English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We investigate a model of continuous-time simple random walk paths in$\mathbb{Z}^d$ undergoing two competing interactions: an attractive one towardsthe large values of a random potential, and a self-repellent one in the spiritof the well-known weakly self-avoiding random walk. We take the potential to bei.i.d.~Pareto-distributed with parameter $\alpha>d$, and we tune the strengthof the interactions in such a way that they both contribute on the same scaleas $t\to\infty$.  Our main results are (1) the identification of the logarithmic asymptotics ofthe partition function of the model in terms of a random variational formula,and, (2) the identification of the path behaviour that gives the overwhelmingcontribution to the partition function for $\alpha>2d$: the random-walk pathfollows an optimal trajectory that visits each of a finite number of randomlattice sites for a positive random fraction of time. We prove a law of largenumbers for this behaviour, i.e., that all other path behaviours give strictlyless contribution to the partition function. The joint distribution of thevariational problem and of the optimal path can be expressed in terms of alimiting Poisson point process arising by a rescaling of the random potential.The latter convergence is in distribution and is in the spirit of a standardextreme-value setting for a rescaling of an i.i.d. potential in large boxes,like in \cite{KLMS09}.

Weakly self-avoiding walk in a pareto-distributed random potential