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KPZ physics and phase transition in a classical single random walker under continuous measurement

We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate $γ$ 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 $γ$, 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, $N^αf(t/N^{α/β})$, with roughness and growth exponents corresponding to the Kardar-Parisi-Zhang (KPZ) universality class, i.e $α^{\rm{1D}}_{\rm{KPZ}}=1/2$ and $β^{\rm{1D}}_{\rm{KPZ}}=1/3$ respectively. When $γ$ is increased outside this regime, we find numerically in 1D a crossover from the KPZ class to a new universality class characterized by exponents $α^{1\rm{D}}_{\text{M}}\approx 1$ and $β^{1\rm{D}}_{\text{M}}\approx 1.4$. In 3D, varying $γ$ beyond a critical value $γ^c_{\rm{M}}$ leads to a phase transition from a smooth phase that we identify as the Edwards-Wilkinson (EW) class to a new universality class with $α^{3\rm{D}}_{\text{M}}\approx1$.