Triangle Seminars
Tuesday, 24 Apr 2007
Universality in critical unitary random matrix ensembles and Painleve equations
Tom Claeys
(Leuven)
Abstract:
The eigenvalues of large unitary random matrices show universal local behavior, only depending on the so-called scaling regime. It is known that in the bulk of the spectrum, local correlations of the eigenvalues are given in terms of the sine kernel, while at the edge one obtains the Airy
kernel. In certain critical random matrix ensembles, other limiting correlation kernels can occur. Near singular interior points, the limiting kernel is related to the Hastings-McLeod solution of the Painleve II equation. Near singular edge points on the other hand, one obtains a kernel related to the second member of the Painleve I hierarchy. We
describe how one can obtain these kernels rigorously in double scaling limits using the Riemann-Hilbert approach.
The eigenvalues of large unitary random matrices show universal local behavior, only depending on the so-called scaling regime. It is known that in the bulk of the spectrum, local correlations of the eigenvalues are given in terms of the sine kernel, while at the edge one obtains the Airy
kernel. In certain critical random matrix ensembles, other limiting correlation kernels can occur. Near singular interior points, the limiting kernel is related to the Hastings-McLeod solution of the Painleve II equation. Near singular edge points on the other hand, one obtains a kernel related to the second member of the Painleve I hierarchy. We
describe how one can obtain these kernels rigorously in double scaling limits using the Riemann-Hilbert approach.
Posted by: brunel
Wednesday, 25 Apr 2007
Some lessons from higher spins
📍 London
Augusto Sagnotti
(Pisa)
Topology changing transitions in N=4
📍 London
Timothy J. Hollowood
(Swansea)
Abstract:
N=4 SYM is known to have a confinement-deconfinement type phase
transition in finite volume as the temperature is raised. This phase transition has been conjectured to smoothly become the Hawking-Page transition between hot AdS space and an AdS black hole as the 't Hooft coupling becomes larger. I show that this phase transition at weak coupling is actually a topology changing transition for the VEVs of the scalar fields and Polyakov loop. This means that the high temperature phase cannot be, as previously thought, the black hole in the dual. I then argue for the existence of a new second order phase transition at a higher temperature to a new phase which has the right symmetries to be identified with the black hole. This is work based on the recent paper hep-th/0703100 with Umut Gursoy, Sean Hartnoll and Prem Kumar.
N=4 SYM is known to have a confinement-deconfinement type phase
transition in finite volume as the temperature is raised. This phase transition has been conjectured to smoothly become the Hawking-Page transition between hot AdS space and an AdS black hole as the 't Hooft coupling becomes larger. I show that this phase transition at weak coupling is actually a topology changing transition for the VEVs of the scalar fields and Polyakov loop. This means that the high temperature phase cannot be, as previously thought, the black hole in the dual. I then argue for the existence of a new second order phase transition at a higher temperature to a new phase which has the right symmetries to be identified with the black hole. This is work based on the recent paper hep-th/0703100 with Umut Gursoy, Sean Hartnoll and Prem Kumar.
Posted by: KCL