Fabio Martinelli
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Fabio Martinelli has a total of 48 co-authors. Here are the top co-authors by number of shared publications:
- Pietro Caputo
Università Roma Tre, Rome, Italy
16 shared publications - Fabio Lucio Toninelli
TU Wien, Vienna, Austria
15 shared publications - Cristina Toninelli
University Paris Dauphine-PSL and CNRS
10 shared publications - Nicoletta Cancrini
Univ. l’Aquila
7 shared publications - Alistair Sinclair
University of California, Berkeley, USA
6 shared publications - Alessandra Faggionato
Department of Mathematics, Sapienza University of Rome, Roma, Italy
5 shared publications
Latest Publications
- Open Access
- 13 July 2026
- Published by Society for Industrial & Applied Mathematics (SIAM)
- p. 497-515
Kinetically Constrained Spin Models: A Short Survey
Continuing the legacy of the International Congress of Mathematicians, these proceedings bring together presentations delivered by invited speakers at the 2026 Congress held in Philadelphia. Publis... Continuing the legacy of the International Congress of Mathematicians, these proceedings bring together presentations delivered by invited speakers at the 2026 Congress held in Philadelphia. Publis...
- 13 July 2026
- Published by Society for Industrial & Applied Mathematics (SIAM)
- p. 497-515
Open AccessKinetically Constrained Spin Models: A Short Survey
Continuing the legacy of the International Congress of Mathematicians, these proceedings bring together presentations delivered by invited speakers at the 2026 Congress held in Philadelphia. Publis... Continuing the legacy of the International Congress of Mathematicians, these proceedings bring together presentations delivered by invited speakers at the 2026 Congress held in Philadelphia. Publis...
- 6 October 2022
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 185 (3-4) , 993-1037
Sharp threshold for the FA-2f kinetically constrained model
- 6 October 2022
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 185 (3-4) , 993-1037
Sharp threshold for the FA-2f kinetically constrained model
- Open Access
Sharp threshold for the FA-2f kinetically constrained model
The Fredrickson-Andersen 2-spin facilitated model on $\mathbb{Z}^d$ (FA-2f) is a paradigmatic interacting particle system with kinetic constraints (KCM) featuring dynamical facilitation, an important mechanism in condensed matter physics. In FA-2f a site may change its state only if at least two of its nearest neighbours are empty. Although the process is reversible w.r.t. a product Bernoulli measure, it is not attractive and features degenerate jump rates and anomalous divergence of characteristic time scales as the density $q$ of empty sites tends to $0$. A natural random variable encoding the above features is $\tau_0$, the first time at which the origin becomes empty for the stationary process. Our main result is the sharp threshold \[\tau_0=\exp\Big(\frac{d\cdot\lambda(d,2)+o(1)}{q^{1/(d-1)}}\Big)\quad \text{w.h.p.}\] with $\lambda(d,2)$ the sharp threshold constant for 2-neighbour bootstrap percolation on $\mathbb{Z}^d$, the monotone deterministic automaton counterpart of FA-2f. This is the first sharp result for a critical KCM and it compares with Holroyd's 2003 result on bootstrap percolation and its subsequent improvements. It also settles various controversies accumulated in the physics literature over the last four decades. Furthermore, our novel techniques enable completing the recent ambitious program on the universality phenomenon for critical KCM and establishing sharp thresholds for other two-dimensional KCM.Open AccessSharp threshold for the FA-2f kinetically constrained model
The Fredrickson-Andersen 2-spin facilitated model on $\mathbb{Z}^d$ (FA-2f) is a paradigmatic interacting particle system with kinetic constraints (KCM) featuring dynamical facilitation, an important mechanism in condensed matter physics. In FA-2f a site may change its state only if at least two of its nearest neighbours are empty. Although the process is reversible w.r.t. a product Bernoulli measure, it is not attractive and features degenerate jump rates and anomalous divergence of characteristic time scales as the density $q$ of empty sites tends to $0$. A natural random variable encoding the above features is $\tau_0$, the first time at which the origin becomes empty for the stationary process. Our main result is the sharp threshold \[\tau_0=\exp\Big(\frac{d\cdot\lambda(d,2)+o(1)}{q^{1/(d-1)}}\Big)\quad \text{w.h.p.}\] with $\lambda(d,2)$ the sharp threshold constant for 2-neighbour bootstrap percolation on $\mathbb{Z}^d$, the monotone deterministic automaton counterpart of FA-2f. This is the first sharp result for a critical KCM and it compares with Holroyd's 2003 result on bootstrap percolation and its subsequent improvements. It also settles various controversies accumulated in the physics literature over the last four decades. Furthermore, our novel techniques enable completing the recent ambitious program on the universality phenomenon for critical KCM and establishing sharp thresholds for other two-dimensional KCM. - Open Access
