444 research outputs found
On Hubbard-Stratonovich Transformations over Hyperbolic Domains
We discuss and prove validity of the Hubbard-Stratonovich (HS) identities
over hyperbolic domains which are used frequently in the studies on disordered
systems and random matrices. We also introduce a counterpart of the HS identity
arising in disordered systems with "chiral" symmetry. Apart from this we
outline a way of deriving the nonlinear -model from the gauge-invariant
Wegner orbital model avoiding the use of the HS transformations.Comment: More accurate proofs are given; a few misprints are corrected; a
misleading reference and a footnote in the end of section 2.2 are remove
Random Energy Model with complex replica number, complex temperatures and classification of the string's phases
The results by E. Gardner and B.Derrida have been enlarged for the complex
temperatures and complex numbers of replicas. The phase structure is found.
There is a connection with string models and their phase structure is analyzed
from the REM's point of view.Comment: 11 pages,revte
Induced vs Spontaneous Breakdown of S-matrix Unitarity: Probability of No Return in Quantum Chaotic and Disordered Systems
We investigate systematically sample-to sample fluctuations of the
probability of no return into a given entrance channel for wave
scattering from disordered systems. For zero-dimensional ("quantum chaotic")
and quasi one-dimensional systems with broken time-reversal invariance we
derive explicit formulas for the distribution of , and investigate
particular cases. Finally, relating to violation of S-matrix unitarity
induced by internal dissipation, we use the same quantity to identify the
Anderson delocalisation transition as the phenomenon of spontaneous breakdown
of S-matrix unitarity.Comment: This is the published version, with a few modifications added to the
last par
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
The decay of photoexcited quantum systems: a description within the statistical scattering model
The decay of photoexcited quantum systems (examples are photodissociation of
molecules and autoionization of atoms) can be viewed as a half-collision
process (an incoming photon excites the system which subsequently decays by
dissociation or autoionization). For this reason, the standard statistical
approach to quantum scattering, originally developed to describe nuclear
compound reactions, is not directly applicable. Using an alternative approach,
correlations and fluctuations of observables characterizing this process were
first derived in [Fyodorov YV and Alhassid Y 1998 Phys. Rev. A 58, R3375]. Here
we show how the results cited above, and more recent results incorporating
direct decay processes, can be obtained from the standard statistical
scattering approach by introducing one additional channel.Comment: 7 pages, 2 figure
On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
Given any fixed positive semi-definite diagonal matrix
we derive the explicit formula for the density of complex eigenvalues for
random matrices of the form } where the random unitary
matrices are distributed on the group according to the Haar
measure.Comment: 10 pages, 1 figur
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 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 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
Scaling and the center of band anomaly in a one-dimensional Anderson model with diagonal disorder
We resolve the problem of the violation of single parameter scaling at the
zero energy of the Anderson tight-binding model with diagonal disorder. It
follows from the symmetry properties of the tight-binding Hamiltonian that this
spectral point is in fact a boundary between two adjacent bands. The states in
the vicinity of this energy behave similarly to states at other band
boundaries, which are known to violate single parameter scaling.Comment: revised version, 4 pages, 2 figures, revte
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