We consider the zero-average Gaussian free field on a certain class of finite \(d\)-regular graphs for fixed \(d\ge 3\). This class includes \(d\)-regular expanders of large girth and typical realisations of random \(d\)-regular graphs. We show that the level set of the zero-average Gaussian free field above level \(h\) has a giant component in the whole supercritical phase, that is for all \(h<h_\star\), with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [AC20b], where it was shown that a linear fraction of vertices is in mesoscopic components if \(h<h_\star\), and together with the description of the subcritical phase from [AC20b] establishes a fully-fledged percolation phase transition for the model.

We investigate the uniform boundedness of the fronts of the solutions to the randomized Fisher-KPP equation and to its linearization, the parabolic Anderson model. It has been known that for the standard (i.e. deterministic) Fisher-KPP equation, as well as for the special case of a randomized Fisher-KPP equation with so-called ignition type nonlinearity, one has a uniformly bounded (in time) transition front. Here, we show that this property of having a uniformly bounded transition front fails to hold for the general randomized Fisher-KPP equation. Nevertheless, we establish that this property does hold true for the parabolic Anderson model.

We study the simple random walk on the configuration model with given degree sequence \((d_1^n, \dots ,d_n^n)\) and investigate the connected components of its vacant set at level \(u>0\). We show that the size of the maximal connected component exhibits a phase transition at level \(u^*\) which can be related with the critical parameter of random interlacements on a certain Galton-Watson tree. We further show that there is a critical window of size \(n^{-1/3}\) around \(u^*\) in which the largest connected components of the vacant set have a metric space scaling limit resembling the one of the critical Erdős-Rényi random graph.

We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed \(p\)-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then the resulting function has trivial geometry, that is only two critical points. This contrasts with the situation of no or weak external field where these functions typically have an exponential number of critical points. We give an explicit threshold \(h_c\) for the magnitude of the external fieldnecessary for trivialization and conjecture \(h_c\) to be sharp. The Kac-Rice formula is our main tool. Our work extends [Fyo15], which identified the trivial regime for the special case of pure \(p\)-spin Hamiltonians with random external field.

We consider the zero-average Gaussian free field on a certain class of finite \(d\)-regular graphs for fixed \(d\geq 3\). This class includes \(d\)-regular expanders of large girth and typical realisations of random \(d\)-regular graphs. We show that the level set of the zero-average Gaussian free field above level \(h\) exhibits a phase transition at level \(h_\star\), which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite \(d\)-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level \(h\) does not contain any connected component of larger than logarithmic size whenever \(h>h_\star\), and on the contrary, whenever \(h<h_\star\), a linear fraction of the vertices is contained in connected components of the level set above level \(h\) having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase \(h<h_\star\), as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level \(h\). The proofs in this article make use of results from the accompanying paper [AC1].

We study level-set percolation of the Gaussian free field on the infinite \(d\)-regular tree for fixed \(d\geq 3\). Denoting by \(h_\star\) the critical value, we obtain the following results: for \(h>h_\star\) we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level \(h\); for \(h<h_\star\) we prove that the number of vertices connected over distance \(k\) above level \(h\) to a fixed vertex grows exponentially in \(k\) with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level \(h\), at least away from the critical value \(h_\star\). Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value \(h_\star\) and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [AC2].

We consider branching random walk in spatial random branching environment (BRWRE) in dimension one, as well as related differential equations: the Fisher-KPP equation with random branching and its linearized version, the parabolic Anderson model (PAM). When the random environment is bounded, we show that after recentering and scaling, the position of the maximal particle of the BRWRE, the front of the solution of the PAM, as well as the front of the solution of the randomized Fisher-KPP equation fulfill quenched invariance principles. In addition, we prove that at time t the distance between the median of the maximal particle of the BRWRE and the front of the solution of the PAM is in \(O(\ln t)\). This partially transfers results from [Bra78] to the setting of BRWRE.

We study Markov processes with values in the space of general two-dimensional arrays whose distribution is exchangeable. The results of this paper are inspired by the theory of exchangeable dynamical random graphs developed by H. Crane.

In [CW17], it was shown that the clock process associated with the Metropolis dynamics of the Random Energy Model converges to an \(\alpha\)-stable process, after being scaled by a random, Hamiltonian dependent, normalisation. We prove here that this random normalisation can be replaced by a deterministic one.

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the 'oriented percolation universality class'. If the influence of the random medium on the walk is small in space-time regions where the medium is 'typical', we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining.
Such random walks occur naturally as the spatial embedding of an ancestral lineage in spatial population models with local regulation. We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high, thus partly settling a question from Depperschmidt (2008) on the behaviour of their ancestral lines.

