Zero-temperature Ising Model Research Articles

Order-disorder transition in the zero-temperature Ising model on random graphs.

The zero-temperature Ising model is known to reach a fully ordered ground state in sufficiently dense random graphs. In sparse random graphs, the dynamics gets absorbed in disordered local minima at magnetization close to zero. Here, we find that the nonequilibrium transition between the ordered and the disordered regime occurs at an average degree that slowly grows with the graph size. The system shows bistability: The distribution of the absolute magnetization in the reached absorbing state is bimodal, with peaks only at zero and unity. For a fixed system size, the average time to absorption behaves nonmonotonically as a function of average degree. The peak value of the average absorption time grows as a power law of the system size. These findings have relevance for community detection, opinion dynamics, and games on networks.

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

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.

Self-organized Segregation on the Grid

We consider an agent-based model in which two types of agents interact locally over a graph and have a common intolerance threshold $\tau$ for changing their types with exponentially distributed waiting times. The model is equivalent to an unperturbed Schelling model of self-organized segregation, an Asynchronous Cellular Automata (ACA) with extended Moore neighborhoods, or a zero-temperature Ising model with Glauber dynamics, and has applications in the analysis of social and biological networks, and spin glasses systems. Some rigorous results were recently obtained in the theoretical computer science literature, and this work provides several extensions. We enlarge the intolerance interval leading to the formation of large segregated regions of agents of a single type from the known size $\epsilon>0$ to size $\approx 0.134$. Namely, we show that for $0.433 < \tau < 1/2$ (and by symmetry $1/2<\tau<0.567$), the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood. We further extend the interval leading to large segregated regions to size $\approx 0.312$ considering "almost segregated" regions, namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows. In this case, we show that for $0.344 < \tau \leq 0.433$ (and by symmetry for $0.567 \leq \tau<0.656$) the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood. The exponential bounds that we provide also imply that complete segregation, where agents of a single type cover the whole grid, does not occur with high probability for $p=1/2$ and the range of tolerance considered.

Universal persistence exponent in transition to antiferromagnetic order in coupled logistic maps

We study coupled logistic maps on a one-dimensional lattice. We coarse grain the system by labeling the local state + if the value of the variable is above the fixed point and - if the value is below the (nonzero) fixed point, x(*)=1-1/μ. We find that the system exhibits a transition from a global state, in which moving defects occur, to a global state, which is frozen in time (modulo-2). All such states display nonzero value of persistence. Besides, we also observe a state which displays long-range antiferromagnetic order. Onset of such a state is accompanied with a power-law behavior of persistence P(t) as a function of time at the critical line. Usually, persistence transitions show nonuniversal exponent. However, here persistence exponent for synchronous dynamics is 3/8 over the entire lower critical curve. This exponent is the same as one observed for asynchronous Glauber dynamics simulation of zero-temperature Ising model in one dimension. The expected value for synchronous dynamics is 3/4. We present a plausible argument for this scaling.

Hydrodynamical Limits and Geometric Measure Theory: Mean Curvature Limits from a Threshold Voter Model

We consider hydrodynamical limits for a simple threshold voter model for a microscopically evolving random interface. This model, which is a zero-temperature Ising model, was studied by Spohn in a 1+1 setting. The model leads to motion by a certain anisotropic mean curvature. Here we develop this model through some notions of geometric measure theory, dispensing with the 1+1 restriction.