Freezing transition in decaying Burgers turbulence and random matrix dualities

We reveal a phase transition with decreasing viscosity ν at ν=ν c >0 in one-dimensional decaying Burgers turbulence with a power-law–correlated random profile of Gaussian-distributed initial velocities . The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities, continuously dependent on ν, 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 the random matrix theory. They are essentially non–mean-field in nature as also demonstrated by the shock size distribution computed numerically and different from the short-range–correlated Kida model. We also provide some insights for the finite-viscosity behaviour of velocities in the latter model.