Antal A. Járai

On the monoid of singular braids

Abstract We show that the singular braid monoid on 3 strands embeds in the integral group ring of the braid group proving a special case of a conjecture originating from J.S. Birman (1993). The statement is proven by developing general results connected to the conjecture, including an extension of Artins braiding theorem to a special class of singular braids. We prove two corollaries which are also in strong connection with the conjecture and discuss the difficulties arising when the number of strands is greater than 3.

Approaching criticality via the zero dissipation limit in the abelian avalanche model

The discrete height abelian sandpile model was introduced by Bak, Tang, Wiesenfeld and Dhar as an example for the concept of self-organized criticality. When the model is modified to allow grains to disappear on each toppling, it is called bulk-dissipative. We provide a detailed study of a continuous height version of the abelian sandpile model, called the abelian avalanche model, which allows an arbitrarily small amount of dissipation to take place on every toppling. We prove that for non-zero dissipation, the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. We show that in the whole non-zero dissipation regime, the model is not critical, i.e., spatial covariances of local observables decay exponentially. We then study the zero dissipation limit and prove that the self-organized critical model is recovered, both for the stationary measure and for the dynamics. We obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality. We rigorously establish the mean-field value of this exponent for

Inequalities for critical exponents in

d > 4

-dimensional sandpiles

d>4.

The lowest crossing in two-dimensional critical percolation

We use the lace expansion to analyse networks of mutually-avoiding self-avoiding walks, having the topology of a graph. The networks are defined in terms of spread-out self-avoiding walks that are permitted to take large steps. We study the asymptotic behaviour of networks in the limit of widely separated network branch points, and prove Gaussian behaviour for sufficiently spread-out networks on

Anchored burning bijections on finite and infinite graphs

{{\mathbb{Z}}^{d}}

The Sandpile Cellular Automaton

in dimensions

Phase transition in a sequential assignment problem on graphs

d\,>\,4