Abstract
We consider the random walk loop soup on the discrete half-plane and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to 1/2. The absence of percolation at intensity 1/2 was shown in a previous work. We also show that in the supercritical regime, one can keep only the loops up to some large enough upper bound on the diameter and still have percolation.
Highlights
We will consider discrete loops on Z2, that is to say finite paths to the nearest neighbours on Z2 that return to the origin and visit at least two vertices
In [3] are considered loops parametrised by continuous time rather than discrete time. μZ2 has a continuous analogue, the measure μC on the Brownian loops on C
Let Ptz,z (·) be the standard Brownian bridge probability measure from z to z of length t. μC is a measure on continuous time-parametrised loops on C defined as μC(·) :=
Summary
We will consider discrete (rooted) loops on Z2, that is to say finite paths to the nearest neighbours on Z2 that return to the origin and visit at least two vertices. Given α > 0 we will denote by LZα2 respectively LCα the Poisson ensemble of intensity αμZ2 respectively αμC, called random walk respectively Brownian loop soup. The outer boundaries of outermost clusters in a sub-critical Brownian loop soup were identified to be a Conformal. There the authors consider a Brownian loop soup in the half-plane and a continuous path cutting the half-plane, parametrised by the half-plane capacity. For such a path the half-plane capacity at time t equals 2t. That is to say the critical intensity parameter for the two-dimensional Brownian loop soups and random walk loop soups is the same.
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