The ultimate goal of these lectures is to present the ergodic theory of the Burgers equation with random force in noncompact setting. Developing such a theory has been considered a hard problem since the first publication on the randomly forced inviscid Burgers [EKMS00]. It was solved in a recent work [BCK] for the forcing of Poissonian type. The Burgers equation is a basic fluid dynamic model,...

We report a Monte Carlo simulation study of the properties of highly asymmetric binary hard-sphere mixtures. This system is treated within an effective fluid approximation in which the large particles interact through a depletion potential [R. Roth, Phys. Rev. E 62 5360 (2000)] designed to capture the effects of a virtual sea of small particles. We generalize this depletion potential to include...

Model systems in which fluid particles move in a disordered matrix of immobile obstacles have been found to be a reasonable representation of a colloidal fluid confined in a disordered porous medium. For systems consisting of hard-sphere particles, the obstacle matrix partitions the space available to the fluid particles into voids of finite volume (“traps”) and a percolating void that extends ...

We study the distribution of the percolation time T of 2-neighbour bootstrap percolation on [n] with initial set A ∼ Bin([n], p). We determine T up to a constant factor with high probability for all p above the critical probability for percolation, and to within a 1 + o(1) factor for a large range of p.

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d > 6 for sufficient spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V 2/3 and below by a small consta...

We estimate locations of the regions of the percolation and of the non-percolation in the plane (λ, β): the Poisson rate – the inverse temperature, for interacted particle systems in finite dimension Euclidean spaces. Our results about the percolation and about the nonpercolation are obtained under different assumptions. The intersection of two groups of the assumptions reduces the results to t...

This work analyzes a percolation model on the diamond hierarchical lattice (DHL), where the percolation transition is retarded by the inclusion of a probability of erasing specific connected structures. It has been inspired by the recent interest on the existence of other universality classes of percolation models. The exact scale invariance and renormalization properties of DHL leads to recurr...

1 Executive Summary We have continued with a review on the bond percolation and the introduced the coupling of the bond percolation process, then defined the critical phenomenon and the percolation probability. There is a theorem about the percolation probability is a non decreasing function and we bounded it between 1/3 and 2/3 for the two dimensional case. 2 Bond Percolation 2.1 Preliminaries...