Coalescing and branching simple symmetric exclusion process
Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph $G=(V,E)$ dual to the biased voter model on $G$. Our main goal are tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as $|V|\rightarrow \infty$. Our results allow us to recover very directly and improve to $\ell^p$-mixing, $p\ge 2$, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson-Andersen one spin facilitated (FA-$1$f) KCM on the discrete $d$-dimensional torus. In view of applications to the more complex FA-$j$f KCM, $j>1$, we also extend part of the analysis to an analogous process with a more general product state space.Open AccessCoalescing and branching simple symmetric exclusion process
Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph $G=(V,E)$ dual to the biased voter model on $G$. Our main goal are tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as $|V|\rightarrow \infty$. Our results allow us to recover very directly and improve to $\ell^p$-mixing, $p\ge 2$, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson-Andersen one spin facilitated (FA-$1$f) KCM on the discrete $d$-dimensional torus. In view of applications to the more complex FA-$j$f KCM, $j>1$, we also extend part of the analysis to an analogous process with a more general product state space. - Open Access
Universality for critical KCM: finite number of stable directions
In this paper we consider kinetically constrained models (KCM) on $\mathbb Z^2$ with general update families $\mathcal U$. For $\mathcal U$ belonging to the so-called "critical class" our focus is on the divergence of the infection time of the origin for the equilibrium process as the density of the facilitating sites vanishes. In a recent paper Mar\^ech\'e and two of the present authors proved that if $\mathcal U$ has an infinite number of "stable directions", then on a doubly logarithmic scale the above divergence is twice the one in the corresponding $\mathcal U$-bootstrap percolation. Here we prove instead that, contrary to previous conjectures, in the complementary case the two divergences are the same. In particular, we establish the full universality partition for critical $\mathcal U$. The main novel contribution is the identification of the leading mechanism governing the motion of infected critical droplets. It consists of a peculiar hierarchical combination of mesoscopic East-like motions.Open AccessUniversality for critical KCM: finite number of stable directions
In this paper we consider kinetically constrained models (KCM) on $\mathbb Z^2$ with general update families $\mathcal U$. For $\mathcal U$ belonging to the so-called "critical class" our focus is on the divergence of the infection time of the origin for the equilibrium process as the density of the facilitating sites vanishes. In a recent paper Mar\^ech\'e and two of the present authors proved that if $\mathcal U$ has an infinite number of "stable directions", then on a doubly logarithmic scale the above divergence is twice the one in the corresponding $\mathcal U$-bootstrap percolation. Here we prove instead that, contrary to previous conjectures, in the complementary case the two divergences are the same. In particular, we establish the full universality partition for critical $\mathcal U$. The main novel contribution is the identification of the leading mechanism governing the motion of infected critical droplets. It consists of a peculiar hierarchical combination of mesoscopic East-like motions. - Open Access
Towards a universality picture for the relaxation to equilibrium of kinetically constrained models
Recent years have seen a great deal of progress in our understanding of bootstrap percolation models, a particular class of monotone cellular automata. In the two dimensional lattice there is now a quite satisfactory understanding of their evolution starting from a random initial condition, with a strikingly beautiful universality picture for their critical behaviour. Much less is known for their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM each vertex is resampled (independently) at rate one by tossing a p-coin iff it can be infected in the next step by the bootstrap model. In particular infection can also heal, hence the non-monotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own as they feature some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics. In this paper we pave the way towards proving universality results for KCM similar to those for bootstrap percolation. Our novel and general approach establishes a close connection between the critical scaling of characteristic time scales for KCM and the scaling of the critical length in critical bootstrap models. Although the full proof of universality for KCM is deferred to a forthcoming paper, here we apply our general method to the Friedrickson-Andersen k-facilitated models, amongst the most studied KCM, and to the Gravner-Griffeath model. In both cases our results are close to optimal.
