arXiv subjects

Mattia Radice

Publications and source records attributed to Mattia Radice.

14 recordsLinked to original sources

arXiv2025

First-passage statistics of random walks: a general approach via Riemann-Hilbert problems

We study first-passage statistics for one-dimensional random walks $S_n$ with independent and identically distributed jumps starting from the origin. We focus on the joint distribution of the first-passage time $τ_b$ and first-passage position $S_{τ_b}$ beyond a threshold $b\geq0$, as well as the distribution of $S_n$ for the walks that do not cross $b$ up to step $n$. By solving suitable Riemann-Hilbert problems, we are able to obtain exact and semi-explicit general formulae for the quantities of interest. Notably, such formulae are written solely in terms of the characteristic function of the jumps. In contrast with previous results, our approach is universally valid, applicable to both continuous and discrete, symmetric and asymmetric jump distributions. We complement our theoretical findings with explicit examples.

cond-mat.stat-mech↗

arXiv2024

Optimal conditions for first passage of jump processes with resetting

We investigate the first passage time beyond a barrier located at $b\geq0$ of a random walk with independent and identically distributed jumps, starting from $x_0=0$. The walk is subject to stochastic resetting, meaning that after each step the evolution is restarted with fixed probability $r$. We consider a resetting protocol that is an intermediate situation between a random walk ($r=0$) and an uncorrelated sequence of jumps all starting from the origin ($r=1$), and derive a general condition for determining when restarting the process with $0<r<1$ is more efficient than restarting after each jump. If the mean first passage time of the process in absence of resetting is larger than that of the sequence of jumps, this condition is sufficient to establish the existence of an optimal $0<r^*<1$ that represents the best strategy, outperforming both $r=0$ and $r=1$. Our findings are discussed by considering two important examples of jump processes, for which we draw the phase diagram illustrating the regions of the parameter space where resetting with some $0<r^*<1$ is optimal.

cond-mat.stat-mech↗

arXiv2024

Optimizing Leapover Lengths of Lévy Flights with Resetting

We consider a one-dimensional search process under stochastic resetting conditions. A target is located at $b\geq0$ and a searcher, starting from the origin, performs a discrete-time random walk with independent jumps drawn from a heavy-tailed distribution. Before each jump, there is a given probability $r$ of restarting the walk from the initial position. The efficiency of a "myopic search" - in which the search stops upon crossing the target for the first time - is usually characterized in terms of the first-passage time $τ$. On the other hand, great relevance is encapsulated by the leapover length $l = x_τ - b$, which measures how far from the target the search ends. For symmetric heavy-tailed jump distributions, in the absence of resetting the average leapover is always infinite. Here we show instead that resetting induces a finite average leapover $\ell_b(r)$ if the mean jump length is finite. We compute exactly $\ell_b(r)$ and determine the condition under which resetting allows for nontrivial optimization, i.e., for the existence of $r^*$ such that $\ell_b(r^*)$ is minimal and smaller than the average leapover of the single jump.

cond-mat.stat-mech↗

arXiv2023

First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion

We study the statistics of random functionals $\mathcal{Z}=\int_{0}^{\mathcal{T}}[x(t)]^{γ-2}dt$, where $x(t)$ is the trajectory of a one-dimensional Brownian motion with diffusion constant $D$ under the effect of a logarithmic potential $V(x)=V_0\ln(x)$. The trajectory starts from a point $x_0$ inside an interval entirely contained in the positive real axis, and the motion is evolved up to the first-exit time $\mathcal{T}$ from the interval. We compute explicitly the PDF of $\mathcal{Z}$ for $γ=0$, and its Laplace transform for $γ\neq0$, which can be inverted for particular combinations of $γ$ and $V_0$. Then we consider the dynamics in $(0,\infty)$ up to the first-passage time to the origin, and obtain the exact distribution for $γ>0$ and $V_0>-D$. By using a mapping between Brownian motion in logarithmic potentials and heterogeneous diffusion, we extend this result to functionals measured over trajectories generated by $\dot{x}(t)=\sqrt{2D}[x(t)]^θη(t)$, where $θ<1$ and $η(t)$ is a Gaussian white noise. We also emphasize how the different interpretations that can be given to the Langevin equation affect the results. Our findings are illustrated by numerical simulations, with good agreement between data and theory.

cond-mat.stat-mech↗

arXiv2023

Effects of mortality on stochastic search processes with resetting

We study the first-passage time to the origin of a mortal Brownian particle, with mortality rate $ μ$, diffusing in one dimension. The particle starts its motion from $ x>0 $ and it is subject to stochastic resetting with constant rate $ r $. We first unveil the relation between the probability of reaching the target and the mean first-passage time of the corresponding problem in absence of mortality, which allows us to deduce under which conditions the former can be increased by adjusting the restart rate. We then consider the first-passage time conditioned on the event that the particle reaches the target before dying, and provide exact expressions for the mean and the variance as functions of $ r $, corroborated by numerical simulations. By studying the impact of resetting for different mortality regimes, we also show that, if the average lifetime $ τ_μ=1/μ$ is long enough with respect to the diffusive time scale $ τ_D=x^2/(4D) $, there exist both a resetting rate $ r_μ^* $ that maximizes the probability and a rate $ r_m $ that minimizes the mean first-passage time. However, the two never coincide for positive $ μ$, making the optimization problem highly nontrivial.

