226 research outputs found
Distribution of velocities in an avalanche
For a driven elastic object near depinning, we derive from first principles
the distribution of instantaneous velocities in an avalanche. We prove that
above the upper critical dimension, d >= d_uc, the n-times distribution of the
center-of-mass velocity is equivalent to the prediction from the ABBM
stochastic equation. Our method allows to compute space and time dependence
from an instanton equation. We extend the calculation beyond mean field, to
lowest order in epsilon=d_uc-d.Comment: 4 pages, 2 figure
Hessian spectrum at the global minimum of high-dimensional random landscapes
Using the replica method we calculate the mean spectral density of the
Hessian matrix at the global minimum of a random dimensional
isotropic, translationally invariant Gaussian random landscape confined by a
parabolic potential with fixed curvature . Simple landscapes with
generically a single minimum are typical for , and we show that
the Hessian at the global minimum is always {\it gapped}, with the low spectral
edge being strictly positive. When approaching from above the transitional
point separating simple landscapes from 'glassy' ones, with
exponentially abundant minima, the spectral gap vanishes as .
For the Hessian spectrum is qualitatively different for 'moderately
complex' and 'genuinely complex' landscapes. The former are typical for
short-range correlated random potentials and correspond to 1-step
replica-symmetry breaking mechanism. Their Hessian spectra turn out to be again
gapped, with the gap vanishing on approaching from below with a larger
critical exponent, as . At the same time in the 'most complex'
landscapes with long-ranged power-law correlations the replica symmetry is
completely broken. We show that in that case the Hessian remains gapless for
all values of , indicating the presence of 'marginally stable'
spatial directions. Finally, the potentials with {\it logarithmic} correlations
share both 1RSB nature and gapless spectrum. The spectral density of the
Hessian always takes the semi-circular form, up to a shift and an amplitude
that we explicitly calculate.Comment: 28 pages, 1 figure; a brief summary of main results is added to the
introductio
Freezing Transition in Decaying Burgers Turbulence and Random Matrix Dualities
We reveal a phase transition with decreasing viscosity 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
Log-correlated Random Energy Models with extensive free energy fluctuations: pathologies caused by rare events as signatures of phase transitions
We address systematically an apparent non-physical behavior of the free
energy moment generating function for several instances of the logarithmically
correlated models: the Fractional Brownian Motion with Hurst index
(fBm0) (and its bridge version), a 1D model appearing in decaying Burgers
turbulence with log-correlated initial conditions, and finally, the
two-dimensional logREM introduced in [Cao et al., Phys.Rev.Lett.,118,090601]
based on the 2D Gaussian free field (GFF) with background charges and directly
related to the Liouville field theory. All these models share anomalously large
fluctuations of the associated free energy, with a variance proportional to the
log of the system size. We argue that a seemingly non-physical vanishing of the
moment generating function for some values of parameters is related to the
termination point transition (a.k.a pre-freezing). We study the associated
universal log corrections in the frozen phase, both for log-REMs and for the
standard REM, filling a gap in the literature. For the above mentioned
integrable instances of logREMs, we predict the non-trivial free energy
cumulants describing non-Gaussian fluctuations on the top of the Gaussian with
extensive variance. Some of the predictions are tested numerically.Comment: 17 pages, 4 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
Exponential number of equilibria and depinning threshold for a directed polymer in a random potential
By extending the Kac-Rice approach to manifolds of finite internal dimension,
we show that the mean number
of all possible equilibria
(i.e. force-free configurations, a.k.a. equilibrium points) of an elastic line
(directed polymer), confined in a harmonic well and submitted to a quenched
random Gaussian potential in dimension , grows exponentially
with its
length . The growth rate is found to be directly related to the
generalised Lyapunov exponent (GLE) which is a moment-generating function
characterising the large-deviation type fluctuations of the solution to the
initial value problem associated with the random Schr\"odinger operator of the
1D Anderson localization problem. For strong confinement, the rate is small
and given by a non-perturbative (instanton, Lifshitz tail-like) contribution to
GLE. For weak confinement, the rate is found to be proportional to the
inverse Larkin length of the pinning theory. As an application, identifying the
depinning with a landscape "topology trivialization" phenomenon, we obtain an
upper bound for the depinning threshold , in the presence of an applied
force, for elastic lines and -dimensional manifolds, expressed through the
mean modulus of the spectral determinant of the Laplace operators with a random
potential. We also discuss the question of counting of stable equilibria.
Finally, we extend the method to calculate the asymptotic number of equilibria
at fixed energy (elastic, potential and total), and obtain the (annealed)
distribution of the energy density over these equilibria (i.e. force-free
configurations). Some connections with the Larkin model are also established.Comment: v1: 6 pages main text + 14 pages supplemental material, 10 figures.
v2: LaTeX, 79 pages, 18 eps figures, new material (Sections 6, 10, 11 &
Appendices C, E, F, G
Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes
We study three instances of log-correlated processes on the interval: the
logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the
Gaussian log-correlated potential in presence of edge charges, and the
Fractional Brownian motion with Hurst index (fBM0). In previous
collaborations we obtained the probability distribution function (PDF) of the
value of the global minimum (equivalently maximum) for the first two processes,
using the {\it freezing-duality conjecture} (FDC). Here we study the PDF of the
position of the maximum through its moments. Using replica, this requires
calculating moments of the density of eigenvalues in the -Jacobi
ensemble. Using Jack polynomials we obtain an exact and explicit expression for
both positive and negative integer moments for arbitrary and
positive integer in terms of sums over partitions. For positive moments,
this expression agrees with a very recent independent derivation by Mezzadri
and Reynolds. We check our results against a contour integral formula derived
recently by Borodin and Gorin (presented in the Appendix A from these authors).
The duality necessary for the FDC to work is proved, and on our expressions,
found to correspond to exchange of partitions with their dual. Performing the
limit and to negative Dyson index , we obtain the
moments of and give explicit expressions for the lowest ones. Numerical
checks for the GUE polynomials, performed independently by N. Simm, indicate
encouraging agreement. Some results are also obtained for moments in Laguerre,
Hermite-Gaussian, as well as circular and related ensembles. The correlations
of the position and the value of the field at the minimum are also analyzed.Comment: 64 page, 5 figures, with Appendix A written by Alexei Borodin and
Vadim Gorin; The appendix H in the ArXiv version is absent in the published
versio
- …