Improved intolerance intervals and size bounds for a Schelling-type spin system

Abstract

A spin system with long-range interacting particles is considered. The initial states of the particles are chosen uniformly at random and they are located at the nodes of a flat torus . Each node of the torus is connected to the nodes located in an l ∞-ball of radius w in the toroidal space centered at itself and it is assumed that h is exponentially larger than w 2. According to the states of the neighboring particles and based on the value of a common intolerance threshold τ, every particle is labeled ‘stable’ or ‘unstable’. Furthermore, unstable particles that can potentially become stable by flipping their states are labeled as ‘p-stable’. Finally, p-stable particles that remain so for a random, independent and identically distributed waiting time, will flip their states and become stable. When the waiting times have exponential distributions and τ ⩽ 1/2, this model is equivalent to a Schelling model of self-organized segregation in an open system, a zero-temperature Ising model with Glauber dynamics, or an asynchronous cellular automaton with extended Moore neighborhoods. Previous work has shown that if the intolerance parameter of the model τ ∈ (≈0.488, ≈0.512)\\{1/2}, then for a sufficiently large neighborhood of interaction N = (2w + 1)2, any particle will end up in an exponentially large ‘monochromatic’ region of particles of the same state, with high probability (w.h.p.). In this paper the above results are extended to the larger interval τ ∈ (≈0.433, ≈0.567)\\{1/2}, and the bounds on the size of the monochromatic ball are also improved by exponential factors. Furthermore, it is shown that when particles are placed on the infinite lattice rather than on a flat torus, for the values of τ mentioned above, sufficiently large N, and after a sufficiently long evolution time, any particle is contained in a large monochromatic ball of size exponential in N, w.h.p.