Open AccessTowards a universality picture for the relaxation to equilibrium of kinetically constrained models
Recent years have seen a great deal of progress in our understanding of bootstrap percolation models, a particular class of monotone cellular automata. In the two dimensional lattice there is now a quite satisfactory understanding of their evolution starting from a random initial condition, with a strikingly beautiful universality picture for their critical behaviour. Much less is known for their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM each vertex is resampled (independently) at rate one by tossing a p-coin iff it can be infected in the next step by the bootstrap model. In particular infection can also heal, hence the non-monotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own as they feature some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics. In this paper we pave the way towards proving universality results for KCM similar to those for bootstrap percolation. Our novel and general approach establishes a close connection between the critical scaling of characteristic time scales for KCM and the scaling of the critical length in critical bootstrap models. Although the full proof of universality for KCM is deferred to a forthcoming paper, here we apply our general method to the Friedrickson-Andersen k-facilitated models, amongst the most studied KCM, and to the Gravner-Griffeath model. In both cases our results are close to optimal.
- Open Access
Universality results for kinetically constrained spin models in two dimensions
Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as $\mathcal U$-bootstrap percolation. KCM also display some of the peculiar features of the so-called "glassy dynamics", and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics. We consider two-dimensional KCM with update rule $\mathcal U$, and focus on proving universality results for the mean infection time of the origin, in the same spirit as those recently established in the setting of $\mathcal U$-bootstrap percolation. We first identify what we believe are the correct universality classes, which turn out to be different from those of $\mathcal U$-bootstrap percolation. We then prove universal upper bounds on the mean infection time within each class, which we conjecture to be sharp up to logarithmic corrections. In certain cases, including all supercritical models, and the well-known Duarte model, our conjecture has recently been confirmed in [MMT]. In fact, in these cases our upper bound is sharp up to a constant factor in the exponent. For certain classes of update rules, it turns out that the infection time of the KCM diverges much faster than for the corresponding $\mathcal U$-bootstrap process when the equilibrium density of infected sites goes to zero. This is due to the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the monotone bootstrap dynamics.Open AccessUniversality results for kinetically constrained spin models in two dimensions
Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as $\mathcal U$-bootstrap percolation. KCM also display some of the peculiar features of the so-called "glassy dynamics", and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics. We consider two-dimensional KCM with update rule $\mathcal U$, and focus on proving universality results for the mean infection time of the origin, in the same spirit as those recently established in the setting of $\mathcal U$-bootstrap percolation. We first identify what we believe are the correct universality classes, which turn out to be different from those of $\mathcal U$-bootstrap percolation. We then prove universal upper bounds on the mean infection time within each class, which we conjecture to be sharp up to logarithmic corrections. In certain cases, including all supercritical models, and the well-known Duarte model, our conjecture has recently been confirmed in [MMT]. In fact, in these cases our upper bound is sharp up to a constant factor in the exponent. For certain classes of update rules, it turns out that the infection time of the KCM diverges much faster than for the corresponding $\mathcal U$-bootstrap process when the equilibrium density of infected sites goes to zero. This is due to the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the monotone bootstrap dynamics. - 15 July 2017
