Optimal strategies for the growth of dual-seeded lattice structures

TL;DR

This paper develops an optimal control framework for the growth of lattice structures modeled by a 2D zero-temperature Ising model, aiming to efficiently steer the system to a uniform positive state from dual initial clusters.

Contribution

It introduces a Markov decision process approach to optimize external perturbations for lattice growth, comparing different geometries and identifying optimal strategies in various regimes.

Findings

Optimal strategies depend on growth geometry and regime.

Markov decision process effectively models lattice growth control.

Strategies significantly improve convergence to uniform positive state.

Abstract

Optimal growth of structures governed by spatially stochastic dynamics arises in many scientific settings, for example in processes such as solution-based crystallization and the formation of microbial biofilms on patterned substrates or microfluidic networks. In this work, we investigate lattice growth using a two-dimensional, zero-temperature stochastic model of short-range spin interactions. Our goal is to determine how external perturbations can be optimized to steer the system efficiently toward the uniformly positive state, starting from two initial clusters of positive sites. To achieve this, we cast the problem as a Markov decision process adapted for a two-dimensional Ising model with zero-temperature dynamics. Within this framework, we compare alternative growth geometries and identify the structure of optimal strategies across three representative regimes.