Kardar-Parisi-Zhang Physics and Phase Transition in a Classical Single Random Walker under Continuous Measurement
Tony Jin, D.G. Martin
Abstract
We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate $\ensuremath{\gamma}$ on a discrete lattice. Although such a continuous measurement cannot affect physical observables, it has a nontrivial effect on the probability distribution of the random walker. At small $\ensuremath{\gamma}$, we show analytically that the time evolution of the latter can be mapped to the stochastic heat equation. In this limit, the width of the log-probability thus follows a Family-Vicsek scaling law, ${N}^{\ensuremath{\alpha}}f(t/{N}^{\ensuremath{\alpha}/\ensuremath{\beta}})$, with roughness and growth exponents corresponding to the Kardar-Parisi-Zhang (KPZ) universality class, i.e., ${\ensuremath{\alpha}}_{\mathrm{KPZ}}^{1\mathrm{D}}=1/2$ and ${\ensuremath{\beta}}_{\mathrm{KPZ}}^{1\mathrm{D}}=1/3$, respectively. When $\ensuremath{\gamma}$ is increased outside this regime, we find numerically in 1D a crossover from the KPZ class to a new universality class characterized by exponents ${\ensuremath{\alpha}}_{M}^{1\mathrm{D}}\ensuremath{\approx}1$ and ${\ensuremath{\beta}}_{M}^{1\mathrm{D}}\ensuremath{\approx}1.4$. In 3D, varying $\ensuremath{\gamma}$ beyond a critical value ${\ensuremath{\gamma}}_{M}^{c}$ leads to a phase transition from a smooth phase that we identify as the Edwards-Wilkinson class to a new universality class with ${\ensuremath{\alpha}}_{M}^{3\mathrm{D}}\ensuremath{\approx}1$.