arXiv subjects

Fabio Toninelli

Publications and source records attributed to Fabio Toninelli.

At least 19 recordsLinked to original sources

arXiv2026

Mixing times for Glauber dynamics of lozenge tilings of the hexagon

We prove that the continuous-time, single-flip Glauber dynamics for lozenge tilings of the size-$N$ hexagon mix in time $N^{2+o(1)}$. This was predicted to hold on fairly general domains of diameter $N$ (on the basis of the ``Lifshitz law'' heuristic) but had previously only been established in domains such that the associated limit shape has no frozen facets. To access the hexagon, we introduce a multi-scale comparison argument between the height function of the random tiling evolving under the Glauber dynamics and the limit shape of a volume-tilted tiling (whose tilting parameter varies suitably in time).

math.PR↗

arXiv2025

Dimers with layered disorder

We study the dimer model on the square grid, with quenched random edge weights. Randomness is chosen to have a layered structure, similar to that of the celebrated McCoy-Wu disordered Ising model. Disorder has a highly non-trivial effect and it produces an essential singularity of the free energy, with $e^{-\sqrt{{\rm distance}}}$ decay of dimer-dimer correlations, at a point of the ``liquid'' (or ``massless'') phase where the homogeneous dimer model has instead a real analytic free energy and correlations decaying like $1/({\rm distance})^2$. Moreover, at a point where the homogeneous model has a transition between a massive (gaseous) and massless (liquid) phase, the critical exponent 3/2 (Pokrovsky-Talapov law), characteristic of the transition between the two regimes, is modified by disorder into an exponent that ranges continuously between 3/2 and infinity.

math.PR↗

arXiv2024

Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension

The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) that describes, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with a conserved scalar quantity. In space dimension d = 2, the SBE is critical, being formally scale invariant under diffusive scaling. As such, it falls outside of the domain of applicability of the theories of Regularity Structures and paracontrolled calculus. In apparent contrast with the formal scale invariance, we fully prove the conjecture first appeared in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] according to which the 2d-SBE is logarithmically superdiffusive, i.e. its diffusion coefficient diverges like $(\log t)^{2/3}$ as $t\to\infty$, thus removing subleading diverging multiplicative corrections in [D. De Gaspari & L. Haunschmid-Sibitz, Electron. J. Probab., 2024] and in [H.-T. Yau, Ann. of Math., 2004] for 2d-ASEP. We precisely identify the constant prefactor of the logarithm and show it is proportional to $\lambda^{4/3}$, for $\lambda>0$ the coupling constant, which, intriguingly, turns out to be exactly the same as for the one-dimensional Stochastic Burgers/KPZ equation. More importantly, we prove that, under super-diffusive space-time rescaling, the SBE has an explicit Gaussian fixed point in the Renormalization Group sense, by deriving a superdiffusive central limit-type theorem for its solution. This is the first scaling limit result for a critical singular SPDE, beyond the weak coupling regime, and is obtained via a refined control, on all length-scales, of the resolvent of the generator of the SBE. We believe our methods are well-suited to study other out-of-equilibrium driven diffusive systems at the critical dimension, such as 2d-ASEP, which, we conjecture, have the same large-scale Fixed Point as SBE.

math.PR↗

arXiv2024

Weak coupling limit of KPZ with rougher than white noise

We consider the KPZ equation in $1$ spatial dimension with noise that is rougher than white by an exponent $\gamma>1/4$. Under a weak coupling limit, formally removing the nonlinearity from the equation, we show using regularity structures that the renormalised solutions converge to a Gaussian limit that is different from the solution of the linear part of the equation. The regime of this effect has a nontrivial overlap with the subcritical regime $\gamma<1/2$.

math.PR↗

arXiv2024

Convergence of dynamical stationary fluctuations

We present a general black box theorem that ensures convergence of a sequence of stationary Markov processes, provided a few assumptions are satisfied. This theorem relies on a control of the resolvents of the sequence of Markov processes, and on a suitable characterization of the resolvents of the limit. One major advantage of this approach is that it circumvents the use of the Boltzmann-Gibbs principle: for instance, we deduce in a rather simple way that the stationary fluctuations of the one-dimensional zero-range process converge to the stochastic heat equation. More importantly, it allows to establish results that were probably out of reach of existing methods: using the black box result, we are able to prove that the stationary fluctuations of a discrete model of ordered interfaces, that was considered previously in the statistical physics literature, converge to a system of reflected stochastic PDEs.

