Speaker
Yan Fyodorov
(Nottingham University)
Description
Freezing transition with decreasing temperature is a generic
property of equilibrium statistical mechanics models whose
random energy landscapes are logarithmically correlated in
space. The extreme value statistics plays an important role in
elucidating the nature of such a transition. In the present work
we reveal a similar transition to take place in one-dimensional
decaying Burgers turbulence with a power-law correlated
random profile of Gaussian-distributed initial velocities
< v(x; 0)v(x'; 0) > ~ |x -x'|^{-2}, with the role of
temperature played by viscosity. The low-viscosity phase
exhibits non-Gaussian one-point probability density of
velocities, continuously dependent on viscosity, reflecting a
spontaneous one step replica symmetry breaking (RSB) in
the associated statistical mechanics problem. We obtain the
low orders cumulants analytically. Our results, which are
checked numerically, are based on combining insights in the
mechanism of the freezing transition in random logarithmic
potentials with an extension of duality relations discovered
recently in Random Matrix Theory. The presentation is
based on the work Y.V. Fyodorov, P. Le Doussal and A. Rosso
Europh. Lett. v.90 (2010) 60004.
