The spatial distribution of persistent spins at zero-temperature in the pure two-dimensional Ising model is investigated numerically. A persistence correlation length, ξ(t) ∼ t Z is identified such that for length scales r << ξ(t) the persistent spins form a fractal with dimension d f ; for length scales r >> ξ(t) the distribution of persistent spins is homogeneous. The zero-temperature persistence exponent, θ, is found to satisfy the scaling relation θ = Z(2 − d f ) with θ = 0.209 ± 0.002, Z = 1/2 and d f ∼ 1.58. 2The 'persistence' problem has attracted considerable interest in recent years [1][2][3][4][5][6][7][8][9]. In its most general form, it is concerned with the fraction of space which persists in its initial state up to some later time.Hence, in the non-equilibrium dynamics of spin systems at zero-temperature we are interested in the fraction of spins, P (t), that persist in the same state as at t = 0 up to some later time t. For the pure ferromagnetic two-dimensional Ising model, P (t) has been found to decay algebraically [1-4]where θ = 0.209 ± 0.002 [5]. Similar algebraic decay has been found in numerous other systems displaying persistence [9]. Most of the recent theoretical effort has gone into obtaining the numerical value of θ for different models.Very recently, Manoj and Ray [10] have studied the spatial correlation of persistent sites in the 1d A + A → 0 model. They found that the set of persistent sites in their 1d model forms a fractal over sufficiently small length scales.In this letter we present the results of an extensive numerical study of the spatial distribution of persistent spins in the pure 2d Ising model at zero-temperature. As we will see, the 2d Ising model exhibits behaviour very similar to that found by Manoj and Ray [10] in their simple 1d model.The Hamiltonian for our model is given bywhere S i = ±1 are Ising spins situated on every site of a square lattice with periodic boundary conditions; the summation in Eqn. (2) runs over all nearest-neighbour pairs only.The data presented in this work were obtained for a lattice with dimensions 1000×1000 (= N ). 3Each simulation run begins at t = 0 with a random (±1) starting configuration of the spins and then we update the lattice via single spin flip zero-temperature Glauber dynamics [5].The rule we use is: always flip if the energy change is negative, never flip if the energy change is positive and flip at random if the energy change is zero.For each spin S i we defineHence, if n i (t) = 1 for all t ≥ 0 spin S i is persistent at time t; n i (t) = 0 otherwise.The total number, n(t), of spins which have never flipped until time t is then given by n(t) = i n i (t), and the persistence probability by [1]where < . . . > indicates averages over different initial conditions and histories. We averaged over at least 100 different initial conditions and histories for each run.To investigate the spatial correlations in this model, we follow Manoj and Ray [10] and study the 2-point correlator defined bywhere < . . . > now also includes ...

We study the zero-temperature persistence phenomenon in the random bond +/-J Ising model on a square lattice via extensive numerical simulations. We find strong evidence for "blocking" regardless of the amount disorder present in the system. The fraction of spins which never flips displays interesting nonmonotonic, double-humped behavior as the concentration of ferromagnetic bonds is varied from zero to one. The peak is identified with the onset of the zero-temperature spin glass transition in the model. The residual persistence is found to decay algebraically and the persistence exponent theta(p) approximately = 0.9 over the range 0.1< or =p< or =0.9. Our results are completely consistent with the result of Gandolfi, Newman, and Stein for infinite systems that this model has "mixed" behavior, namely positive fractions of spins that flip finitely and infinitely often, respectively. [Gandolfi, Newman and Stein, Commun. Math. Phys. 214, 373 (2000).].