Anthony P. Metcalfe, Paris 6

Location :

INRIA,  
meeting room « verte (green) », 
5th floor, 23 avenue d'Italie, 
75013 PARIS

Access: https://intranet.paris-rocquencourt.inria.fr/pratique/pre-antenne-parisienne/comment-se-rendre-a-la-pre-antenne-paris-italie/ 

Abstract :

Probability distributions on sets of interlaced particles arise naturally in the study of many 
systems, most famously when considering the eigenvalues of the minors of the Gaussian Unitary 
Ensemble (GUE). In this talk we will consider a related system of interlaced particles, namely 
standard Gelfand-Tsetlin patterns.

A standard Gelfand-Tsetlin pattern of size n is a triangluar array of particles with n particles on the 
top level, n-1 particles on the first level beneath the top, n-2 on the second etc. These particles 
interlace in the sense that between any two consecutive particles on a level, there is exactly 
one particle on the level beneath.

Imposing the uniform measure on the set of all Gelfand-Tsetlin patterns that arise from a fixed top 
row, we show that the particles in the pattern have a determinantal structure and calculate the 
correlation kernel. Letting the size of the pattern increase, we show, under some regularity 
assumptions, that locally the particles on a fixed level in the bulk of the pattern behave 
asymptotically like a determinantal random point field, with correlation kernel given by the sine 
kernel. 

This is achieved using techniques from saddle point analysis.

Host :

INRIA