# Mathematical Physics and Probability Reading seminar

(Covering topics related to random matrices, representation theory, integrable systems and interacting stochastic particle systems)

## Term 1

3rd October. Roger Tribe (Warwick). A short non-solution to the KPZ problem.

Abstract. I will go through a very short formal derivation of the Tracy-Widom distribution for the fluctuations of the KPZ interface and descirbe the remaining gap between the formal steps and the rigorous proof.

10th October. Will FitzGerald (Warwick). Reflected Brownian motions and random matrix theory.

Abstract. Systems of Brownian motions with one-sided reflection in the KPZ universality class are described by distributions from random matrix theory. We will show how this connection can be obtained directly from Schutz-type transition probabilities. This can be extended in the presence of a wall, drifts and for particle systems connected to last passage percolation.

References.

Kurt Johansson, A multi-dimensional Markov chain and the Meixner ensemble
https://projecteuclid.org/euclid.afm/1485907101
Jon Warren, Dyson's Brownian motions, intertwining and interlacing
https://projecteuclid.org/euclid.ejp/1464818490
Alexei Borodin, Patrik Ferrari, Michael Prahofer, Tomohiro Sasamoto and Jon Warren, Maximum of Dyson Brownian motion and non-colliding systems with a boundary
https://projecteuclid.org/euclid.ecp/1465234756

17th October Will FitzGerald (Warwick). Reflected Brownian motions and random matrix theory-II.

24th October. Theodoros Assiotis (Warwick). Matrix Bougerol identity and the Hua-Pickrell measures.

Abstract: I will talk about how one can extend to the matrix setting two celebrated one-dimensional identities of Bougerol and Dufresne related to exponential functionals of Brownian motion.

31st October. Elia Bisi (Warwick). Point-to-line last passage percolation via symplectic Schur functions.

Abstract. We discuss a new formula, in terms of symplectic Schur functions, for the point-to-line last passage percolation with exponentially distributed waiting times. We then show how to derive, in the scaling limit, Sasamoto’s Fredholm determinant formula for the GOE Tracy-Widom distribution. If time permits, we also go through the last passage percolation model in the point-to-half-line geometry, where the asymptotic distribution is instead the marginal of the $Airy_{2 \to 1}$ process.

7th November. Elia Bisi (Warwick). Point-to-line last passage percolation via symplectic Schur functions-II. (Point-to-line polymers and orthogonal Whittaker functions.)

Abstract. We study a one dimensional directed polymer model in an inverse-gamma random environment, known as the log-gamma polymer, in the point-to-line geometry. Via the use of A.N.Kirillov’s geometric Robinson-Schensted-Knuth correspondence, we compute the Laplace transform of the partition function as an integral of orthogonal Whittaker functions. In the zero temperature limit, we recover the formula we discussed in the previous talk for the distribution of the point-to-line last passage percolation with exponentially distributed waiting times.

14th November. No seminar

21st November. Theodoros Assiotis (Warwick). 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.

28th November

5th December