1,720,975 research outputs found
A field-theoretical approach to the spin glass transition: models with long but finite interaction range
No full textThe Kac limit for diluted spin glasses
No full textWe study diluted spin glass models in arbitrary dimension, where each spin interacts with a finite number of other spins chosen at random with a probability decaying to zero over some distance ¿-1. For systems with pairwise interactions we show that the infinite-volume free energy converges to that of the mean-field Viana–Bray model,1 in the Kac limit ¿¿0. For p-spin like models we get only one bound: the free-energy is bounded from above by the one of the mean-field diluted p-spin
Multi-level pinning problems for random walks and self-avoiding lattice paths
No full textWe consider a generalization of the classical pinning problem for integer-valued random walks conditioned to stay non-negative. More specifically, we take pinning potentials of the form j 0 εjNj, where Nj is the number of visits to the state j and {εj} is a non-negative sequence. Partly motivated by similar problems for low-temperature contour models in statistical physics, we aim at finding a sharp characterization of the threshold of the wetting transition, especially in the regime where the variance σ2 of the single step of the random walk is small. Our main result says that, for natural choices of the pinning sequence {εj }, localization (respectively delocalization) occurs if σ−2 j 0(j + 1)εj δ−1 (respectively δ), for some universal δ 0 a large enough parameter. This generalization is directly relevant for applications to the above mentioned contour models
On the approach to equilibrium for a polymer with adsorption and repulsion
No full textWe consider paths of a one–dimensional simple random walk conditioned to come back to
the origin after L steps, L ∈ 2N. In the pinning model each path η has a weight λN( ́),
where λ > 0 and N(η) is the number of zeros in η. When the paths are constrained to
be non–negative, the polymer is said to satisfy a hard–wall constraint. Such models are
well known to undergo a localization/delocalization transition as the pinning strength λ is
varied. In this paper we study a natural “spin flip” dynamics for these models and derive
several estimates on its spectral gap and mixing time. In particular, for the system with
the wall we prove that relaxation to equilibrium is always at least as fast as in the free case
(i.e. λ = 1 without the wall), where the gap and the mixing time are known to scale as
L−2 and L2 log L, respectively. This improves considerably over previously known results.
For the system without the wall we show that the equilibrium phase transition has a clear
dynamical manifestation: for λ > 1 relaxation is again at least as fast as the diffusive free
case, but in the strictly delocalized phase (λ < 1) the gap is shown to be O(L−5/2), up to
logarithmic corrections. As an application of our bounds, we prove stretched exponential
relaxation of local functions in the localized regime
Disorder relevance at marginality and critical point shift
No full textRecently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is relevant or irrelevant in the Harris criterion sense: the question addressed is whether quenched disorder leads to a critical behavior which is different from the one observed in the pure, i.e. annealed, system. The Harris criterion prediction is based on the sign of the specific heat exponent of the pure system, but it yields no prediction in the case of vanishing exponent. This case is called marginal, and the physical literature is divided on what one should observe for marginal disorder, notably there is no agreement on whether a small amount of disorder leads or not to a difference between the critical point of the quenched system and the one for the pure system. In [Comm. Pure Appl. Math. 63 (2010) 233-265] we have proven that the two critical points differ at marginality of at least exp(- c/β4), where c > 0 and β2 is the disorder variance, for β ε (0,1) and Gaussian IID disorder. The purpose of this paper is to improve such a result: we establish in particular that the exp(-c/β4) lower bound on the shift can be replaced by exp(-c(b)/βb), c(b) > 0 for b > 2 (b = 2 is the known upper bound and it is the result claimed in [J. Stat. Phys. 6 (1992) 1189-1213]), and we deal with very general distribution of the IID disorder variables. The proof relies on coarse graining estimates and on a fractional moment change of measure argument based on multi-body potential modifications of the law of the disorder. © Association des Publications de l'Institut Henri Poincaré, 2011
Hierarchical pinning models, quadratic maps and quenched disorder
No full textWe consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by Derrida et al. (J Stat Phys 66:1189-1213, 1992), which can be re-interpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {Rn}n=1,2, ... , which in absence of disorder (i. e., when the initial condition is degenerate) reduces to a particular case of the well-known logistic map. The large-n limit of the sequence of random variables 2-n log Rn, a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter α ε (0, 1), related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured in Derrida et al. (J Stat Phys 66:1189-1213, 1992) that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2 1/2 we find the correct scaling form (for weak disorder) of the critical point shift. © Springer-Verlag 2009
Quasi-polynomial mixing of the 2D stochastic Ising model with "plus" boundary up to criticality
No full textWe considerably improve upon the recent result of [37] on the mixing time of Glauber dynamics for the 2D Ising model in a box of side L at low temperature and with random boundary conditions whose distribution P stochastically dominates the extremal plus phase. An important special case is when P is concentrated on the homogeneous all-plus configuration, where the mixing time T-MIX is conjectured to be polynomial in L. In [37] it was shown that for a large enough inverse temperature beta and any epsilon > 0 there exists c = c(beta, epsilon) such that lim(L ->infinity) P(T-MIX >= exp(cL(epsilon))) = 0. In particular, for the all-plus boundary conditions and beta large enough, T-MIX <= exp(cL(epsilon)). Here we show that the same conclusions hold for all beta larger than the critical value beta(c) and with exp(cL(epsilon)) replaced by L-c log L (i.e. quasi-polynomial mixing). The key point is a modification of the inductive scheme of [37] together with refined equilibrium estimates that hold up to criticality, obtained via duality and random-line representation tools for the Ising model. In particular, we establish new precise bounds on the law of Peierls contours which complement the Brownian bridge picture established e.g. in [20, 22, 23]
The KAC limit for finite-range spin glasses
Full text linkWe consider a finite-range spin glass model in arbitrary dimension, where the strength of the two-body coupling decays to zero over some distance ¿-1. We show that, under mild assumptions on the interaction potential, the infinite-volume free energy of the system converges to that of the Sherrington-Kirkpatrick one, in the Kac limit ¿¿0. This could be a first step toward an expansion around mean-field theory, for spin glass systems
Replica bounds for diluted non-Poissonian spin systems
Full text linkn this paper we extend replica bounds and free energy subadditivity arguments to diluted spin-glass models on graphs with arbitrary, non-Poissonian degree distribution. The new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method. As a byproduct we obtain self-averaging identities that generalize the Ghirlanda–Guerra ones to the multi-overlap case
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