arXiv subjects

Cristina Toninelli

Publications and source records attributed to Cristina Toninelli.

At least 37 records · Page 2Linked to original sources

arXiv2012

Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality

On the rooted $k$-ary tree we consider a 0-1 kinetically constrained spin model in which the occupancy variable at each node is re-sampled with rate one from the Bernoulli(p) measure iff all its children are empty. For this process the following picture was conjectured to hold. As long as $p$ is below the percolation threshold $p_c=1/k$ the process is ergodic with a finite relaxation time while, for $p>p_c$, the process on the infinite tree is no longer ergodic and the relaxation time on a finite regular sub-tree becomes exponentially large in the depth of the tree. At the critical point $p=p_c$ the process on the infinite tree is still ergodic but with an infinite relaxation time. Moreover, on finite sub-trees, the relaxation time grows polynomially in the depth of the tree. The conjecture was recently proved by the second and forth author except at criticality. Here we analyse the critical and quasi-critical case and prove for the relevant time scales: (i) power law behaviour in the depth of the tree at $p=p_c$ and (ii) power law scaling in $(p_c-p)^{-1}$ when $p$ approaches $p_c$ from below. Our results, which are very close to those obtained recently for the Ising model at the spin glass critical point, represent the first rigorous analysis of a kinetically constrained model at criticality.

math.PR↗

arXiv2012

Finite size effects in the dynamics of glass-forming liquids

We present a comprehensive theoretical study of finite size effects in the relaxation dynamics of glass-forming liquids. Our analysis is motivated by recent theoretical progress regarding the understanding of relevant correlation length scales in liquids approaching the glass transition. We obtain predictions both from general theoretical arguments and from a variety of specific perspectives: mode-coupling theory, kinetically constrained and defect models, and random first order transition theory. In the latter approach, we predict in particular a non-monotonic evolution of finite size effects across the mode-coupling crossover due to the competition between mode-coupling and activated relaxation. We study the role of competing relaxation mechanisms in giving rise to non-monotonic finite size effects by devising a kinetically constrained model where the proximity to the mode-coupling singularity can be continuously tuned by changing the lattice topology. We use our theoretical findings to interpret the results of extensive molecular dynamics studies of four model liquids with distinct structures and kinetic fragilities. While the less fragile model only displays modest finite size effects, we find a more significant size dependence evolving with temperature for more fragile models, such as Lennard-Jones particles and soft spheres. Finally, for a binary mixture of harmonic spheres we observe the predicted non-monotonic temperature evolution of finite size effects near the fitted mode-coupling singularity, suggesting that the crossover from mode-coupling to activated dynamics is more pronounced for this model. Finally, we discuss the close connection between our results and the recent report of a non-monotonic temperature evolution of a dynamic length scale near the mode-coupling crossover in harmonic spheres.

cond-mat.stat-mech↗

arXiv2012

Universality in one-dimensional hierarchical coalescence processes

Motivated by several models introduced in the physics literature to study the nonequilibrium coarsening dynamics of one-dimensional systems, we consider a large class of "hierarchical coalescence processes" (HCP). An HCP consists of an infinite sequence of coalescence processes ${ξ^{(n)}(\cdot)}_{n\ge1}$: each process occurs in a different "epoch" (indexed by $n$) and evolves for an infinite time, while the evolution in subsequent epochs are linked in such a way that the initial distribution of $ξ^{(n+1)}$ coincides with the final distribution of $ξ^{(n)}$. Inside each epoch the process, described by a suitable simple point process representing the boundaries between adjacent intervals (domains), evolves as follows. Only intervals whose length belongs to a certain epoch-dependent finite range are active, that is, they can incorporate their left or right neighboring interval with quite general rates. Inactive intervals cannot incorporate their neighbors and can increase their length only if they are incorporated by active neighbors. The activity ranges are such that after a merging step the newly produced interval always becomes inactive for that epoch but active for some future epoch. Without making any mean-field assumption we show that: (i) if the initial distribution describes a renewal process, then such a property is preserved at all later times and all future epochs; (ii) the distribution of certain rescaled variables, for example, the domain length, has a well-defined and universal limiting behavior as $n\to \infty$ independent of the details of the process (merging rates, activity ranges$,...$). This last result explains the universality in the limiting behavior of several very different physical systems (e.g., the East model of glassy dynamics or the Paste-all model) which was observed in several simulations and analyzed in many physics papers. The main idea to obtain the asymptotic result is to first write down a recursive set of nonlinear identities for the Laplace transforms of the relevant quantities on different epochs and then to solve it by means of a transformation which in some sense linearizes the system.

