Related papers: Diffusion and spectral dimension on Eden tree
We study simple random walk on the uniform spanning tree on Z^2 . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean…
We calculate the spectral dimension of a wide class of tree-like fractals by solving the random walk problem through a new analytical technique, based on invariance under generalized cutting-decimation transformations. These fractals are…
We relate the fractal dimension of the backbone, and the spectral dimension of Eden trees to the dynamical exponent z. In two dimensions, it gives fractal dimension of backbone equal to 4/3 and spectral dimension of trees equal to 5/4. In…
The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of…
We analyse how simple local constraints in two dimensions lead a defect to exhibit robust, non-transient, and tunable, subdiffusion. We uncover a rich dynamical phenomenology realised in ice- and dimer-type models. On the microscopic scale…
In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree $n$ with an independent copy of a graph $G_n$ and gluing the inserted graphs along the…
We present a simple yet rigorous approach to the determination of the spectral dimension of random trees, based on the study of the massless limit of the Gaussian model on such trees. As a byproduct, we obtain evidence in favor of a new…
A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in…
We discuss the scaling properties of free branched polymers. The scaling behaviour of the model is classified by the Hausdorff dimensions for the internal geometry: d_L and d_H, and for the external one: D_L and D_H. The dimensions d_H and…
It has recently been shown that networks possessing scale-free and fractal properties may exhibit a bifractal nature, in which local structures are described by two different fractal dimensions. In this study, we investigate random walks on…
We analyze random walk through fractal environments, embedded in 3-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight…
We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an infinite…
We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant…
We consider a random walk on a Galton-Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive…
The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and…
The spectral dimension has been widely used to understand transport properties on regular and fractal lattices. Nevertheless, it has been little studied for complex networks such as scale-free and small world networks. Here we study the…
We consider random walks (RWs) and self-avoiding walks (SAWs) on disordered lattices directly at the percolation threshold. Applying numerical simulations, we study the scaling behavior of the models on the incipient percolation cluster in…
We define generic ensembles of infinite trees. These are limits as $N\to\infty$ of ensembles of finite trees of fixed size $N$, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices…
We measure the fractal dimension of loop-erased random walk (LERW) in 3 dimensions, and estimate that it is 1.62400 +- 0.00005. LERW is closely related to the uniform spanning tree and the abelian sandpile model. We simulated LERW on both…
Consider the long-range percolation model on the integer lattice $\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise $x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$, independently of the…