This term, all seminars take place Thursdays at 2pm, room MS.04 (Zeeman Building), unless indicated otherwise.

Schedule for

Daisuke Shiraishi, 08.02.2018


25.04.2019 Olga Izyumtseva (QMUL)
Self-intersection local time for compactly perturbed Wiener processes
We consider Gaussian processes obtained as the compact perturbations of Wiener process. Increments of compactly perturbed Wiener process on small time intervals have similar behaviour to the behaviour of increments of Wiener process. The law of iterated logarithm and asymptotics of small ball probabilities can be established for it. The main advantage of introduced class of Gaussian processes is the possibility to construct Rosen renormalization for the self-intersection local times in the planar case. We present the corresponding statement in terms of Fourier-Wiener transform.
The talk is based on the joint works with Andrey Dorogovtsev.

02.05.2019 Nicolas Dirr (Cardiff)
A Stochastic porous medium equation with divergence form noise
We construct nonnegative martingale solutions to the stochastic porous medium equation in one space dimension by introducing a (spatial) semi-discretization and establishing convergence, based on energy estimates.
09.05.2019 Erik Slivken (Paris)
Neighborhood Growth on the Hamming plane
We introduce a generalized growth model on two-dimensional Hamming graphs that accounts for long-range interactions. We start with a collection of occupied sites on Z_+^2. The decision to add a point at a site is made by counting the number of currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. This process can be viewed as a generalization of bootstrap percolation. We study a of number of extremal quantities. Some natural, like the smallest spanning set or the slowest spanning set; others related to the probability that a rectangle will be spanned by a given density of initially occupied sites. This is joint work with Janko Gravner, J.E. Paguyo, and David Sivakoff.

No seminars (Warwick-Cergy meeting)

30.05.2019 Matthew Dickson (Warwick)
Large Deviations and Loop Soups: HYL Interactions
Bosons are quantum particles whose collective wave-functions are symmetric under the transposition of particles. By using the Feynman-Kac formulae, boson gases can be described by random ``loop soups." These are marked point processes where the marks are random continuous loops with prescribed integer parameter times. We will describe the empirical loop-type counts using large deviation techniques, and see how the famous Bose-Einstein condensation (BEC) phenomena manifests itself from this perspective. Large deviation analysis will also allow us to derive the thermodynamic pressure for a loop-HYL model based upon a hard-sphere interaction approximation derived by Huang, Yang and Luttinger (1953). We also derive the condensate density for this interacting model.
Contact: Stefan A
06.06.2019 Sabine Boegli (Imperial College)
Schroedinger operator with non-zero accumulation points of complex eigenvalues
We consider Schroedinger operators on the whole Euclidean space or on the half-space, subject to real Robin boundary conditions. I will present the construction of a non-real potential that decays at infinity so that the corresponding Schroedinger operator has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum. This proves that the Lieb-Thirring inequalities, crucial in quantum mechanics for the proof of stability of matter, do no longer hold in the non-selfadjoint case.
Contact: Daniel
13.06.2019 Eleanor Archer (University of Warwick)
Brownian motion on looptree-type structures
The study of percolation models on random planar maps gives important insights into the behaviour of random surfaces. In this talk, we will start by reviewing some background on random maps, with particular emphasis on the Brownian universality class and associated percolation models. It turns out that in this regime, critical percolation clusters are closely related to a class of random objects called looptrees, which will be the central objects of this talk. In particular, we will outline some of their key volume growth properties, and discuss a scaling limit result for random walks on looptrees, including giving a construction of the limiting diffusion. We will conclude by exploring applications of these looptree results, for example to outerplanar maps, and to the study of random walks on critical percolation clusters.
Contact: Daniel
20.06.2019 Nikos Zygouras (University of Warwick)
On the Kardar-Parisi-Zhang equation and universality
It was proposed by Kardar, Parisi and Zhang in the 1980s that a large class of randomly growing interfaces exhibit universal fluctuations described mathematically by a nonlinear stochastic partial differential equation, which is now known as the Kardar-Parisi-Zhang or KPZ equation. Examples of physical systems which fall in this class are percolation of liquid in porous media, growth of bacteria colonies, currents in one dimensional traffic or liquid systems, liquid crystals etc. Understanding this physical proposal has given a remarkable impetus into mathematics, motivating the development of new mathematical theories and surprising links between different disciplines.
The fluctuations of systems within the KPZ class are governed by exponents and distributions that lie outside the realm of the classical central limit theorem. In dimension one these are linked to laws emerging from random matrix theory. In higher dimensions, however, even a prediction on the exponents elude us.
I will review some of our current understanding and challenges around the one dimensional KPZ and present some first steps in dimension two.

27.06.2019 Chenjie Fan (University of Chicago)
On well-posedness of the stochastic non-linear Schrödinger equation
We will present some recent work on the well posedness of stochastic mass critical NLS. This is joint work with Weijun Xu.
Contact: Vedran
18.07.2019 Inés Armendáriz (University of Buenos Aires)
Gaussian random permutations and the boson point process
We construct an infinite volume spatial random permutation associated to a Gaussian Hamiltonian. These measures are parametrized by the point density and the temperature. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Bose-Einstein condensation occurs for dimensions 3 or larger, and above a critical density. Each finite cycle of the permutation induces a loop of points. For subcritical densities we define the spatial random permutation as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of doubly-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2010). For dimensions greater than or equal to 3 and supercritical densities, we define the spatial permutation as the superposition of independent realizations of the Gaussian loop soup at critical density and Gaussian random interlacements at the remaining density. The resulting measure satisfies a Markov property and it is Gibbs for the Gaussian Hamiltonian. We show that the point marginal has the same distribution as the boson point process introduced by Shirai-Takahashi (2003) in the subcritical case, and by Tamura-Ito (2007) in the supercritical case.
Joint work with P.A. Ferrari and S. Yuhjtman.

Contact: Stefan G