Minimizing the Profligacy of Searches with Reset

TL;DR

This paper introduces a framework for optimizing search strategies by balancing expected cost and success probability, revealing complex phase transitions in diffusion with resetting.

Contribution

It defines the profligacy of search and derives the optimal strategy, uncovering intricate phase behaviors including continuous, discontinuous, and re-entrant transitions.

Findings

Optimal resetting amount depends nontrivially on parameters.

Identifies classical and non-standard phase transitions.

Reveals complex behaviors like tricritical points and re-entrant transitions.

Abstract

We introduce the profligacy of a search process as a competition between its expected cost and the probability of finding the target. The arbiter of the competition is a parameter $\lambda$ that represents how much a searcher invests into increasing the chance of success. Minimizing the profligacy with respect to the search strategy specifies the optimal search. We show that in the case of diffusion with stochastic resetting, the amount of resetting in the optimal strategy has a highly nontrivial dependence on model parameters resulting in classical continuous transitions, discontinuous transitions and tricritical points as well as non-standard discontinuous transitions exhibiting re-entrant behavior and overhangs.