Ioan Manolescu

Inhomogeneous bond percolation on square, triangular and hexagonal lattices

The star–triangle transformation is used to obtain an equivalence extending over the set of all (in)homogeneous bond percolation models on the square, triangular and hexagonal lattices. Among the consequences are box-crossing (RSW) inequalities for such models with parameter-values at which the transformation is valid. This is a step toward proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of inhomogeneous systems. The proofs extend to certain isoradial models to which previous methods do not apply.

Universality for bond percolation in two dimensions

All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the critical point (assuming the exponents exist). This is proved using the star–triangle transformation and the box-crossing property. The exponents in question are the one-arm exponent ρ, the 2j-alternating-arms exponents ρ2j for j≥1, the volume exponent δ, and the connectivity exponent η. By earlier results of Kesten, this implies universality also for the near-critical exponents β, γ, ν, Δ (assuming these exist) for any of these models that satisfy a certain additional hypothesis, such as the homogeneous bond percolation models on these three lattices.

Planar lattices do not recover from forest fires

Self-destructive percolation with parameters p,δ is obtained by taking a site percolation configuration with parameter p, closing all sites belonging to infinite clusters, then opening every closed site with probability δ, independently of the rest. Call θ(p,δ) the probability that the origin is in an infinite cluster in the configuration thus obtained. For two-dimensional lattices, we show the existence of δ>0 such that, for any p>pc, θ(p,δ)=0. This proves the conjecture of van den Berg and Brouwer [Random Structures Algorithms 24 (2004) 480–501], who introduced the model. Our results combined with those of van den Berg and Brouwer [Random Structures Algorithms 24 (2004) 480–501] imply the nonexistence of the infinite parameter forest-fire model. The methods herein apply to site and bond percolation on any two-dimensional planar lattice with sufficient symmetry.

On the probability that self-avoiding walk ends at a given point

We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Z^d for d>1. We show that the probability that a walk of length n ends at a point x tends to 0 as n tends to infinity, uniformly in x. Also, for any fixed x in Z^d, this probability decreases faster than n^{-1/4 + epsilon} for any epsilon >0. When |x|= 1, we thus obtain a bound on the probability that self-avoiding walk is a polygon.

The phase transitions of the random-cluster and Potts models on slabs with

We prove sharpness of the phase transition for the random-cluster model with

q \geq 1

q \geq 1

are sharp

on graphs of the form

Bond percolation on isoradial graphs: criticality and universality

\mathcal{S} := \mathcal{G} \times S

SCALING LIMITS AND INFLUENCE OF THE SEED GRAPH IN PREFERENTIAL ATTACHMENT TREES

, where

The phase transitions of the planar random-cluster and Potts models with q \ge 1 are sharp

\mathcal{G}