Timetable for Term 2: Tuesdays, 13:15-14:30 at B3.02
18.01.2022. Dom Brockington (Warwick), Brownian Motions in random velocity fields.
Abstract. We study a one-dimensional particle that is transported by a Gaussian velocity field that is white in time and rapidly decorrelating in space. The model arises as a simplified description of the trajectory of a particle in a turbulent fluid, called the Kraichnan model. The density of a cloud of such particles moving in the same environment, but each with its own independent molecular diffusivity, is expected to have fluctuations described by the KPZ equation when observed far from the origin. We will describe the regimes in which the KPZ equation is expected to appear and provide some partial results.
25.01.2022. Yuchen Liao (Warwick), Fredholm determinants and integrable differential equations I.
Abstract: The interplay between Fredholm determinants/Pfaffians and (tau functions of) hierarchies of integrable differential equations has been an active research topic in mathematical physics for the past 40 years
since the work of Jimbo-Miwa-Ueno, Brézin-Hikami, Tracy-Widom and many others. In this talk I will discuss some recent progress along this direction made by Alexandre Krajenbrink and collaborators.
I will mainly discuss a unifying construction relating Fredholm determinants for a large class of operators (of the so-called Hankel composition type) to the solution of certain first-order systems of nonlinear differential equations, Known as the Zakharov-Shabat systems. Primary examples include the Tracy-Widom (Airy) distributions for soft-edge scaling limits of random Hermitian matrices and the Rider-Sinclair distribution appearing as the scaling limit of the largest real eigenvalue of real Ginibre matrices. Time permitting, I will also discuss extensions of the construction to the hard edge of random matrices and finite-temperature generalization of the Airy kernel.
01.02.2022. Yuchen Liao (Warwick), Fredholm determinants and integrable differential equations II.
08.02.2022. Yuchen Liao (Warwick), Fredholm determinants and integrable differential equations III.
15.02.2022. Oleg Zaboronski (Warwick). I will be reading Krajenbrink, Le Doussal's paper on weak noise theory for the Kardar-Parisi-Zhang equation (KPZ) and the non-linear Schroedinger equation (NLS), reference [1] on the list. The provisional plan is: (1) Give a heuristic derivation of the large deviations principle for KPZ to explain the emergence of the imaginary time NLS; (2) Construct the Lax pair and the integrals of motion; (3) Define the reflection coefficients for the auxiliary linear problem. Solve the NLS in terms of the Fredholm determinant. The corresponding operator is built out of the reflection coefficients as a Hankel convolution. We have already seen the operators of this type in Yuchen's lectures. (4) Calculate the reflection coefficients for some particular initial conditions.
22.02.2022. Oleg Zaboronski (Warwick). I will continue reading Krajenbrink, Le Doussal's paper on weak noise theory for the Kardar-Parisi-Zhang equation (KPZ) and the non-linear Schroedinger equation (NLS), reference [1] on the list. Today I will show how to solve the imaginary time NLS in terms of resolvents of certain Hankel compositions.
01.03.2022. Matteo Mucciconi (Warwick). KPZ fixed point and KP equation.Abstract. I am going to report results by Quastel and Remenik [4], who discovered that marginals of the KPZ fixed point solve a matrix version of the KP equation. Additional connections between solutions of solvable KPZ models and the KP equation are presented.
08.03.2022. Roger Tribe (Warwick) Asymptotics for Fredholm Determinants and Fredholm Pfaffians via Kac's probabilistic method.
Abstract. This is follow up to the talk in Dario's (applied probability) seminar, giving some of the gory details on how Kac
read off asymptotics for a Fredholm Determinant via a representation in terms of a
random walk. Will FitzGerald, Oleg and I pushed this method to treat a class of
Fredholm Pfaffians.
15.03.2022. Matteo Mucciconi (Warwick) KPZ equation and KP equation.
Abstract : In my last seminar we discussed how certain explicit Fredholm determinant solutions of the KPZ equation imply that the probability distribution of the height function satisfies the KP equation. This fact certainly has consequences and in this seminar we will discuss some of them. In particular I will discuss how the KP equation can be used to compute the large deviation function of the KPZ equation under certain regimes. These results were presented by Le Doussal in
22.03.2022. Matteo Mucciconi (Warwick) KPZ equation and KP equation-II.
Provisional Plan for term 2
References
[1] Krajenbrink, Le Doussal, The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation;
[2] Ablowitz, Kaup, Newell, Segur, The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems;
[3] Krajenbrink, FROM PAINLEVE; TO ZAKHAROV-SHABAT AND BEYOND: FREDHOLM DETERMINANTS AND INTEGRO-DIFFERENTIAL HIERARCHIES;
[4] Quastel, Remenik, KP GOVERNS RANDOM GROWTH OFF A ONE DIMENSIONAL SUBSTRATE;
[5] Krajenbrink, Le Doussal, Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions.