math.PR↗

arXiv2012

Fredrickson-Andersen one spin facilitated model out of equilibrium

We consider the Fredrickson and Andersen one spin facilitated model (FA1f) on an infinite connected graph with polynomial growth. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability $p\in[0,1]$ or $q=1-p$ respectively, provided that at least one of its nearest neighbours is empty. We study the non-equilibrium dynamics started from an initial distribution $ν$ different from the stationary product $p$-Bernoulli measure $μ$. We assume that, under $ν$, the mean distance between two nearest empty sites is uniformly bounded. We then prove convergence to equilibrium when the vacancy density $q$ is above a proper threshold $\bar q<1$. The convergence is exponential or stretched exponential, depending on the growth of the graph. In particular it is exponential on $\bbZ^d$ for $d=1$ and stretched exponential for $d>1$. Our result can be generalized to other non cooperative models.

math.PR↗

arXiv2012

The East model: recent results and new progresses

The East model is a particular one dimensional interacting particle system in which certain transitions are forbidden according to some constraints depending on the configuration of the system. As such it has received particular attention in the physics literature as a special case of a more general class of systems referred to as kinetically constrained models, which play a key role in explaining some features of the dynamics of glasses. In this paper we give an extensive overview of recent rigorous results concerning the equilibrium and non-equilibrium dynamics of the East model together with some new improvements.

math.PR↗

arXiv2012

Finite size scaling of the dynamical free-energy in a kinetically constrained model

We determine the finite size corrections to the large deviation function of the activity in a kinetically constrained model (the Fredrickson-Andersen model in one dimension), in the regime of dynamical phase coexistence. Numerical results agree with an effective model where the boundary between active and inactive regions is described by a Brownian interface.

cond-mat.stat-mech↗

arXiv2011

Activity phase transition for constrained dynamics

We consider two cases of kinetically constrained models, namely East and FA-1f models. The object of interest of our work is the activity A(t) defined as the total number of configuration changes in the interval [0,t] for the dynamics on a finite domain. It has been shown in [GJLPDW1,GJLPDW2] that the large deviations of the activity exhibit a non-equilibirum phase transition in the thermodynamic limit and that reducing the activity is more likely than increasing it due to a blocking mechanism induced by the constraints. In this paper, we study the finite size effects around this first order phase transition and analyze the phase coexistence between the active and inactive dynamical phases in dimension 1. In higher dimensions, we show that the finite size effects are also determined by the dimension and the choice of boundary conditions.

cond-mat.stat-mech↗

arXiv2010

Kinetically Constrained Models

In this chapter we summarize recent developments in the study of kinetically constrained models (KCMs) as models for glass formers. After recalling the definition of the KCMs which we cover we study the possible occurrence of ergodicity breaking transitions and discuss in some detail how, before any such transition occurs, relaxation timescales depend on the relevant control parameter (density or temperature). Then we turn to the main issue: the prediction of KCMs for dynamical heterogeneities. We focus in particular on multipoint correlation functions and susceptibilities, and decoupling in the transport coefficients. Finally we discuss the recent view of KCMs as being at first order coexistence between an active and an inactive space-time phase.

cond-mat.stat-mech↗

arXiv2009

Bootstrap percolation and kinetically constrained models on hyperbolic lattices

We study bootstrap percolation (BP) on hyperbolic lattices obtained by regular tilings of the hyperbolic plane. Our work is motivated by the connection between the BP transition and the dynamical transition of kinetically constrained models, which are in turn relevant for the study of glass and jamming transitions. We show that for generic tilings there exists a BP transition at a nontrivial critical density, $0<ρ_c<1$. Thus, despite the presence of loops on all length scales in hyperbolic lattices, the behavior is very different from that on Euclidean lattices where the critical density is either zero or one. Furthermore, we show that the transition has a mixed character since it is discontinuous but characterized by a diverging correlation length, similarly to what happens on Bethe lattices and random graphs of constant connectivity.

