A (2+1)-dimensional growth process with explicit stationary measures

We introduce a class of (2+1)-dimensional stochastic growth processes, that can be seen as irreversible random dynamics of discrete interfaces. "Irreversible" means that the interface has an average non-zero drift. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. The model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting Hammersley processes. When the dynamical asymmetry parameter $(p-q)$ equals zero, the infinite-volume Gibbs measures $\pi_\rho$ (with given slope $\rho$) are stationary and reversible. When $p\ne q$, $\pi_\rho$ are not reversible any more but, remarkably, they are still stationary. In such stationary states, we find that the average height function at any given point $x$ grows linearly with time $t$ with a non-zero speed: $\mathbb E Q_x(t):=\mathbb E(h_x(t)-h_x(0))= V(\rho) t$ while the typical fluctuations of $Q_x(t)$ are smaller than any power of $t$ as $t\to\infty$. In the totally asymmetric case of $p=0,q=1$ and on the hexagonal lattice, the dynamics coincides with the "anisotropic KPZ growth model" introduced by A. Borodin and P. L. Ferrari. For a suitably chosen, "integrable", initial condition (that is very far from the stationary state), they were able to determine the hydrodynamic limit and a CLT for interface fluctuations on scale $\sqrt{\log t}$, exploiting the fact that in that case certain space-time height correlations can be computed exactly.Comment: 37 pages, 13 figures. v3: some references added, introduction expanded, minor changes in the bul

Disordered pinning models and copolymers: beyond annealed bounds

We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by Garel et al. [Europhys. Lett. 8 (1989) 9--13], pinning and wetting models in various dimensions, and the Poland--Scheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of noninteger moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, for example, if the disorder distribution is Gaussian. In particular, for pinning models with loop exponent $0<\alpha<1/2$ this implies the existence of a transition from weak to strong disorder. For the copolymer model, under a (restrictive) condition on the law of the underlying renewal, we show that the critical point coincides with the one predicted via renormalization group arguments in the theoretical physics literature. A stronger result holds for a "reduced wetting model" introduced by Bodineau and Giacomin [J. Statist. Phys. 117 (2004) 801--818]: without restrictions on the law of the underlying renewal, the critical point coincides with the corresponding renormalization group prediction.Comment: Published in at http://dx.doi.org/10.1214/07-AAP496 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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.Comment: 24 page

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 $\gamma^{-1}$. 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 $\gamma\to0$. This could be a first step toward an expansion around mean field theory, for spin glass systems.Comment: 4 pages; references updated, typos correcte

On the irrelevant disorder regime of pinning models

Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that is now rigorous. In this work, we exploit interpolation and replica coupling methods to obtain sharper results on the irrelevant disorder regime of pinning models. In particular, in this regime, we compute the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and quenched averaged correlation length exponents coincide, while, in general, they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals.Comment: Published in at http://dx.doi.org/10.1214/09-AOP454 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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 $\gamma^{-1}$. 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 $\gamma\to0$. 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 $\gamma$, based on a large deviation functional for overlap profiles. This will be developed in future work.Comment: 19 pages, revtex

Stationary measure of the driven two-dimensional q-Whittaker particle system on the torus

We consider a q-deformed version of the uniform Gibbs measure on dimers on the periodized hexagonal lattice (equivalently, on interlacing particle configurations, if vertical dimers are seen as particles) and show that it is invariant under a certain irreversible q-Whittaker dynamic. Thereby we provide a new non-trivial example of driven interacting two-dimensional particle system, or of (2+1)-dimensional stochastic growth model, with explicit stationary measure. We emphasize that this measure is far from being a product Bernoulli measure. These Gibbs measures and dynamics both arose earlier in the theory of Macdonald processes. The q=0 degeneration of the Gibbs measures reduce to the usual uniform dimer measures with given tilt, the degeneration of the dynamics originate in the study of Schur processes and the degeneration of the results contained herein were recently treated in work of the second author.Comment: 12 pages, 4 figure

A one-dimensional coagulation-fragmentation process with a dynamical phase transition

We introduce a reversible Markovian coagulation-fragmentation process on the set of partitions of $\{1,\ldots,L\}$ into disjoint intervals. Each interval can either split or merge with one of its two neighbors. The invariant measure can be seen as the Gibbs measure for a homogeneous pinning model \cite{cf:GBbook}. Depending on a parameter $\lambda$, the typical configuration can be either dominated by a single big interval (delocalized phase), or be composed of many intervals of order $1$ (localized phase), or the interval length can have a power law distribution (critical regime). In the three cases, the time required to approach equilibrium (in total variation) scales very differently with $L$. In the localized phase, when the initial condition is a single interval of size $L$, the equilibration mechanism is due to the propagation of two "fragmentation fronts" which start from the two boundaries and proceed by power-law jumps

Stochastic heat equation limit of a (2+1)d growth model

We determine a $q\to 1$ limit of the two-dimensional $q$-Whittaker driven particle system on the torus studied previously in [Corwin-Toninelli, arXiv:1509.01605]. This has an interpretation as a $(2+1)$-dimensional stochastic interface growth model, that is believed to belong to the so-called anisotropic Kardar-Parisi-Zhang (KPZ) class. This limit falls into a general class of two-dimensional systems of driven linear SDEs which have stationary measures on gradients. Taking the number of particles to infinity we demonstrate Gaussian free field type fluctuations for the stationary measure. Considering the temporal evolution of the stationary measure, we determine that along characteristics, correlations are asymptotically given by those of the $(2+1)$-dimensional additive stochastic heat equation. This confirms (for this model) the prediction that the non-linearity for the anisotropic KPZ equation in $(2+1)$-dimension is irrelevant.Comment: 24 pages, 1 figur