Universal survival probability for a correlated random walk and applications to records

Abstract

We consider a model of space-continuous one-dimensional random walk with simple correlation between the steps: the probability that two consecutive steps have same sign is $q$ with $0\le q\le 1$. The parameter $q$ allows thus to control the persistence of the random walk. We compute analytically the survival probability of a walk of $n$ steps, showing that it is independent of the jump distribution for any finite $n$. This universality is a consequence of the Sparre-Andersen theorem for random walks with uncorrelated and symmetric steps. We then apply this result to derive the distribution of the step at which the random walk reaches its maximum and the record statistics of the walk, which show the same universality. In particular, we show that the distribution of the number of records for a walk of $n\gg 1$ steps is the same as for a random walk with $n_{\mathrm{eff}}(q)=n/(2(1-q))$ uncorrelated and symmetrically distributed steps. We also show that in the regime where $n\to \infty$ and $q\to 1$ with $y=n(1-q)$, this model converges to the run-and-tumble particle, a persistent random walk often used to model the motion of bacteria. Our theoretical results are confirmed by numerical simulations.

arXiv comment28 pages, 10 figures

References in corpus (29)

• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Diffusive transport without detailed balance in motile bacteria: Does microbiology need statistical physics?
• Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension
• Dynamics of a self-propelled particle in a harmonic trap
• Run-and-tumble particle in one-dimensional confining potential: Steady state, relaxation and first passage properties
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• Records in stochastic processes -- Theory and applications
• Universal Record Statistics of Random Walks and Lévy Flights
• Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records
• Jamming and Attraction of Interacting Run-and-Tumble Random Walkers
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Record statistics and persistence for a random walk with a drift
• Record dynamics and the observed temperature plateau in the magnetic creep rate of type II superconductors
• Generalised `Arcsine' laws for run-and-tumble particle in one dimension
• Non-crossing run-and-tumble particles on a line
• Intermittent quakes and record dynamics in the thermoremanent magnetization of a spin-glass
• Record statistics of a strongly correlated time series: random walks and Lévy flights
• Universal Order Statistics of Random Walks
• Record Statistics for Multiple Random Walks
• Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension
• Record Statistics of Continuous Time Random Walk
• Active hard-spheres in infinitely many dimensions
• Survival Probability of Random Walks and Lévy Flights on a Semi-Infinite Line
• Linear response in aging glassy systems, intermittency and the Poisson statistics of record fluctuations
• Exact statistics of record increments of random walks and Lévy flights
• Statistics of the Number of Records for Random Walks and Lévy Flights on a $1D$ Lattice
• Gap statistics close to the quantile of a random walk
• Record statistics for a discrete-time random walk with correlated steps

Cited by in corpus (15)

• Condensation transition in the late-time position of a Run-and-Tumble particle
• Record statistics for random walks and Lévy flights with resetting
• Distribution of the time of the maximum for stationary processes
• Survival probability of a run-and-tumble particle in the presence of a drift
• Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
• Statistical properties of avalanches via the c-record process
• Active Random Walks in One and Two Dimensions
• Extremal statistics for a resetting Brownian motion before its first-passage time
• First passage statistics of active random walks on one and two dimensional lattices
• Survival and extreme statistics of work, heat, and entropy production in steady-state heat engines
• Record Ages of Scale Invariant non-Markovian Random Walks
• Extremal statistics for a one-dimensional Brownian motion with a reflective boundary
• Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous and discrete time stochastic processes
• First-passage and extreme value statistics for overdamped Brownian motion in a linear potential
• First-return time in fractional kinetics