cond-mat.stat-mech↗

arXiv2008

Facilitated oriented spin models:some non equilibrium results

We analyze the relaxation to equilibrium for kinetically constrained spin models (KCSM) when the initial distribution $ν$ is different from the reversible one, $μ$. This setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase. We concentrate on two basic oriented KCSM: the East model on $\bbZ$, for which the constraint requires that the East neighbor of the to-be-update vertex is vacant and the model on the binary tree introduced in \cite{Aldous:2002p1074}, for which the constraint requires the two children to be vacant. While the former model is ergodic at any $p\neq 1$, the latter displays an ergodicity breaking transition at $p_c=1/2$. For the East we prove exponential convergence to equilibrium with rate depending on the spectral gap if $ν$ is concentrated on any configuration which does not contain a forever blocked site or if $ν$ is a Bernoulli($p'$) product measure for any $p'\neq 1$. For the model on the binary tree we prove similar results in the regime $p,p' p_c$ and $p\leq p_c$ convergence to equilibrium cannot occur for all local functions. Finally we present a very simple argument (different from the one in \cite{Aldous:2002p1074}) based on a combination of combinatorial results and ``energy barrier'' considerations, which yields the sharp upper bound for the spectral gap of East when $p\uparrow 1$.

math.PR↗

arXiv2007

Facilitated spin models: recent and new results

Facilitated or kinetically constrained spin models (KCSM) are a class of interacting particle systems reversible w.r.t. to a simple product measure. Each dynamical variable (spin) is re-sampled from its equilibrium distribution only if the surrounding configuration fulfills a simple local constraint which \emph{does not involve} the chosen variable itself. Such simple models are quite popular in the glass community since they display some of the peculiar features of glassy dynamics, in particular they can undergo a dynamical arrest reminiscent of the liquid/glass transitiom. Due to the fact that the jumps rates of the Markov process can be zero, the whole analysis of the long time behavior becomes quite delicate and, until recently, KCSM have escaped a rigorous analysis with the notable exception of the East model. In these notes we will mainly review several recent mathematical results which, besides being applicable to a wide class of KCSM, have contributed to settle some debated questions arising in numerical simulations made by physicists. We will also provide some interesting new extensions. In particular we will show how to deal with interacting models reversible w.r.t. to a high temperature Gibbs measure and we will provide a detailed analysis of the so called one spin facilitated model on a general connected graph.

math.PR↗

arXiv2007

A new class of cellular automata with a discontinuous transition

This paper has been withdrawn by the authors due to a mistake in the proof and a corresponding incorrect result. A correct rigorous analysis of a similar model is presented in ``Spiral Model: a cellular automaton with a discontinuous glass transition'', arXiv:0709.0378.

cond-mat.stat-mech↗

arXiv2007

Statistical Physics of Group Testing

This paper provides a short introduction to the group testing problem, and reviews various aspects of its statistical physics formulation. Two main issues are discussed: the optimal design of pools used in a two-stage testing experiment, like the one often used in medical or biological applications, and the inference problem of detecting defective items based on pool diagnosis. The paper is largely based on: M. Mézard and C. Toninelli, arXiv:0706.3104, and M. Mézard and M. Tarzia {\it Phys. Rev. E} {\bf 76}, 041124 (2007).

cond-mat.stat-mech↗

arXiv2007

Spiral model, jamming percolation and glass-jamming transitions

The Spiral Model (SM) corresponds to a new class of kinetically constrained models introduced in joint works with D.S. Fisher [8,9]. They provide the first example of finite dimensional models with an ideal glass-jamming transition. This is due to an underlying jamming percolation transition which has unconventional features: it is discontinuous (i.e. the percolating cluster is compact at the transition) and the typical size of the clusters diverges faster than any power law, leading to a Vogel-Fulcher-like divergence of the relaxation time. Here we present a detailed physical analysis of SM, see [5] for rigorous proofs. We also show that our arguments for SM does not need any modification contrary to recent claims of Jeng and Schwarz [10].

