arXiv subjects

Fabio Lucio Toninelli

Publications and source records attributed to Fabio Lucio Toninelli.

45 records · Page 3Linked to original sources

arXiv2008

On the approach to equilibrium for a polymer with adsorption and repulsion

We consider paths of a one-dimensional simple random walk conditioned to come back to the origin after L steps (L an even integer). In the 'pinning model' each path ηhas a weight λ^{N(η)}, 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 λis 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 (λ=1, no wall), where the gap and the mixing time are known to scale as L^{-2} and L^2\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\geq 1 the 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.

math.PR↗

arXiv2007

A replica-coupling approach to disordered pinning models

We consider a renewal process τ={τ_0,τ_1,...} on the integers, where the law of τ_i-τ_{i-1} has a power-like tail P(τ_i-τ_{i-1}=n)=n^{-(α+1)}L(n) with α\ge0 and L(.) slowly varying. We then assign a random, n-dependent reward/penalty to the occurrence of the event that the site n belongs to tau. This class of problems includes, among others, (1+d)-dimensional models of pinning of directed polymers on a one-dimensional random defect, (1+1)-dimensional models of wetting of disordered substrates, and the Poland-Scheraga model of DNA denaturation. By varying the average of the reward, the system undergoes a transition from a localized phase where τoccupies a finite fraction of N to a delocalized phase where the density of τvanishes. In absence of disorder the transition is of first order for α>1 and of higher order for α<1. Moreover, for αranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small amount of disorder is known to modify the order of transition as soon as α>1/2. In physical terms, disorder is relevant in this situation, in agreement with the heuristic Harris criterion. On the other hand, for 0<α<1/2 it has been proven recently by K. Alexander that, if disorder is sufficiently weak, critical exponents are not modified by randomness: disorder is irrelevant. In this work, generalizing techniques which in the framework of spin glasses are known as replica coupling and interpolation, we give a new, simpler proof of the main results of [2]. Moreover, we (partially) justify a small-disorder expansion worked out in [9] for α<1/2, showing that it provides a free energy upper bound which improves the annealed one.

math-ph↗

arXiv2005

Estimates on path delocalization for copolymers at selective interfaces

We consider a directed random walk model of a random heterogeneous polymer in the proximity of an interface separating two selective solvents. This model exhibits a localization/delocalization transition. A positive value of the free energy corresponds to the localized regime and strong results on the polymer path behavior are known in this case. We focus on the interior of the delocalized phase, which is characterized by the free energy equal to zero, and we show in particular that in this regime there are O(log N) monomers in the unfavorable solvent (N is the length of the polymer). The previously known result was o(N). Our approach is based on concentration bounds on suitably restricted partition functions. The same idea allows also to interpolate between different types of disorder in the weak coupling limit. In this way we show the universal nature of this limit, previously considered only for binary disorder.

math.PR↗

arXiv2005

A field-theoretical approach to the spin glass transition: models with long but finite interaction range

We study spin glasses with Kac type interaction potential for small but finite inverse interaction range $γ$. Using the theoretical setup of coupled replicas, through the replica method we argue that the probability of overlap profiles can be expressed for small $γ$ through a large-deviation functional. This result is supported by rigorous arguments, showing that the large-deviation functional provides at least upper bounds for the probability. Finally we analyze the rate function, in the vicinity of the critical point $T_c=1$ of mean field theory, and we study the free energy cost of overlap interfaces, assuming the validity of a gradient expansion for the rate functional.

cond-mat.dis-nn↗

arXiv2005

The Ising-Sherrington-Kirpatrick model in a magnetic field at high temperature

We study a spin system on a large box with both Ising interaction and Sherrington-Kirpatrick couplings, in the presence of an external field. Our results are: (i) existence of the pressure in the limit of an infinite box. When both Ising and Sherrington-Kirpatrick temperatures are high enough, we prove that: (ii) the value of the pressure is given by a suitable replica symmetric solution, and (iii) the fluctuations of the pressure are of order of the inverse of the square of the volume with a normal distribution in the limit. In this regime, the pressure can be expressed in terms of random field Ising models.

math.PR↗

arXiv2004

Finite-range spin glasses in the Kac limit: free energy and local observables

We study a finite range spin glass model in arbitrary dimension, where the intensity of the coupling between spins decays to zero over some distance $γ^{-1}$. We prove that, under a positivity condition for the interaction potential, the infinite-volume free energy of the system converges to that of the Sherrington-Kirkpatrick model, in the Kac limit $γ\to0$. We study the implication of this convergence for the local order parameter, i.e., the local overlap distribution function and a family of susceptibilities to it associated, and we show that locally the system behaves like its mean field analogue. Similar results are obtained for models with $p$-spin interactions. Finally, we discuss a possible approach to the problem of the existence of long range order for finite $γ$, based on a large deviation functional for overlap profiles. This will be developed in future work.

cond-mat.dis-nn↗

arXiv2003

The Kac limit for finite-range spin glasses

We consider a finite range spin glass model in arbitrary dimension, where the strength of the two-body coupling decays to zero over some distance $γ^{-1}$. 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 $γ\to0$. This could be a first step toward an expansion around mean field theory, for spin glass systems.

cond-mat.stat-mech↗

arXiv2003

Replica bounds for diluted non-Poissonian spin systems

In 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.

cond-mat.dis-nn↗

arXiv2003

The high temperature region of the Viana-Bray diluted spin glass model

In this paper, we study the high temperature or low connectivity phase of the Viana-Bray model. This is a diluted version of the well known Sherrington-Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy, and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington-Kirkpatrick model is discussed.

cond-mat.dis-nn↗