Persistence in nonequilibrium surface growth

Abstract

Persistence probabilities of the interface height in ( 1+1 ) - and ( 2+1 ) -dimensional atomistic, solid-on-solid, stochastic models of surface growth are studied using kinetic Monte Carlo simulations, with emphasis on models that belong to the molecular beam epitaxy (MBE) universality class. Both the initial transient and the long-time steady-state regimes are investigated. We show that for growth models in the MBE universality class, the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady-state regimes. For the MBE universality class, the positive and negative persistence exponents in the steady-state are found to be theta(S)(+) =0.66+/-0.02 and theta(S)(-) =0.78+/-0.02, respectively, in ( 1+1 ) dimensions, and theta(S)(+) =0.76+/-0.02 and theta(S)(-) =0.85+/-0.02, respectively, in ( 2+1 ) dimensions. The noise reduction technique is applied on some of the ( 1+1 ) -dimensional models in order to obtain accurate values of the persistence exponents. We show analytically that a relation between the steady-state persistence exponent and the dynamic growth exponent, found earlier to be valid for linear models, should be satisfied by the smaller of the two steady-state persistence exponents in the nonlinear models. Our numerical results for the persistence exponents are consistent with this prediction. We also find that the steady-state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state. The dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form.

Persistence and first-passage properties in nonequilibrium systems

“…[228,230]). In addition, the relation θ(H) = 1 − H which is valid for symmetric processes (where the probability distribution of x has the x → −x symmetry) can be generalized to non-symmetric processes [231]. In fact, below we present the more general result valid for non-symmetric processes and recover the result θ(H) = 1 − H as a special case when the x → −x symmetry is restored.…”

Volatility, persistence, and survival in financial markets

“…The current work is, in some sense, a continuation of our earlier work on understanding various stochastic phenomena (i.e. thermal step fluctuations [23,24,25] and nonequilibrium surface growth [16]) from the first-passage statistics perspective -here the stochastic process under consideration being an economic phenomenon (i.e. stock price fluctuations) rather than physical phenomena as in the past.…”

“…One way to explore the temporal evolution of a stochastic system such as a fluctuating stock price, denoted by x(t), is by measuring the persistence probability, P (t). That is the probability of the stochastic variable x(t) not reaching its original value corresponding to the starting time t 0 up to a later time t 0 + t. This concept, closely related to the first-passage probability, has been successfully implemented in surface growth phenomena [15,16] and has been used to determine the universality class and the nonlinear features of the underlying dynamical process through the exponent θ associated with the power-law decay P (t) ∼ t −θ of the persistence probability at large times. Alternatively, one is interested in measuring the survival probability S(t) which is defined as the probability of the stock price remaining above a reference value up to time t. In contrast with the persistence probability, we show that the survival probability depends independently on both the total measurement time t m and the time between successive recordings, i.e., the sampling time δt.…”

“…On the other hand we investigate the persistence of price fluctuations described by the persistence exponent θ. The bridge between these two analyses is provided by the second order Hurst exponent H 2 associated with the correlation function of the stock price, which has been shown [15,16] to be simply related to the persistence exponent through H 2 = 1−θ. In this study we verify that this simple relation is satisfied by both low and high frequency fluctuating financial stocks.…”

Persistence of Brownian motion in a shear flow

“…This probability is defined as the probability that the stochastic variable x(t) does not return to the origin up to time t. In many cases, it decays according to a power law P (t) ∼ t −θ , thus defining the nontrivial persistence exponent θ . This exponent has been studied experimentally and theoretically for fluctuating interfaces [1][2][3], critical dynamics [4], granular media [5,6], disordered environments [7][8][9], and polymer dynamics [10].…”