cond-mat.stat-mech↗

arXiv2007

Spiral Model: a cellular automaton with a discontinuous glass transition

We introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be either empty or occupied by a single particle and the dynamics follows a deterministic updating rule at discrete times which allows only emptying sites. We prove that the threshold density $ρ_c$ for convergence to a completely empty configuration is non trivial, $0<ρ_c<1$, contrary to standard bootstrap percolation. Furthermore we prove that in the subcritical regime, $ρ<ρ_c$, emptying always occurs exponentially fast and that $ρ_c$ coincides with the critical density for two-dimensional oriented site percolation on $\bZ^2$. This is known to occur also for some cellular automata with oriented rules for which the transition is continuous in the value of the asymptotic density and the crossover length determining finite size effects diverges as a power law when the critical density is approached from below. Instead for our model we prove that the transition is {\it discontinuous} and at the same time the crossover length diverges {\it faster than any power law}. The proofs of the discontinuity and the lower bound on the crossover length use a conjecture on the critical behaviour for oriented percolation. The latter is supported by several numerical simulations and by analytical (though non rigorous) works through renormalization techniques. Finally, we will discuss why, due to the peculiar {\it mixed critical/first order character} of this transition, the model is particularly relevant to study glassy and jamming transitions. Indeed, we will show that it leads to a dynamical glass transition for a Kinetically Constrained Spin Model. Most of the results that we present are the rigorous proofs of physical arguments developed in a joint work with D.S.Fisher.

cond-mat.stat-mech↗

arXiv2007

Group Testing with Random Pools: optimal two-stage algorithms

We study Probabilistic Group Testing of a set of N items each of which is defective with probability p. We focus on the double limit of small defect probability, p<<1, and large number of variables, N>>1, taking either p->0 after $N\to\infty$ or $p=1/N^β$ with $β\in(0,1/2)$. In both settings the optimal number of tests which are required to identify with certainty the defectives via a two-stage procedure, $\bar T(N,p)$, is known to scale as $Np|\log p|$. Here we determine the sharp asymptotic value of $\bar T(N,p)/(Np|\log p|)$ and construct a class of two-stage algorithms over which this optimal value is attained. This is done by choosing a proper bipartite regular graph (of tests and variable nodes) for the first stage of the detection. Furthermore we prove that this optimal value is also attained on average over a random bipartite graph where all variables have the same degree, while the tests have Poisson-distributed degrees. Finally, we improve the existing upper and lower bound for the optimal number of tests in the case $p=1/N^β$ with $β\in[1/2,1)$.

cs.DS↗

arXiv2007

Hydrodynamic Limit for a Particle System with degenerate rates

We study the hydrodynamic limit for some conservative particle systems with degenerate rates, namely with nearest neighbor exchange rates which vanish for certain configurations. These models belong to the class of {\sl kinetically constrained lattice gases} (KCLG) which have been introduced and intensively studied in physics literature as simple models for the liquid/glass transition. Due to the degeneracy of rates for KCLG there exists {\sl blocked configurations} which do not evolve under the dynamics and in general the hyperplanes of configurations with a fixed number of particles can be decomposed into different irreducible sets. As a consequence, both the Entropy and Relative Entropy method cannot be straightforwardly applied to prove the hydrodynamic limit. In particular, some care should be put when proving the One and Two block Lemmas which guarantee local convergence to equilibrium. We show that, for initial profiles smooth enough and bounded away from zero and one, the macroscopic density profile for our KCLG evolves under the diffusive time scaling according to the porous medium equation. Then we prove the same result for more general profiles for a slightly perturbed dynamics obtained by adding jumps of the Symmetric Simple Exclusion. The role of the latter is to remove the degeneracy of rates and at the same time they are properly slowed down in order not to change the macroscopic behavior. The equilibrium fluctuations and the magnitude of the spectral gap for this perturbed model are also obtained.

math.PR↗

arXiv2006

On the universality of jamming percolation: reply to the comment of Jeng and Schwarz

In Phys. Rev. Lett. 96, 035702 (2006) we introduced a class of kinetically constrained models which display a dynamical glass transition. We focused on a particular example: the "knights" model. As correctly pointed out by Jeng and Schwarz cond-mat/0612484, we overlooked some additional directed frozen structures of the knights model which are not simple directed percolation (DP) paths: these "thicker" directed structures lower the critical density. Here we argue that, nevertheless, the full directed processes are in the DP universality class and the T-junctions between perpendicular segments of these give rise to a jamming percolation transition with the universal properties discussed in our previous work. Moreover, we present other models for which all our previous results, included the value of the critical density, hold rigorously.

cond-mat.stat-mech↗