4,780 research outputs found
Is the Riemann zeta function in a short interval a 1-RSB spin glass ?
Fyodorov, Hiary & Keating established an intriguing connection between the
maxima of log-correlated processes and the ones of the Riemann zeta function on
a short interval of the critical line. In particular, they suggest that the
analogue of the free energy of the Riemann zeta function is identical to the
one of the Random Energy Model in spin glasses. In this paper, the connection
between spin glasses and the Riemann zeta function is explored further. We
study a random model of the Riemann zeta function and show that its two-overlap
distribution corresponds to the one of a one-step replica symmetry breaking
(1-RSB) spin glass. This provides evidence that the local maxima of the zeta
function are strongly clustered.Comment: 20 pages, 1 figure, Minor corrections, References update
Stochastic Stability: a Review and Some Perspectives
A review of the stochastic stability property for the Gaussian spin glass
models is presented and some perspectives discussed.Comment: 12 pages, typos corrected, references added. To appear in Journal of
Statistical Physics, Special Issue for the 100th Statistical Mechanics
Meetin
Short-range spin glasses and Random Overlap Structures
Properties of Random Overlap Structures (ROSt)'s constructed from the
Edwards-Anderson (EA) Spin Glass model on with periodic boundary
conditions are studied. ROSt's are random matrices whose entries
are the overlaps of spin configurations sampled from the Gibbs measure. Since
the ROSt construction is the same for mean-field models (like the
Sherrington-Kirkpatrick model) as for short-range ones (like the EA model), the
setup is a good common ground to study the effect of dimensionality on the
properties of the Gibbs measure. In this spirit, it is shown, using translation
invariance, that the ROSt of the EA model possesses a local stability that is
stronger than stochastic stability, a property known to hold at almost all
temperatures in many spin glass models with Gaussian couplings. This fact is
used to prove stochastic stability for the EA spin glass at all temperatures
and for a wide range of coupling distributions. On the way, a theorem of Newman
and Stein about the pure state decomposition of the EA model is recovered and
extended.Comment: 27 page
Universal crossing probability in anisotropic systems
Scale-invariant universal crossing probabilities are studied for critical
anisotropic systems in two dimensions. For weakly anisotropic standard
percolation in a rectangular-shaped system, Cardy's exact formula is
generalized using a length-rescaling procedure. For strongly anisotropic
systems in 1+1 dimensions, exact results are obtained for the random walk with
absorbing boundary conditions, which can be considered as a linearized
mean-field approximation for directed percolation. The bond and site directed
percolation problem is itself studied numerically via Monte Carlo simulations
on the diagonal square lattice with either free or periodic boundary
conditions. A scale-invariant critical crossing probability is still obtained,
which is a universal function of the effective aspect ratio r_eff=c r where
r=L/t^z, z is the dynamical exponent and c is a non-universal amplitude.Comment: 7 pages, 4 figure
A unified stability property in spin glasses
Gibbs' measures in the Sherrington-Kirkpatrick type models satisfy two
asymptotic stability properties, the Aizenman-Contucci stochastic stability and
the Ghirlanda-Guerra identities, which play a fundamental role in our current
understanding of these models. In this paper we show that one can combine these
two properties very naturally into one unified stability property
Two-Parton Contribution to the Heavy-Quark Forward-Backward Asymmetry in NNLO QCD
Forward-backward asymmetries, , are important observables for the
determination of the neutral-current couplings of heavy quarks in inclusive
heavy quark production, . In view of the
measurement perspectives on at a future linear collider, precise
predictions of are required for massive quarks. We compute the
contribution of the final state to to order \as^2 in
the QCD coupling. We provide general formulae, and we show that this
contribution to is infrared-finite. We evaluate these two-parton
contributions for and quarks on and near the resonance, and for
quarks above threshold. Moreover, near the threshold we obtain, by
expanding in the heavy-quark velocity , an expression for to order \as^2 and NNLL in . This quantity is equal, to this
order in , to the complete forward-backward asymmetry .Comment: latex, 26 pages, 2 tables, 17 figure
On the distribution of maximum value of the characteristic polynomial of GUE random matrices
Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random N×N matrices H from the Gaussian Unitary Ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of DN(x):=−log|det(xI−H)| as N→∞ and x∈(−1,1). We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher-Hartwig asymptotics due to Krasovsky \cite{K07} with the heuristic {\it freezing transition} scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the Circular Unitary Ensemble, the present GUE case poses new challenges. In particular we show how the conjectured {\it self-duality} in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of DN(x)
Interpolating the Sherrington-Kirkpatrick replica trick
The interpolation techniques have become, in the past decades, a powerful
approach to lighten several properties of spin glasses within a simple
mathematical framework. Intrinsically, for their construction, these schemes
were naturally implemented into the cavity field technique, or its variants as
the stochastic stability or the random overlap structures. However the first
and most famous approach to mean field statistical mechanics with quenched
disorder is the replica trick. Among the models where these methods have been
used (namely, dealing with frustration and complexity), probably the best known
is the Sherrington-Kirkpatrick spin glass: In this paper we are pleased to
apply the interpolation scheme to the replica trick framework and test it
directly to the cited paradigmatic model: interestingly this allows to obtain
easily the replica-symmetric control and, synergically with the broken replica
bounds, a description of the full RSB scenario, both coupled with several minor
theorems. Furthermore, by treating the amount of replicas as an
interpolating parameter (far from its original interpretation) this can be
though of as a quenching temperature close to the one introduce in
off-equilibrium approaches and, within this viewpoint, the proof of the
attended commutativity of the zero replica and the infinite volume limits can
be obtained.Comment: This article is dedicated to David Sherrington on the occasion of his
seventieth birthda
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