Controlled stochastic growth of structures in the zero-temperature Ising model

Abstract

The optimization of structure growth in spatially stochastic systems poses a compelling challenge in a wide range of applications, including biological pattern formation and solution-based crystallization. In this talk, we investigate control strategies for a two-dimensional zero-temperature Ising model on a finite square lattice evolving under Metropolis dynamics. We consider a setting in which an external controller aims to drive the system from a configuration containing one or two small droplets of +-spins toward the all-plus configuration by flipping selected spins at prescribed times. To analyze this control problem, we formulate it as a Markov decision process (MDP), a classical framework for sequential decision-making problems under uncertainty. To compute an optimal policy, we construct a simplified model by reducing the configuration space to the local minima of the Hamiltonian. Leveraging structural properties of this simplified model, we characterize the optimal policy by solving the Bellman equations in a recursive manner. Finally, we present simulation results demonstrating the performance of the optimal policy and illustrating the induced growth mechanism. Moreover, we show that its geometric features persist at low but positive temperatures.

About Maike

Maike is a PhD student at the University of Twente. Her research focusses on the control of spatially interacting Markov processes. Apart from that, she is an enthusiastic violinist.