Alessandra Faggionato

We consider a continuous-time random walk on the $d$-dimensional torus $\mathbb{T}^d_{N}=\mathbb{Z}^d/N \mathbb{Z}^d$, possibly with long-range, but finite, jumps. The law of the jumps is regulated by a random environment $ξ$ yielding a stationary and ergodic field of random conductances. The complex mobility matrix $σ_N^ξ(ω)$ measures the linear response of the random walk to a $\cos(ωt)$-type oscillating external field. By investigating the homogenization properties of the medium, and assuming in addition that the conductances have finite second moment, we show that, for almost every realization of the environment $ξ$, the complex mobility matrix $σ_N^ξ(ω)$ converges as $N\to+\infty$ to a deterministic limiting matrix $σ(ω)$ and provide different characterizations of $σ(ω)$.

We consider random resistor networks with nodes given by a point process on $\mathbb{R}^d$ and with random conductances. The length range of the electrical filaments can be unbounded. We assume that the randomness is stationary and ergodic w.r.t. the action of the group $\mathbb{G}$, given by $\mathbb{R}^d$ or $\mathbb{Z}^d$. This action is covariant w.r.t. translations on the Euclidean space. Under minimal assumptions we prove that a.s. the suitably rescaled directional conductivity of the resistor network along the principal directions of the effective homogenized matrix $D$ converges to the corresponding eigenvalue of $D$ times the intensity of the point process. More generally, we prove a quenched scaling limit of the directional conductivity along any vector $e\in {\rm Ker}(D) \cup {\rm Ker}(D)^\perp$. Our results cover plenty of models including e.g. the standard conductance model on $\mathbb{Z}^d$ (also with long filaments), the Miller-Abrahams resistor network for conduction in amorphous solids (to which we can now extend the bounds in agreement with Mott's law previously obtained in \cite{CP1,FM,FSS} for Mott's random walk), resistor networks on the supercritical cluster in lattice and continuum percolations, resistor networks on crystal lattices and on Delaunay triangulations.

We consider continuous-time random walks on a random locally finite subset of $$\mathbb {R}^d$$R d with random symmetric jump probability rates. The jump range can be unbounded. We assume some second-moment conditions and that the above randomness is left invariant by the action of the group $$\mathbb {G}=\mathbb {R}^d$$G = R d or $$\mathbb {G}=\mathbb {Z}^d$$G = Z d . We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space–time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic equation is non-random and governed by the effective homogenized matrix D of the single random walk, which can be degenerate. The above result covers a very large family of models including e.g. simple exclusion processes built from random conductance models on $$\mathbb {Z}^d$$Z d and on crystal lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters.

We consider continuous-time random walks on a random locally finite subset of $\mathbb{R}^d$ with random symmetric jump probability rates. The jump range can be unbounded. We assume some second--moment conditions and that the above randomness is left invariant by the action of the group $\mathbb{G}=\mathbb{R}^d$ or $\mathbb{G}=\mathbb{Z}^d$. We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space-time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic equation is non-random and governed by the effective homogenized matrix $D$ of the single random walk, which can be degenerate. The above result covers a very large family of models including e.g. simple exclusion processes built from random conductance models on $\mathbb{Z}^d$ and on crystal lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters.

We consider random walks on the support of a random purely atomic measure on $\mathbb{R}^d$ with random jump probability rates. The jump range can be unbounded. The purely atomic measure is reversible for the random walk and stationary for the action of the group $\mathbb{G}=\mathbb{R}^d$ or $\mathbb{G}=\mathbb{Z}^d$. By combining two-scale convergence and Palm theory for $\mathbb{G}$-stationary random measures and by developing a cut-off procedure, under suitable second moment conditions we prove for almost all environments the homogenization for the massive Poisson equation of the associated Markov generators. In addition, we obtain the quenched convergence of the $L^2$-Markov semigroup and resolvent of the diffusively rescaled random walk to the corresponding ones of the Brownian motion with covariance matrix $2D$. For symmetric jump rates, the above convergence plays a crucial role in the derivation of hydrodynamic limits when considering multiple random walks with site-exclusion or zero range interaction. We do not require any ellipticity assumption, neither non-degeneracy of the homogenized matrix $D$. Our results cover a large family of models, including e.g. random conductance models on $\mathbb{Z}^d$ and on general lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, simple random walks on supercritical percolation clusters.

We consider random graphs $\mathcal{G}$ built on a homogeneous Poisson point process on $\mathbb{R}^d$, $d\geq 2$, with points $x$ marked by i.i.d. random variables $E_x$. Fixed a symmetric function $h(\cdot, \cdot)$, the vertexes of $\mathcal{G}$ are given by points of the Poisson point process, while the edges are given by pairs $\{x,y\}$ with $x\not =y$ and $|x-y|\leq h(E_x,E_y)$. We call $\mathcal{G}$ Poisson $h$-generalized Boolean model, as one recovers the standard Poisson Boolean model by taking $h(a,b):=a+b$ and $E_x\geq 0$. Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left-right crossings in a box of size $n$ is lower bounded by $Cn^{d-1}$ apart from an event of exponentially small probability. As special applications, when the marks are non-negative, we consider the Poisson Boolean model and its generalization to $h(a,b)=(a+b)^\gamma$ with $\gamma>0$, the weight-dependent random connection models with max-kernel and with min-kernel and the graph obtained from the Miller-Abrahams random resistor network in which only filaments with conductivity lower bounded by a fixed positive constant are kept.