arXiv subjects

Edward Mottram

Publications and source records attributed to Edward Mottram.

2 recordsLinked to original sources

arXiv2014

A universal exponent for Brownian entropic repulsion

We investigate the extent to which the phenomenon of Brownian entropic repulsion is universal. Consider a Brownian motion conditioned on the event $\mathcal{E}$ -- that its local time is bounded everywhere by 1. This event has probability zero and so must be approximated by events of positive probability. We prove that several natural quantities, in particular the speed of the process, are highly sensitive to the approximation procedure, and hence are not universal. However, we also propose an exponent $κ$ -- which measures the strength of the entropic repulsion by evaluating the probability that a particular point comes close to violating the condition $\mathcal{E}$. We show that $κ=3$ for several natural approximations of $\mathcal{E}$, and conjecture that $κ=3$ is universal in a sense that we make precise.

math.PR↗

arXiv2013

Percolation with constant freezing

We introduce and study a model of percolation with constant freezing (PCF) where edges open at constant rate 1, and clusters freeze at rate αindependently of their size. Our main result is that the infinite volume process can be constructed on any amenable vertex transitive graph. This is in sharp contrast to models of percolation with freezing previously introduced, where the limit is known not to exist. Our interest is in the study of the percolative properties of the final configuration as a function of α. We also obtain more precise results in the case of trees. Surprisingly the algebraic exponent for the cluster size depends on the degree, suggesting that there is no lower critical dimension for the model. Moreover, even for α<α_c, it is shown that finite clusters have algebraic tail decay, which is a signature of self organised criticality. Partial results are obtained on Z^d, and many open questions are discussed.

math.PR↗