Sharp bounds on connectivity functions for random walks and random interlacements
File(s)
Author(s)
Shulzhenko, Yuriy
Type
Thesis
Abstract
Consider the vacant set $\mathcal{V}_N$ induced by a simple random walk on a $d$-dimensional torus with side length $N$. We determine the typical size of the largest vacant cluster in the subcritical regime $u>u_{*}$, and the second largest cluster in the supercritical regime $u<u_{*}$ for all dimensions $d\geq3$. In the case $d\geq4$, these are shown to have a diameter which is logarithmic, meanwhile when $d=3$, an additional, multiplicative, doubly logarithmic term appears, coinciding with other long-range correlation models such as the Gaussian free field level set. Moreover, for $d=3$, we obtain the precise polynomial divergence of cluster sizes as $u\rightarrow u_{*}$. \\
Our findings stem from matching upper and lower bounds which we derive for the truncated two-point connectivity function for the random interlacements vacant set $\mathcal{V}^u$. In the case of the lower bounds, a change-of-measure argument is used in order to apply spatial modulations to the random interlacements, rendering them locally sub- or supercritical, depending on whether we wish to insulate, or facilitate, connections. A novelty in our approach is that extra care is taken to deal with rotational invariance of the resulting connectivity function, requiring new capacity bounds on oblique tubes in the case of $d=3$. \\
For the upper bounds, we invoke a localisation and bootstrapping argument, which effectively handles the obstacle of long-range correlations inherently present in our model. In order to deal with the non-monotonicity appearing in the supercritical regime, a new approach involving sub-collections of the random interlacements is presented, effectively reformulating the required events in a monotonic way in exchange for an acceptable combinatorial cost. As a final highlight, we tailor an algorithmic approach to overcome the structural rigidity enforced by the connectedness of trajectories comprising the random interlacements model.
Our findings stem from matching upper and lower bounds which we derive for the truncated two-point connectivity function for the random interlacements vacant set $\mathcal{V}^u$. In the case of the lower bounds, a change-of-measure argument is used in order to apply spatial modulations to the random interlacements, rendering them locally sub- or supercritical, depending on whether we wish to insulate, or facilitate, connections. A novelty in our approach is that extra care is taken to deal with rotational invariance of the resulting connectivity function, requiring new capacity bounds on oblique tubes in the case of $d=3$. \\
For the upper bounds, we invoke a localisation and bootstrapping argument, which effectively handles the obstacle of long-range correlations inherently present in our model. In order to deal with the non-monotonicity appearing in the supercritical regime, a new approach involving sub-collections of the random interlacements is presented, effectively reformulating the required events in a monotonic way in exchange for an acceptable combinatorial cost. As a final highlight, we tailor an algorithmic approach to overcome the structural rigidity enforced by the connectedness of trajectories comprising the random interlacements model.
Version
Open Access
Date Issued
2024-09-13
Date Awarded
01/12/2024
License URL
Advisor
Rodriguez, Pierre-François
Goswami, Subhajit
Li, Xue-Mei
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/S023925/1
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)