Consider the subgraph of the discrete \(d\)-dimensional torus of size length \(N\), \(d\geq 3\), induced by the range of the simple random walk on the torus run until the time \(uN^d\). We prove that for all \(d\geq 3\) and \(u>0\), the mixing time for the random walk on this subgraph is of order \(N^2\) with probability at least \(1 - Ce^{-(\log N)^2}\).

For \(d\ge 3\) we construct a new coupling of the trace left by a random walk on a large \(d\)-dimensional discrete torus with the random interlacements on \(\mathbb{Z}^d\). This coupling has the advantage of working up to \emph{macroscopic} subsets of the torus. As an application, we show a sharp phase transition for the diameter of the component of the vacant set on the torus containing a given point. The threshold where this phase transition takes place coincides with the critical value \(u_\star(d)\) of random interlacements on \(\mathbb Z^d\). Our main tool is a variant of the \emph{soft-local time} coupling technique of [PT12].

We study the Metropolis dynamics of the simplest mean-field spin glass model, the Random Energy Model. We show that this dynamics exhibits aging by showing that the properly rescaled time change process between the Metropolis dynamics and a suitably chosen `fast' Markov chain converges in distribution to a stable subordinator. The rescaling might depend on the realization of the environment, but we show that its exponential growth rate is deterministic.

We give a complete classification of scaling limits of randomly trapped random walks and associated clock processes on \(\mathbb Z^d\), \(d\ge 2\). Namely, under the hypothesis that the discrete skeleton of the randomly trapped random walk has a slowly varying return probability, we show that the scaling limit of its clock process is either deterministic linearly growing or a stable subordinator. In the case when the discrete skeleton is a simple random walk on \(\mathbb Z^d\), this implies that the scaling limit of the randomly trapped random walk is either Brownian motion or the Fractional Kinetics process, as conjectured in [BCCR13].

We introduce a general model of trapping for random walks on graphs. We give the possible scaling limits of these Randomly Trapped Random Walks on Z. These scaling limits include the well known Fractional Kinetics process, the Fontes-Isopi-Newman singular diffusion as well as a new broad class we call Spatially Subordinated Brownian Motions. We give sufficient conditions for convergence and illustrate these on two important examples.

We consider directed random walk on the infinite percolation cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the `ancestral lineage' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) with a regeneration approach. Via an analysis of joint renewals of two independent walks on the same cluster, we obtain furthermore a quenched central limit theorem (i.e. for almost any realisation of the cluster) in dimensions 1+d with d ≥ 2.

We review and comment recent research on random interlacements model introduced by A.-S. Sznitman in [Szn10]. A particular emphasis is put on motivating the definition of the model via natural questions concerning geometrical/percolative properties of random walk trajectories on finite graphs, as well as on presenting some important techniques used in random interlacements' literature in the most accessible way. This text is an expanded version of the lecture notes for the mini-course given at the XV Brazilian School of Probability in 2011.

We prove a shape theorem for the internal (graph) distance on the interlacement set I^u of the random interlacement model on Z^d, d ≥ 3. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.

We consider the simple random walk on a random d-regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. It was shown in [CTW11] that this so-called vacant set exhibits a phase transition at u=u_*: there is a giant component if u<u_* and only small components when u>u_*. In this paper we show the existence of a critical window of size n^-1/3 around u_*. In this window the size of the largest cluster is of order~n^2/3.

We give an asymptotic evaluation of the complexity of spherical p-spin spin-glass models via random matrix theory. This study enables us to obtain detailed information about the bottom of the energy landscape, including the absolute minimum (the ground state), the other local minima, and describe an interesting layered structure of the low critical values for the Hamiltonians of these models. We also show that our approach allows us to compute the related TAP-complexity and extend the results known in the physics literature. As an independent tool, we prove a LDP for the k-th largest eigenvalue of the GOE, extending the results of Ben Arous, Dembo and Guionnett (2001).

We consider a random walk among unbounded random conductances on the two-dimensional integer lattice. When the distribution of the conductances has an infinite expectation and a polynomial tail, we show that the scaling limit of this process is the fractional kinetics process. This extends the results of the paper [BC09] where a similar limit statement was proved in dimension d ≥ 3. To make this extension possible, we prove several estimates on the Green function of the process killed on exiting large balls.

We study the trajectory of a simple random walk on a d-regular graph with d >2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u_* ∈ (0,∞) such that, with high probability as n grows, if u<u_*, then the largest component of the vacant set has a volume of order n, and if u>u_*, then it has a volume of order log n. The critical value u_* coincides with the critical intensity of a random interlacement process on a d-regular tree. We also show that the random interlacements model describes the structure of the vacant set in local neighbourhoods.

We consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail. We prove, that the scaling limit of this process is a Fractional-Kinetics process -- that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator. We further show, that the same process appears in the scaling limit of the non-symmetric Bouchaud's trap model.

We give a new proof of aging for a version of a Glauber dynamics in the Random Energy Model. The proof uses ideas that were developed in [BBC08] for studying the dynamics of a p-spin Sherrington-Kirkpatrick spin glass.

We consider a version of a Glauber dynamics for a p-spin Sherrington-Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for any p ≥ 3 and any inverse temperature β > 0, there exist constants ξ > 0, such that for all exponential time scales, exp(γN), with γ ≤ ξ, the properly rescaled clock process (time-change process), converges to an α-stable subordinator where α = γ/ β²< 1. Moreover, the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system), the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.

Aging has become the paradigm to describe dynamical behavior of glassy systems, and in particular spin glasses. Trap models have been introduced as simple caricatures of effective dynamics of such systems. In this Letter we show that in a wide class of mean field models and on a wide range of time scales, aging occurs precisely as predicted by the REM-like trap model of Bouchaud and Dean. This is the first rigorous result about aging in mean field models except for the REM and the spherical model.

We study the simple random walk on the n-dimensional hypercube, in particular its hitting times of large (possibly random) sets. We give simple conditions on these sets ensuring that the properly-rescaled hitting time is asymptotically exponentially distributed, uniformly in the starting position of the walk. These conditions are then verified for percolation clouds with densities that are much smaller than (n log n)⁻¹. A main motivation behind this paper is the study of the so-called aging phenomenon in the Random Energy Model (REM), the simplest model of a mean-field spin glass. Our results allow us to prove aging in the REM for all temperatures, thereby extending earlier results to their optimal temperature domain.

We study a two-species interacting particle model on a subset of Z with open boundaries. The two species are injected with time dependent rate on the left, resp. right boundary. Particles of different species annihilate when they try to occupy the same site. This model has been proposed as a simple model for the dynamics of an "order book" on a stock market. We consider the hydrodynamic scaling limit for the empirical process and prove a large deviation principle that implies convergence to the solution of a non-linear parabolic equation.

We give the "quenched" scaling limit of Bouchaud's trap model in d ≥ 2. This scaling limit is the Fractional-Kinetics process, that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator.

Let ℓ(n,x) be the local time of a random walk on Z². We prove a strong law of large numbers for the quantity for all α ≧ 0. We use this result to describe the distribution of the local time of a typical point in the range of the random walk.

We give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large two-dimensional tori and for trap dynamics of the Random Energy Model on a broad range of time scales.

These notes cover one of the topics of the class given in the Les Houches Summer School "Mathematical statistical physics" in July 2005. The lectures tried to give a summary of the recent mathematical results about the long-time behaviour of dynamics of (mean-field) spin-glasses and other disordered media. We have chosen here to restrict the scope of these notes to the dynamics of trap models only, but to cover this topic in somewhat more depth.

Let τ_x be a collection of i.i.d. positive random variables with distribution in the domain of attraction of α-stable law with α < 1. The symmetric Bouchaud's trap model on Z is a Markov chain X(t) whose transition rates are given by w_xy = (2τ_x)^-1 if x, y are neighbours in Z. We study the behaviour of two correlation functions: P[X(t_w+t) = X(t_w)] and P[X(t') = X(t_w) ∀t'∈[t_w,t_w+t]]. It is well known that for any of these correlation functions a time-scale t = f(t_w) such that aging occurs can be found. We study these correlation functions on time-scales different from f(t_w), and we describe more precisely the behaviour of a singular diffusion obtained as the scaling limit of Bouchaud's trap model.

We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.

Let E_x be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on Z² is a Markov chain X(t) whose transition rates are given by w_xy= ν exp(βE_x) if x, y are neighbours in Z². We study the behaviour of two correlation functions: P[X(t_w+t) = X(t_w)] and P[X(t') = X(t_w) ∀t'∈[t_w,t_w+t]]. We prove the (sub)aging behaviour of these functions when β > 1.

Let E_i be a collection of i.i.d. exponential random variables. Bouchaud's model on Z is a Markov chain X(t) whose transition rates are given by w_ij= exp(-β((1-a)E_i-aE_j)) if i, j are neighbours in Z. We study the behaviour of two correlation functions: and We prove the (sub)aging behaviour of these functions when β > 1 and a∈[0,1].

We prove the validity of the Critical path analysis for a continuum percolation model close to Golden-Kozlov one. This is obtained in the limit of strong disorder.

A random cluster measure on Z^d that is not translationally invariant is constructed for d > 2, the critical density p_c, and sufficiently large q. The resulting measure is proven to be a Gibbs state satisfying cluster model DLR-equations.