We investigate the splitting probability of a monitored continuous-time quantum walk with two targets and show that, in stark contrast to a classical random walk, it exhibits a nonanalytic, phase-transition-like behavior controlled by the sampling time at the targets. For large systems and sampling times smaller than a critical value $τ_c = 2π/ΔE$, where $ΔE$ is the energy bandwidth, the splitting probability is universal and equal to $1/2$, independent of the initial condition and the sampling time. Above the critical sampling time, a nonuniversal regime emerges in which the splitting probability deviates from $1/2$ and develops a fluctuating pattern of pronounced peaks and dips dependent on both the sampling time and the initial condition. These results follow from a nontrivial mapping of the splitting problem onto a pair of single-target detection problems enabled by the superposition principle.
The random walk with hyperbolic probabilities that we are introducing is an example of stochastic diffusion in a one-dimensional heterogeneous media. Although driven by site-dependent one-step transition probabilities, the process retains some of the features of a simple random walk, shows other traits that one would associate with a biased random walk and, at the same time, presents new properties not related with either of them. In particular, we show how the system is not fully ergodic, as not every statistic can be estimated from a single realization of the process. We further give a geome…
We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability $ π_{0,\underline{x}}(x_0)$ that the process crosses $x$ before 0 starting from a given position $x_0\in[0,x]$ in the regime $x_0\ll x$. This analysis provides in particular a fully explicit determination of the transmission probability ($x_0=0$), in striking contrast with the trivial prediction $ π_{0,\underline{x}}(0)=0$ obtained by taking the continuous limit of the process, which reveals the importance of the microscopic properties of the dynam…
Conditional splitting probabilities for hidden-state inference in drift-diffusive processes
Imperial College London（帝国理工学院）；Laboratoire de Physique de l’Ecole Normale Supérieure (LPENS)（巴黎高等师范学院物理实验室）；Max Planck Insitute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）
We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent $\mathrm{Beta}(μ,μ)$ distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter $η$. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary …
We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find t…