math.PR↗

arXiv2024

Lecture notes on stationary critical and super-critical SPDEs

The goal of these lecture notes is to present recent results regarding the large-scale behaviour of critical and super-critical non-linear stochastic PDEs, that fall outside the realm of the theory of Regularity Structures. These include the two-dimensional Anisotropic KPZ equation, the stochastic Burgers equation in dimension $d\ge 2$ and the stochastic Navier-Stokes equation with divergence-free noise in dimension $d=2$. Rather than providing complete proofs, we try to emphasise the main ideas, and some crucial aspects of our approach: the role of the generator equation and of the Fluctuation-Dissipation Theorem to identify the limit process; Wiener chaos decomposition with respect to the stationary measure and its truncation; and the so-called Replacement Lemma, which controls the weak coupling limit of the equations in the critical dimension and identifies the limiting diffusivity. For pedagogical reasons, we will focus exclusively on the stochastic Burgers equation. The notes are based on works in collaboration with Dirk Erhard and Massimiliano Gubinelli.

math.PR↗

arXiv2023

Local dimer dynamics in higher dimensions

We consider local dynamics of the dimer model (perfect matchings) on hypercubic boxes $[n]^d$. These consist of successively switching the dimers along alternating cycles of prescribed (small) lengths. We study the connectivity properties of the dimer configuration space equipped with these transitions. Answering a question of Freire, Klivans, Milet and Saldanha, we show that in three dimensions any configuration admits an alternating cycle of length at most 6. We further establish that any configuration on $[n]^d$ features order $n^{d-2}$ alternating cycles of length at most $4d-2$. We also prove that the dynamics of dimer configurations on the unit hypercube of dimension $d$ is ergodic when switching alternating cycles of length at most $4d-4$. Finally, in the planar but non-bipartite case, we show that parallelogram-shaped boxes in the triangular lattice are ergodic for switching alternating cycles of lengths 4 and 6 only, thus improving a result of Kenyon and R\'emila, which also uses 8-cycles. None of our proofs make reference to height functions.

math.CO↗

arXiv2023

Gaussian Fluctuations for the stochastic Burgers equation in dimension $d\geq 2$

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a $d$-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in [H. van Beijeren, R. Kutner and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985]. In both the critical $d=2$ and super-critical $d\geq 3$ cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For $d\ge3$ the scaling adopted is the classical diffusive one, while in $d=2$ it is the weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way.

math.PR↗

arXiv2022

Kinetically constrained models out of equilibrium

We study the full class of kinetically constrained models in arbitrary dimension and out of equilibrium, in the regime where the density $q$ of facilitating sites in the equilibrium measure (but not necessarily in the initial measure) is close to $1$. For these models, we establish exponential convergence to equilibrium in infinite volume and linear time precutoff in finite volume with appropriate boundary condition. Our results are the first out-of-equilibrium results that hold for any model in the so-called critical class, which is covered in its entirety by our treatment, including e.g. the Fredrickson-Andersen 2-spin facilitated model. In addition, they generalise, unify and sometimes simplify several previous works in the field. As byproduct, we recover and generalise exponential tails for the connected component of the origin in the upper invariant trajectory of perturbed cellular automata and in the set of eventually infected sites in subcritical bootstrap percolation models. Our approach goes through the study of cooperative contact processes, last passage percolation, Toom contours, as well as a very convenient coupling between contact processes and kinetically constrained models.

math.PR↗

arXiv2022

Non-reversible stationary states for majority voter and Ising dynamics on trees

We study three Markov processes on infinite, unrooted, regular trees: the stochastic Ising model (also known as the Glauber heat bath dynamics of the Ising model), a majority voter dynamic, and a coalescing particle model. In each of the three cases the tree exhibits a preferred direction encoded into the model. For all three models, our main result is the existence of a stationary but non-reversible measure. For the Ising model, this requires imposing that the inverse temperature is large and choosing suitable non-uniform couplings, and our theorem implies the existence of a stationary measure which looks nothing like a low-temperature Gibbs measure. The interesting aspect of our results lies in the fact that the analogous processes do not have non-Gibbsian stationary measures on $\mathbb Z^d$, owing to the amenability of that graph. In fact, no example of a stochastic Ising model with a non-reversible stationary state was known to date.

