The longest excursion of stochastic processes in nonequilibrium systems

We consider the excursions, i.e. the intervals between consecutive zeros, of stochastic processes that arise in a variety of nonequilibrium systems and study the temporal growth of the longest one l_{\max}(t) up to time t. For smooth processes, we find a universal linear growth < l_{\max}(t) > \simeq Q_{\infty} t with a model dependent amplitude Q_\infty. In contrast, for non-smooth processes with a persistence exponent \theta, we show that < l_{\max}(t) > has a linear growth if \theta < \theta_c while < l_{\max}(t) > \sim t^{1-\psi} if \theta > \theta_c. The amplitude Q_{\infty} and the exponent \psi are novel quantities associated to nonequilibrium dynamics. These behaviors are obtained by exact analytical calculations for renewal and multiplicative processes and numerical simulations for other systems such as the coarsening dynamics in Ising model as well as the diffusion equation with random initial conditions.