed size) of the available data that is randomly refreshed throughout the algorithm. Inspired by the Approximate Bayesian Computation (ABC) literature, the subsampling process is guided by

the fidelity to the observed data, as measured by summary statistics. The resulting algorithm, Informed Sub-Sampling MCMC (ISS-MCMC), is a generic and flexible approach which, contrary to existing scalable methodologies, preserves the simplicity of the Metropolis-Hastings algorithm. Even though exactness is lost, i.e. the chain distribution approximates the posterior, we study and quantify theoretically this bias and show on a diverse set of examples that it yields excellent performances when the computational budget is limited. This is joint work with Florian Maire (Montreal) and Pierre Alquier (INSAE, Paris).

Reference:

Maire, F., Friel, N. & Alquier, P. Stat Comput (2018). https://doi.org/10.1007/s11222-018-9817-3

**Speaker:** Ken Duffy (Maynooth)

**Title:** Reliable communication over noisy channels by Guessing Random Additive Noise Decoding (GRAND)

**Date:** Wednesday, 30th January 2019

**Time: ** 2pm

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

**Abstract:** In 1948 Claude Shannon published his remarkable paper "A Mathematical Theory of Communication", which formed the basis for the digital communication revolution that was to follow, and gave rise to the field of Information Theory. As part of that ground-breaking work, he identified the greatest rate at which data can be communicated over a noisy channel, and provided an algorithm for achieving it. Despite its mathematical elegance, his algorithm is impractical, and much research in the intervening 70 years has focused on identifying practical approaches that enable reliable communication at high rates. That work is ongoing and, for example, Polar Codes, first introduced by Erdal Arikan in 2009, have recently been adopted into the 5G cellular standard.

In this talk we revisit this classical problem through the lens of a new universal channel decoding algorithm called GRAND (Guessing Random Additive Noise Decoding), which we introduced in 2018. GRAND has unusual theoretical and practical features that set it apart from earlier approaches. This talk will provide an introduction to the problem of reliable communication, before delving into the gory detail of the mathematics that underpins the algorithm's theoretical analysis, a probabilistic topic called Guesswork. Analysis of GRAND provides an alternate means for proving Shannon's original coding theorem, giving rise to several new insights along the way.

The talk is based on a work programme with Muriel Medard and her group (MIT). Open theoretical questions will be discussed as well as, time permitting, ongoing engineering efforts in collaboration with Anantha Chandrakasan (MIT), Rabia Yazicigil (BU) and their groups.

**Speaker:** Jonas Arista (UCD)

**Title:** Loop-erased walks and random matrices

**Date:** Wednesday, 6th February 2019

**Time:** 2pm

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

**Abstract:** It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There is a generalisation of the reflection principle for more general (e.g. planar) processes, due to S. Fomin, in which the non-intersection condition is replaced by a condition involving loop-erased paths. We show that in the context of independent Brownian motions in suitable planar domains, this also has close connections to random matrices. We also extend Fomin's identity to the affine setting and show that in this case, by considering independent Brownian motions in an annulus, one obtains a novel interpretation of the circular orthogonal ensemble. Joint work with Neil O’Connell.

**Speaker:** Neil Dobbs

**Title:** Kneading dough badly

**Date: ** Wednesday, 13th February 2019

**Time:** 2pm

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

**Abstract:** A stretch-and-fold operation can be modelled, in dimension one, by a quadratic map. Iterating the map, one can study the properties of the trajectory of a randomly chosen initial point. These turn out to depend rather strongly on the quadratic parameter. I will present some classical and some more recent results and maybe even an open question.

**Speaker:** Andrew Smith (UCD)

**Title:** Robust Loss Distributions

**Date:** Wednesday, 20th February 2019

**Time:** 2pm

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

**Abstract:** Estimating the distribution of insurance claim amounts is required for many purposes, including the assessment of capital adequacy with regard to high distribution percentiles. The most common methodology is to fit a two-parameter exponential family of positive distributions. For this purpose, the method of maximum likelihood is known to be the most efficient, although other techniques such as the method of moments may be chosen for ease of implementation. Our work looks at the situation when the loss data has come from an unknown member of a list of exponential families. In this case, the methodology includes selecting from the list of families as well estimating the parameters. Maximum likelihood can perform poorly for estimating high percentiles, if the model family is mis-specified. The method of moments is generally more robust, in a minimax sense.

**Speaker:** Larbi Alili (Warwick)

**Title:** Space time inversions and Kelvin Transform

**Date: ** Wednesday, 27th February 2019

**Time:** 2pm

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

**Abstract:** We show that a space time inversion of a strong Markov process X implies the existence of a Kelvin transform of harmonic functions. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions which are often not the spherical ones and excessive functions are given explicitly. We treat in details the examples that include non-colliding Bessel particles, Wishart processes and Dyson Brownian Motion.

**Speaker:** Ben Hambly (Oxford)

**Title:** Particle systems and systemic risk

**Date:** Wednesday, 6th March 2019

**Time:** 2pm

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

**Abstract:** Motivated by some simple financial models for systemic risk we consider a system of diffusing particles which all experience a downward jump when one of them hits 0 and is absorbed. By taking a large particle limit we can derive a McKean-Vlasov equation for the system. We discuss the loss process, the proportion of absorbed particles, and show the variety of behaviour that can occur.

**Speaker:** Christian Korff (Glasgow)

**Title: ** Cylindric symmetric functions and positivity

**Date: ** Wednesday, 27th March 2019

**Time: ** 2pm

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

**Abstract:** The ring of symmetric functions takes centre stage in many areas of algebraic combinatorics and probability. What happens if one evaluates symmetric functions at roots of unity? In this talk we will give an overview of the connection between the latter question and the construction of cylindric symmetric functions, which are given as weighted sums of reverse plane partitions which wrap around a cylinder of circumference n where n is the order of the root of unity. The expansion of cylindric symmetric functions in standard bases of the ring of symmetric functions does not always yield non-negative coefficients but we show that the cylindric symmetric functions form so-called "positive subcoalgebras" of the ring of the symmetric functions when viewed as a Hopf algebra. (These terms will be explained.) Here positivity means that the structure constants of the coalgebra are non-negative integers which we show to be equal to Gromov-Witten invariants and tensor multiplicities of the generalised symmetric group.

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