math.PR↗

arXiv2022

Height function localisation on trees

We study two models of discrete height functions, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone, that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.

math.PR↗

arXiv2022

Weakly non-planar dimers

We study a model of fully-packed dimer configurations (or perfect matchings) on a bipartite periodic graph that is two-dimensional but not planar. The graph is obtained from $\mathbb Z^2$ via the addition of an extensive number of extra edges that break planarity (but not bipartiteness). We prove that, if the weight $\lambda$ of the non-planar edges is small enough, a suitably defined height function scales on large distances to the Gaussian Free Field with a $\lambda$-dependent amplitude, that coincides with the anomalous exponent of dimer-dimer correlations. Because of non-planarity, Kasteleyn's theory does not apply: the model is not integrable. Rather, we map the model to a system of interacting lattice fermions in the Luttinger universality class, which we then analyze via fermionic Renormalization Group methods.

math.PR↗

arXiv2022

The mixing time of the lozenge tiling Glauber dynamics

The broad motivation of this work is a rigorous understanding of reversible, local Markov dynamics of interfaces, and in particular their speed of convergence to equilibrium, measured via the mixing time $T_{mix}$. In the $(d+1)$-dimensional setting, $d\ge2$, this is to a large extent mathematically unexplored territory, especially for discrete interfaces. On the other hand, on the basis of a mean-curvature motion heuristics and simulations, one expects convergence to equilibrium to occur on time-scales of order $\approx \delta^{-2}$ in any dimension, with $\delta\to0$ the lattice mesh. We study the single-flip Glauber dynamics for lozenge tilings of a finite domain of the plane, viewed as $(2+1)$-dimensional surfaces. The stationary measure is the uniform measure on admissible tilings. At equilibrium, by the limit shape theorem, the height function concentrates as $\delta\to0$ around a deterministic profile $\phi$, the unique minimizer of a surface tension functional. Despite some partial mathematical results, the conjecture $T_{mix}=\delta^{-2+o(1)}$ has been proven, so far, only in the situation where $\phi$ is an affine function. In this work, we prove the conjecture under the sole assumption that the limit shape $\phi$ contains no frozen regions (facets).

math.PR↗

arXiv2022

$\sqrt{\log t}$-superdiffusivity for a Brownian particle in the curl of the 2d GFF

The present work is devoted to the study of the large time behaviour of a critical Brownian diffusion in two dimensions, whose drift is divergence-free, ergodic and given by the curl of the 2-dimensional Gaussian Free Field. We prove the conjecture, made in [B. Tóth, B. Valkó, J. Stat. Phys., 2012], according to which the diffusion coefficient $D(t)$ diverges as $\sqrt{\log t}$ for $t\to\infty$. Starting from the fundamental work by Alder and Wainwright [B. Alder, T. Wainright, Phys. Rev. Lett. 1967], logarithmically superdiffusive behaviour has been predicted to occur for a wide variety of out-of-equilibrium systems in the critical spatial dimension $d=2$. Examples include the diffusion of a tracer particle in a fluid, self-repelling polymers and random walks, Brownian particles in divergence-free random environments, and, more recently, the 2-dimensional critical Anisotropic KPZ equation. Even if in all of these cases it is expected that $D(t)\sim\sqrt{\log t}$, to the best of the authors' knowledge, this is the first instance in which such precise asymptotics is rigorously established.

math.PR↗

arXiv2021

Weak coupling limit of the Anisotropic KPZ equation

In the present work, we study the two-dimensional anisotropic KPZ equation (AKPZ), which is formally given by \begin{equation*} \partial_t h=\tfrac12 \Delta h + \lambda ((\partial_1 h)^2)-(\partial_2 h)^2) +\xi\,, \end{equation*} where $\xi$ denotes a space-time white noise and $\lambda>0$ is the so-called coupling constant. The AKPZ equation is a {\it critical} SPDE, meaning that not only it is analytically ill-posed but also the breakthrough path-wise techniques for singular SPDEs [M. Hairer, Ann. Math. 2014] and [M. Gubinelli, P. Imkeller and N. Perkowski, Forum of Math., Pi, 2015] are not applicable. As shown in [G. Cannizzaro, D. Erhard, F. Toninelli, arXiv, 2020], the equation regularised at scale $N$ has a diffusion coefficient that diverges logarithmically as the regularisation is removed in the limit $N\to\infty$. Here, we study the \emph{weak coupling limit} where $\lambda=\lambda_N=\hat\lambda/\sqrt{\log N}$: this is the correct scaling that guarantees that the nonlinearity has a still non-trivial but non-divergent effect. In fact, as $N\to\infty$ the sequence of equations converges to the linear stochastic heat equation \begin{equation*} \partial_t h =\tfrac{\nu_{\rm eff}}{2} \Delta h + \sqrt{\nu_{\rm eff}}\xi\,, \end{equation*} where $\nu_{\rm eff} >1$ is explicit and depends non-trivially on $\hat\lambda$. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearised via Cole-Hopf or any other transformation.

