Freezing Transition in Decaying Burgers Turbulence and Random Matrix Dualities

We reveal a phase transition with decreasing viscosity $\nu$ at \nu=\nu_c>0 in one-dimensional decaying Burgers turbulence with a power-law correlated random profile of Gaussian-distributed initial velocities \sim|x-x'|^{-2}. The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities, continuously dependent on \nu, 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. 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, itself well described by a mean field one step RSB ansatz. We also provide some insights for the finite viscosity behaviour of velocities in the latter model.Comment: Published version, essentially restructured & misprints corrected. 6 pages, 5 figure

Shock statistics in higher-dimensional Burgers turbulence

We conjecture the exact shock statistics in the inviscid decaying Burgers equation in D>1 dimensions, with a special class of correlated initial velocities, which reduce to Brownian for D=1. The prediction is based on a field-theory argument, and receives support from our numerical calculations. We find that, along any given direction, shocks sizes and locations are uncorrelated.Comment: 4 pages, 8 figure

Pre-freezing of multifractal exponents in Random Energy Models with logarithmically correlated potential

Boltzmann-Gibbs measures generated by logarithmically correlated random potentials are multifractal. We investigate the abrupt change ("pre-freezing") of multifractality exponents extracted from the averaged moments of the measure - the so-called inverse participation ratios. The pre-freezing can be identified with termination of the disorder-averaged multifractality spectrum. Naive replica limit employed to study a one-dimensional variant of the model is shown to break down at the pre-freezing point. Further insights are possible when employing zero-dimensional and infinite-dimensional versions of the problem. In particular, the latter version allows one to identify the pattern of the replica symmetry breaking responsible for the pre-freezing phenomenon.Comment: This is published version, 11 pages, 1 figur

Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of 1/f1/f Noises generated by Gaussian Free Fields

We compute the distribution of the partition functions for a class of one-dimensional Random Energy Models (REM) with logarithmically correlated random potential, above and at the glass transition temperature. The random potential sequences represent various versions of the 1/f noise generated by sampling the two-dimensional Gaussian Free Field (2dGFF) along various planar curves. Our method extends the recent analysis of Fyodorov Bouchaud from the circular case to an interval and is based on an analytical continuation of the Selberg integral. In particular, we unveil a {\it duality relation} satisfied by the suitable generating function of free energy cumulants in the high-temperature phase. It reinforces the freezing scenario hypothesis for that generating function, from which we derive the distribution of extrema for the 2dGFF on the $[0,1]$ interval. We provide numerical checks of the circular and the interval case and discuss universality and various extensions. Relevance to the distribution of length of a segment in Liouville quantum gravity is noted.Comment: 25 pages, 12 figures Published version. Misprint corrected, references and note adde

Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks

We use the Real Space Renormalization Group (RSRG) method to study extreme value statistics for a variety of Brownian motions, free or constrained such as the Brownian bridge, excursion, meander and reflected bridge, recovering some standard results, and extending others. We apply the same method to compute the distribution of extrema of Bessel processes. We briefly show how the continuous time random walk (CTRW) corresponds to a non standard fixed point of the RSRG transformation.Comment: 24 pages, 5 figure

One step replica symmetry breaking and extreme order statistics of logarithmic REMs

Building upon the one-step replica symmetry breaking formalism, duly understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclidean-space logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted decorated exponential Poisson point process. The distribution of the random shift is determined solely by the large-distance ("infra-red", IR) limit of the model, and is equal to the free energy distribution at the critical temperature up to a translation. the decoration process is determined solely by the small-distance ("ultraviolet", UV) limit, in terms of the biased minimal process. Our approach provides connections of the replica framework to results in the probability literature and sheds further light on the freezing/duality conjecture which was the source of many previous results for log-REMs. In this way we derive the general and explicit formulae for the joint probability density of depths of the first and second minima (as well its higher-order generalizations) in terms of model-specific contributions from UV as well as IR limits. In particular, we show that the second min statistics is largely independent of details of UV data, whose influence is seen only through the mean value of the gap. For a given log-correlated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of $1/f$-noise.Comment: 44 pages, 6 figure

Finite-Temperature Free Fermions and the Kardar-Parisi-Zhang Equation at Finite Time

We consider the system of $N$ one-dimensional free fermions confined by a harmonic well $V(x) = m\omega^2 {x^2}/{2}$ at finite inverse temperature $\beta = 1/T$. The average density of fermions $\rho_N(x,T)$ at position $x$ is derived. For $N \gg 1$ and $\beta \sim {\cal O}(1/N)$, $\rho_N(x,T)$ is described by a scaling function interpolating between a Gaussian at high temperature, for $\beta \ll 1/N$, and the Wigner semi-circle law at low temperature, for $\beta \gg N^{-1}$. In the latter regime, we unveil a scaling limit, for $\beta {\hbar \omega}= b N^{-1/3}$, where the fluctuations close to the edge of the support, at $x \sim \pm \sqrt{2\hbar N/(m\omega)}$, are described by a limiting kernel $K^{\rm ff}_b(s,s')$ that depends continuously on $b$ and is a generalization of the Airy kernel, found in the Gaussian Unitary Ensemble of random matrices. Remarkably, exactly the same kernel $K^{\rm ff}_b(s,s')$ arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time $t$, with the correspondence $t= b^3$.Marcheurs Browniens répulsifs et matrices aléatoiresParis Sciences et Lettre

Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature

We consider N non-interacting fermions in an isotropic d-dimensional harmonic trap. We compute analytically the cumulative distribution of the maximal radial distance of the fermions from the trap center at zero temperature. While in d = 1 the limiting distribution (in the large N limit), properly centered and scaled, converges to the squared Tracy–Widom distribution of the Gaussian unitary ensemble in random matrix theory, we show that for all d > 1, the limiting distribution converges to the Gumbel law.These limiting forms turn out to be universal, i.e. independent of the details of the trapping potential for a large class of isotropic trapping potentials. We also study the position of the right-most fermion in a given direction in d dimensions and, in the case of a harmonic trap, the maximum momentum, and show that they obey similar Gumbel statistics. Finally, we generalize these results to low but finite temperature