We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate $\gamma$ on a discrete lattice. Although such a continuous measurement cannot affect physical observables, it has a non-trivial effect on the probability distribution of the random walker. At small $\gamma$, we show analytically that the time-evolution of the latter can be mapped to the Stochastic Heat Equation (SHE). In this limit, the width of the log probability thus follows a Family-Vicsek scaling law, $N^{ \alpha}f(t/N^{ \alpha/\beta})$, with roughness and growth exponents corresponding to the Kardar-Parisi-Zhang (KPZ) universality class, i.e $\alpha^{ \rm{ 1D}}_{ \rm{ KPZ}}=1/2$ and $\beta^{ \rm{ 1D}}_{ \rm{ KPZ}}=1/3$ respectively. When $\gamma$ is increased outside this regime, we find numerically in 1D a crossover from the KPZ class to a new universality class characterized by exponents $\alpha^{ 1\rm{ D}}_{ \text{ M}}\approx 1$ and $\beta^{ 1\rm{ D}}_{ \text{ M}}\approx 1.4$. In 3D, varying $\gamma$ beyond a critical value $\gamma^c_{ \rm{ M}}$ leads to a phase transition from a smooth phase that we identify as the Edwards-Wilkinson (EW) class to a new universality class with $\alpha^{ 3\rm{ D}}_{ \text{ M}}\approx1$.
Categories: cond-mat.stat-mech