Related papers: Critical two-point function for long-range models …
We consider the following oriented percolation model of $\mathbb {N} \times \mathbb{Z}^d$: we equip $\mathbb {N}\times \mathbb{Z}^d$ with the edge set $\{[(n,x),(n+1,y)] | n\in \mathbb {N}, x,y\in \mathbb{Z}^d\}$, and we say that each edge…
We extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian…
We investigate phase transitions of two-dimensional Ising models with power-law interactions, using an efficient Monte Carlo algorithm. For slow decay, the transition is of the mean-field type; for fast decay, it belongs to the short-range…
The Pomeau-Manneville map is a paradigmatic intermittent dynamical system exhibiting weak chaos and anomalous dynamics. In this paper we analyse the parameter dependence of superdiffusion for the map lifted periodically onto the real line.…
We consider the Ising model at its critical temperature with external magnetic field $ha^{15/8}$ on the square lattice with lattice spacing $a$. We show that the truncated two-point function in this model decays exponentially with a rate…
We study the cluster-size distribution of supercritical long-range percolation on $\mathbb{Z}^d$, where two vertices $x,y\in\mathbb{Z}^d$ are connected by an edge with probability $\mathrm{p}(\|x-y\|):=p\min(1,\beta\|x-y\|)^{-d\alpha}$ for…
We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions $d=1,2$, where $x$ and $y$ a re connected with probability $\sim\beta/\|x-y\|^{-s}$. We show that when $d<s<2d$ the walk is…
Using the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm, we simulate the two-dimensional (2D) site-diluted Ising model. Since we can tune the critical point of each random sample automatically with the PCC…
The spatial coverage produced by a single discrete-time random walk, with asymmetric jump probability $p\neq 1/2$ and non-uniform steps, moving on an infinite one-dimensional lattice is investigated. Analytical calculations are complemented…
The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by…
Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d_c = 4. In the framework of single-species…
In the neighbourhood of the critical point, the correlation length of the spin-spin correlation function of the two-dimensional Ising model diverges. The correlation function permits a scaling limit in which the separation $N$ between spins…
The main result of this paper is a general central limit theorem for distributions defined by certain renewal type equations. We apply this to weakly self-avoiding random walks. We give good error estimates and Gaussian tail estimates which…
We have studied the conductance distribution function of two-dimensional disordered noninteracting systems in the crossover regime between the diffusive and the localized phases. The distribution is entirely determined by the mean…
We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinsic graph metric). Kesten (1986) famously showed that this is true for the…
We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for $p=p_c+\lambda\delta^{1/\nu}$, with $\nu=4/3$, as the lattice spacing $\delta \to 0$. Our proposed framework extends previous analyses for $p=p_c$, based…
We prove the existence as well as regularity of a finite range decomposition for the resolvent $G_{\alpha} (x-y,m^2) = ((-\Delta)^{\alpha\over 2} + m^{2})^{-1} (x-y) $, for $0<\alpha<2$ and all real $m$, in the lattice ${\mathbf Z}^{d}$ as…
The effect of blocking between different species occurring in one dimension is investigated here numerically in the case of particles following branching and annihilating random walk with two offsprings. It is shown that two-dimensional…
We analyse the critical properties of a weakly diluted (random) Ising model with the long-range interaction decaying with distance $x$ as $\sim x^{-d-\sigma}$ in a $d$-dimensional space. It is known to belong to a new long-range random…
We consider the critical behavior of two-dimensional layered Ising models where the exchange couplings between neighboring layers follow hierarchical sequences. The perturbation caused by the non-periodicity could be irrelevant, relevant or…