- Published by Springer Nature in Bollettino dell'Unione Matematica Italiana
- Vol. 10 (3) , 451-466
Entropic repulsion in $|\nabla \phi |^p$ | ∇ ϕ | p surfaces: a large deviation bound for all $p\ge 1$ p ≥ 1- 15 July 2017
- Published by Springer Nature in Bollettino dell'Unione Matematica Italiana
- Vol. 10 (3) , 451-466
Entropic repulsion in $|\nabla \phi |^p$ | ∇ ϕ | p surfaces: a large deviation bound for all $p\ge 1$ p ≥ 1- Open Access
Mixing length scales of low temperature spin plaquettes models
Plaquette models are short range ferromagnetic spin models that play a key role in the dynamic facilitation approach to the liquid glass transition. In this paper we perform a rigorous study of the thermodynamic properties of two dimensional plaquette models, the square and triangular plaquette models. We prove that for any positive temperature both models have a unique infinite volume Gibbs measure with exponentially decaying correlations. We analyse the scaling of three a priori different static correlation lengths in the small temperature regime, the mixing, cavity and multispin correlation lengths. Finally, using the symmetries of the model we determine an exact self similarity property for the infinite volume Gibbs measure.
Open AccessMixing length scales of low temperature spin plaquettes models
Plaquette models are short range ferromagnetic spin models that play a key role in the dynamic facilitation approach to the liquid glass transition. In this paper we perform a rigorous study of the thermodynamic properties of two dimensional plaquette models, the square and triangular plaquette models. We prove that for any positive temperature both models have a unique infinite volume Gibbs measure with exponentially decaying correlations. We analyse the scaling of three a priori different static correlation lengths in the small temperature regime, the mixing, cavity and multispin correlation lengths. Finally, using the symmetries of the model we determine an exact self similarity property for the infinite volume Gibbs measure.
- Open AccessEntropic repulsion in $|\nabla φ|^p$ surfaces: a large deviation bound for all $p\geq 1$We consider the $(2+1)$-dimensional generalized solid-on-solid (SOS) model, that is the random discrete surface with a gradient potential of the form $|\nabla\phi|^{p}$, where $p\in [1,+\infty]$. We show that at low temperature, for a square region $\Lambda$ with side $L$, both under the infinite volume measure and under the measure with zero boundary conditions around $\Lambda$, the probability that the surface is nonnegative in $\Lambda$ behaves like $\exp(-4\beta\tau_{p,\beta} L H_p(L) )$, where $\beta$ is the inverse temperature, $\tau_{p,\beta}$ is the surface tension at zero tilt, or step free energy, and $H_p(L)$ is the entropic repulsion height, that is the typical height of the field when a positivity constraint is imposed. This generalizes recent results obtained in \cite{CMT} for the standard SOS model ($p=1$).Open AccessEntropic repulsion in $|\nabla φ|^p$ surfaces: a large deviation bound for all $p\geq 1$We consider the $(2+1)$-dimensional generalized solid-on-solid (SOS) model, that is the random discrete surface with a gradient potential of the form $|\nabla\phi|^{p}$, where $p\in [1,+\infty]$. We show that at low temperature, for a square region $\Lambda$ with side $L$, both under the infinite volume measure and under the measure with zero boundary conditions around $\Lambda$, the probability that the surface is nonnegative in $\Lambda$ behaves like $\exp(-4\beta\tau_{p,\beta} L H_p(L) )$, where $\beta$ is the inverse temperature, $\tau_{p,\beta}$ is the surface tension at zero tilt, or step free energy, and $H_p(L)$ is the entropic repulsion height, that is the typical height of the field when a positivity constraint is imposed. This generalizes recent results obtained in \cite{CMT} for the standard SOS model ($p=1$).