Percolation techniques in disordered spin flip dynamics: Relation to the unique invariant measure (original) (raw)

Percolation phase transition on planar spin systems

2021

In this article we study the sharpness of the phase transition for percolation models defined on top of planar spin systems. The two examples that we treat in detail concern the Glauber dynamics for the Ising model and a Dynamic Bootstrap process. For both of these models we prove that their phase transition is continuous and sharp, providing also quantitative estimates on the two point connectivity. The techniques that we develop in this work can be applied to a variety of different dependent percolation models and we discuss some of the problems that can be tackled in a similar fashion. In the last section of the paper we present a long list of open problems that would require new ideas to be attacked.

On the stochastic dynamics of disordered spin models

Journal of statistical physics, 2004

In this article we discuss several aspects of the stochastic dynamics of spin models. The paper has two independent parts. Firstly, we explore a few properties of the multi-point correlations and responses of generic systems evolving in equilibrium with a thermal bath. We propose a fluctuation principle that allows us to derive fluctuation–dissipation relations for many-time correlations and linear responses. We also speculate on how these features will be modified in systems evolving slowly out of equilibrium, such as finite-dimensional ...

Temporally Disordered Bond Percolation on the Directed Square Lattice

Physical Review Letters, 1996

Simple models of directed bond percolation with temporal disorder are introduced and studied via series expansions and Monte Carlo simulations. Series have been derived for the percolation probability on the directed square lattice. Analysis of the series revealed that the critical exponent b and critical point p c change continuously with the strength of the disorder. Monte Carlo simulation confirmed the continuous change of critical exponents. Estimates for the temporal correlationlength exponent n k for weak disorder showed that n k , 2 in apparent violation of the Harris criterion. [S0031-9007 01855-8] PACS numbers: 05.50. + q, 02.50. -r, 05.70.Ln

Freezing into stripe states in two-dimensional ferromagnets and crossing probabilities in critical percolation

Physical Review E, 2009

When a two-dimensional Ising ferromagnet is quenched from above the critical temperature to zero temperature, the system eventually converges to either a ground state or an infinitely long-lived metastable stripe state. By applying results from percolation theory, we analytically determine the probability to reach the stripe state as a function of the aspect ratio and the form of the boundary conditions. These predictions agree with simulation results. Our approach generally applies to coarsening dynamics of nonconserved scalar fields in two dimensions.

Diffusion in disordered lattices and related Heisenberg ferromagnets

2000

We study the diffusion of classical hard-core particles in disordered lattices within the formalism of a quantum spin representation. This analogy enables an exact treatment of non-instantaneous correlation functions at finite particle densities in terms of single spin excitations in disordered ferromagnetic backgrounds. Applications to diluted chains and percolation clusters are discussed. It is found that density fluctuations in the former exhibit a stretched exponential decay while an anomalous power law asymptotic decay is conjectured for the latter.

Coarsening and percolation in a disordered ferromagnet

Physical Review E, 2017

By studying numerically the phase-ordering kinetics of a two-dimensional ferromagnetic Ising model with quenched disorder-either random bonds or random fields-we show that a critical percolation structure forms at an early stage. This structure is then rendered more and more compact by the ensuing coarsening process. Our results are compared to the non-disordered case, where a similar phenomenon is observed, and they are interpreted within a dynamical scaling framework.

One dimensional 1/|j ? i| S percolation models: The existence of a transition forS?2

Communications in Mathematical Physics, 1986

Consider a one-dimensional independent bond percolation model with p~ denoting the probability of an occupied bond between integer sites i and i +j, j > 1. If pj is fixed for j > 2 and lim j2pj > 1, then (unoriented) percolation j~ occurs for Px sufficiently close to 1. This result, analogous to the existence of spontaneous magnetization in tong range one-dimensional Ising models, is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size. Oriented percolation is shown to occur for Pl close to 1 if lim j~pj > 0 for some s < 2. Analogous results are valid for onedimensional site-bond percolation models.

Compact parity-conserving percolation in one dimension

Journal of Physics A: Mathematical and …, 1998

Compact directed percolation is known to appear at the endpoint of the directed percolation critical line of the Domany-Kinzel cellular automaton in 1 + 1 dimension. Equivalently, such transition occurs at zero temperature in a magnetic field H, upon changing the sign of H, in the one-dimensional Glauber-Ising model, with well-known exponents characterising spin-cluster growth. We have investigated here numerically these exponents in the non-equilibrium generalization (NEKIM) of the Glauber model in the vicinity of the parity-conserving phase transition point of the kinks. Critical fluctuations on the level of kinks are found to affect drastically the characteristic exponents of spreading of spins while the hyperscaling relation holds in its form appropriate for compact clusters.

Critical percolation and lack of self-averaging in disordered models

Lack of self-averaging originates in many disordered models from a fragmentation of the phase space where the sizes of the fragments remain sampledependent in the thermodynamic limit. On the basis of new results in percolation theory, we give here an argument in favour of the conjecture that critical two dimensional percolation on the square lattice lacks of self-averaging.