We study the transient between a fully disordered initial condition and a percolating structure in the low-temperature non-conserved order parameter dynamics of the bi-dimensional Ising model. We show that a stable structure of spanning clusters establishes at a time tp L αp . Our numerical results yield αp = 0.50(2) for the square and kagome, αp = 0.33(2) for the triangular and αp = 0.38(5) for the bowtie-a lattices. We generalise the dynamic scaling hypothesis to take into account this new time-scale. We discuss the implications of these results for other nonequilibrium processes.

-We study the ferromagnetic Ising model with long-range interactions in two dimensions. We first present results of a Monte Carlo study which shows that the long-range interactions dominate over the short-range ones in the intermediate regime of interaction range. Based on a renormalization group analysis, we propose a way of computing the influence of the long-range interactions as a dimensional change.Introduction. -In recent years there has been a lot of interest in the statistical physics of classical and quantum systems with long-range interactions, for a review see [1]. The role of quasi-stationary states and ergodicity breaking in long-range interacting systems was investigated in [2] and [3]. In [4] the approaching to equilibrium for long-range quantum systems was examined and there has been a lot of enthusiasm in investigating the entanglement entropy in long-range spin chains [5][6][7]. Very recently an experiment was conducted on a quantum system with tunable long-range interactions [8].In the present study we focus on the Ising model which is probably the most studied model in statistical mechanics, especially in the context of critical phenomena. Most of the studies about the Ising model are concentrated around the short-range case which is exactly solvable in one and two dimensions [9]. In three dimensions the problem was perturbatively studied using the ǫ-expansion technique [10] of the renormalization group (RG) combined with the Borel resummation of the perturbation series, see [11] and references therein. Most recently the problem was revisited by using conformal bootstrap technique [12]. Although now there are little unknown facts around shortrange Ising model the long-range Ising model is still the subject of many contradicting theoretical and numerical studies. We define the long-range Ising model as

Abstract. We study the early time dynamics of the 2d ferromagnetic Ising model instantaneously quenched from the disordered to the ordered, low temperature, phase. We evolve the system with kinetic Monte Carlo rules that do not conserve the order parameter. We confirm the rapid approach to random critical percolation in a time-scale that diverges with the system size but is much shorter than the equilibration time. We study the scaling properties of the evolution towards critical percolation and we identify an associated growing length, different from the curvature driven one. By working with the model defined on square, triangular and honeycomb microscopic geometries we establish the dependence of this growing length on the lattice coordination. We discuss the interplay with the usual coarsening mechanism and the eventual fall into and escape from metastability.arXiv:1705.06508v3 [cond-mat.stat-mech]

We present very accurate numerical estimates of the time and size dependence of the zero-temperature local persistence in the 2d ferromagnetic Ising model. We show that the effective exponent decays algebraically to an asymptotic value θ that depends upon the initial condition. More precisely, we find that θ takes one universal value 0.199(2) for initial conditions with short-range spatial correlations as in a paramagnetic state, and the value 0.033(1) for initial conditions with the long-range spatial correlations of the critical Ising state. We checked universality by working with a square and a triangular lattice, and by imposing free and periodic boundary conditions. We found that the effective exponent suffers from stronger finite size effects in the former case.

In this work we study numerically the final state of the two dimensional ferromagnetic critical Ising model after a quench to zero temperature. Beginning from equilibrium at Tc, the system can be blocked in a variety of infinitely long lived stripe states in addition to the ground state. Similar results have already been obtained for an infinite temperature initial condition and an interesting connection to exact percolation crossing probabilities has emerged. Here we complete this picture by providing a new example of stripe states precisely related to initial crossing probabilities for various boundary conditions. We thus show that this is not specific to percolation but rather that it depends on the properties of spanning clusters in the initial state.