math.PR↗

arXiv2020

Logarithmic superdiffusivity of the 2-dimensional anisotropic KPZ equation

We study an anisotropic variant of the two-dimensional Kardar-Parisi-Zhang equation, that is relevant to describe growth of vicinal surfaces and has Gaussian, logarithmically rough, stationary states. While the folklore belief (based on one-loop Renormalization Group) is that the equation has the same scaling behaviour as the (linear) Edwards-Wilkinson equation, we prove that, on the contrary, the non-linearity induces the emergence of a logarithmic super-diffusivity. This phenomenon is similar in flavour to the super-diffusivity for two-dimensional fluids and driven particle systems.

cond-mat.stat-mech↗

arXiv2020

The stationary AKPZ equation: logarithmic superdiffusivity

We study the two-dimensional Anisotropic KPZ equation (AKPZ) formally given by \begin{equation*} \partial_t H=\frac12\Delta H+\lambda((\partial_1 H)^2-(\partial_2 H)^2)+\xi\,, \end{equation*} where $\xi$ is a space-time white noise and $\lambda$ is a strictly positive constant. While the classical two-dimensional KPZ equation, whose nonlinearity is $|\nabla H|^2=(\partial_1 H)^2+(\partial_2 H)^2$, can be linearised via the Cole-Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field) is superdiffusive: its diffusion coefficient diverges for large times as $\sqrt{\log t}$ up to $\log\log t$ corrections, in a Tauberian sense. Morally, this says that the correlation length grows with time like $t^{1/2}\times (\log t)^{1/4}$. Moreover, we show that if the process is rescaled diffusively ($t\to t/\varepsilon^2, x\to x/\varepsilon, \varepsilon\to0$), then it evolves non-trivially already on time-scales of order approximately $1/\sqrt{|\log\varepsilon|}\ll1$. Both claims hold as soon as the coefficient $\lambda$ of the nonlinearity is non-zero. These results are in contrast with the belief, common in the mathematics community, that the AKPZ equation is diffusive at large scales and, under simple diffusive scaling, converges the two-dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e. the case $\lambda=0$).

math.PR↗

arXiv2014

How quickly can we sample a uniform domino tiling of the 2L x 2L square via Glauber dynamics?

TThe prototypical problem we study here is the following. Given a $2L\times 2L$ square, there are approximately $\exp(4KL^2/π)$ ways to tile it with dominos, i.e. with horizontal or vertical $2\times 1$ rectangles, where $K\approx 0.916$ is Catalan's constant [Kasteleyn '61, Temperley-Fisher '61]. A conceptually simple (even if computationally not the most efficient) way of sampling uniformly one among so many tilings is to introduce a Markov Chain algorithm (Glauber dynamics) where, with rate $1$, two adjacent horizontal dominos are flipped to vertical dominos, or vice-versa. The unique invariant measure is the uniform one and a classical question [Wilson 2004,Luby-Randall-Sinclair 2001] is to estimate the time $T_{mix}$ it takes to approach equilibrium (i.e. the running time of the algorithm). In [Luby-Randall-Sinclair 2001, Randall-Tetali 2000], fast mixin was proven: $T_{mix}=O(L^C)$ for some finite $C$. Here, we go much beyond and show that $c L^2\le T_{mix}\le L^{2+o(1)}$. Our result applies to rather general domain shapes (not just the $2L\times 2L$ square), provided that the typical height function associated to the tiling is macroscopically planar in the large $L$ limit, under the uniform measure (this is the case for instance for the Temperley-type boundary conditions considered in [Kenyon 2000]). Also, our method extends to some other types of tilings of the plane, for instance the tilings associated to dimer coverings of the hexagon or square-hexagon lattices.

math.PR↗