DEPARTMENT OF MATHEMATICS, UPPSALA UNIVERSITY | |

Anthony Metcalfe, Uppsala University, Sweden

Universal edge fluctuations of discrete interlaced particle systems

A discrete Gelfand-Tsetlin pattern is a configuration of particles in

In this talk, we consider the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of a fixed size where the particles on the top row are in deterministic positions. This measure arises naturally as an equivalent description of the uniform probability measure on the set of all tilings of certain polygons with lozenges, for example, the regular hexagon. These systems have a determinantal structure.

We consider the asymptotic behaviour of such systems as the size increases under the assumption that the empirical distribution of the deterministic particles on the top row converges weakly. We consider the asymptotic `shape' of such systems. We provide parameterisations of the asymptotic boundary (the edge) and investigate the local geometric properties of the resulting curves. We also consider the local asymptotic behaviour of particles close to the edge, and we confirm universal edge asymptotic behaviour for `typical' edge points.

Alexander Vasiliev, University of Bergen, Norway

Slit holomorphic stochastic flows

We use general Loewner theory to define general slit Loewner chains in the unit disk, which in the stochastic case lead to slit holomorphic stochastic flows. Radial, chordal and dipolar SLE are classical examples of such flows. Our approach, however, allows to construct new processes of SLE type that possess conformal invariance and some sort of the domain Markov property. The local behavior of these processes is similar to that of classical SLEs.

Joint project with Nam-Gyu Kang, George Ivanov and Alexey Tochin

Henning Sulzbach

Convergence and properties of random self-similar trees

In my talk, I discuss real trees arising as scaling limits for random large discrete trees which satisfy a certain distributional self-similarity. The main application is the analysis of limiting objects in the problem of random dissections of the disk. The topics include convergence of compact metric spaces in the so-called Gromov-Hausdorff-Prokhorov topology as well as characterizations of trees by stochastic fixed-point equations. If time permits, I will also present results about their fractal dimensions. The talk will be based on joint work with Nicolas Broutin, INRIA, Paris.

Dimitris Cheliotis

Triangular matrices and biorthogonal ensembles

We discuss distributional properties of the singular values of certain triangular random matrices. For general i.i.d entries, we study the empirical distribution of the singular values, and for certain choices on the distribution of the entries we show their relation with the biorthogonal Laguerre ensemble.

Rostyslav Kozhan

Multiplicative non-Hermitian perturbations of Jacobi matrices and of classical random matrix ensembles

We fully classify possible eigenvalue configurations of rank-one multiplicative non-Hermitian perturbations of Jacobi matrices. We provide a Hermite-Biehler type theorem for the characteristic polynomial and an application in random matrix theory.

Since the proof employs just the basic linear algebra and complex analysis, the talk should be accessible to PhD or even undergraduate students. This is a joint work with M. Tyaglov.

Achilles Tertikas

On the Hardy constant of non-convex domains

The Hardy constant of a simply connected domain Ω ∈ ℝ

∫

After the work of Ancona where the universal lower bound 1/16 was obtained for simply connected planar domains, there has been a substantial interest on computing or estimating the Hardy constant of simply connected domains. In this talk I will review on some recent work, on computing Hardy constants of non convex simply connected domains. Particular attention will be paid on computing Hardy constants of non convex simply connected planal domains This is part of a joint work with G Barbatis (University of Athens).

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