cond-mat.stat-mech↗

arXiv2022

Non-homogeneous random walks with stochastic resetting: an application to the Gillis model

We consider the problem of the first passage time to the origin of a spatially non-homogeneous random walk with a position-dependent drift, known as the Gillis random walk, in the presence of resetting. The walk starts from an initial site $ x_0 $ and, with fixed probability $ r $, at each step may be relocated to a given site $ x_r $. From a general perspective, we first derive a series of results regarding the first and the second moment of the first hitting time distribution, valid for a wide class of processes, including random walks lacking the property of translational invariance; we then apply these results to the specific model. When resetting is not applied, by tuning the value of a parameter which defines the transition probability of the process, denoted by $ ε$, the recurrence properties of the walk are changed, and we can observe: a transient walk, a null-recurrent walk, or a positive-recurrent walk. When the resetting mechanism is switched on, we study quantitatively in all regimes the improvement of the search efficiency. In particular, in every case resetting allows the system to reach the target with probability one and, on average, in a finite time. If the reset-free system is in the transient or null-recurrent regime, this makes resetting always advantageous and moreover, it assures the existence of an optimal resetting probability $ r^* $ which minimizes the mean first hitting time. Instead, when the system is positive-recurrent, the introduction of resetting is not necessarily beneficial. We explain that in this case there exists a threshold $ r_{\mathrm{th}} $ for the resetting probability $ r $, above which the resetting mechanism yields a larger mean first hitting time with respect to the reset-free system. We provide a study of $ r_{\mathrm{th}} $, which can be zero for some values of the system parameters, meaning that...

math.PR↗

arXiv2022

Extreme value statistics of positive recurrent centrally biased random walks

We consider the extreme value statistics of centrally-biased random walks with asymptotically-zero drift in the ergodic regime. We fully characterize the asymptotic distribution of the maximum for this class of Markov chains lacking translational invariance, with a particular emphasis on the relation between the time scaling of the expected value of the maximum and the stationary distribution of the process.

cond-mat.stat-mech↗

arXiv2022

Diffusion processes with Gamma-distributed resetting and non-instantaneous returns

We consider the dynamical evolution of a Brownian particle undergoing stochastic resetting, meaning that after random periods of time it is forced to return to the starting position. The intervals after which the random motion is stopped are drawn from a Gamma distribution of shape parameter $α$ and scale parameter $r$, while the return motion is performed at constant velocity $v$, so that the time cost for a reset is correlated to the last position occupied during the stochastic phase. We show that for any value of $α$ the process reaches a non-equilibrium steady state and unveil the dependence of the stationary distribution on $v$. Interestingly, there is a single value of $α$ for which the steady state is unaffected by the return velocity. Furthermore, we consider the efficiency of the search process by computing explicitly the mean first passage time. All our findings are corroborated by numerical simulations.

cond-mat.stat-mech↗

arXiv2021

The one-dimensional telegraphic process with noninstantaneous stochastic resetting

In this paper we consider the one-dimensional dynamical evolution of a particle traveling at constant speed and performing, at a given rate, random reversals of the velocity direction. The particle is subject to stochastic resetting, meaning that at random times it is forced to return to the starting point. Here we consider a return mechanism governed by a deterministic law of motion, so that the time cost required to return is correlated to the position occupied at the time of the reset. We show that in such conditions the process reaches a stationary state which, for some kinds of deterministic return dynamics, is independent of the return phase. Furthermore, we investigate the first-passage properties of the system and provide explicit formulas for the mean first-hitting time. Our findings are supported by numerical simulations.

cond-mat.stat-mech↗

arXiv2020

A continuous-time random walk extension of the Gillis model

We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional nonhomogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic process allows to significantly change local, non-local and transport properties in the presence of heavy-tailed waiting-time distributions lacking the first moment: we provide here exact results concerning hitting times, first-time events, survival probabilities, occupation times, the moments spectrum and the statistics of records. Specifically, normal diffusion gives way to subdiffusion and we are witnessing the breaking of ergodicity. Furthermore we also test our theoretical predictions with numerical simulations.

cond-mat.stat-mech↗

arXiv2020

Exploring the Gillis model: a discrete approach to diffusion in logarithmic potentials

Gillis model, introduced more than 60 years ago, is a non-homogeneous random walk with a position dependent drift. Though parsimoniously cited both in the physical and mathematical literature, it provides one of the very few examples of a stochastic system allowing for a number of exact result, although lacking translational invariance. We present old and novel results for such model, which moreover we show represents a discrete version of a diffusive particle in the presence of a logarithmic potential.

cond-mat.stat-mech↗

arXiv2020

Transport properties and ageing for the averaged Lévy-Lorentz gas

We consider a persistent random walk on an inhomogeneous environment where the reflection probability depends only on the distance from the origin. Such an environment is the result of an average over all realizations of disorder of a Lévy-Lorentz (LL) gas. Here we show that this averaged Lévy-Lorentz gas yields nontrivial results even when the related LL gas is trivial. In particular, we investigate its long time transport properties such as the mean square displacement and the statistics of records, as well as the occurrence of ageing phenomena.

cond-mat.stat-mech↗

arXiv2020

Statistics of occupation time and connection to local properties of non-homogeneous random walks

We consider the statistics of occupation times, the number of visits at the origin and the survival probability for a wide class of stochastic processes, which can be classified as renewal processes. We show that the distribution of these observables can be characterized by a single parameter, that is connected to a local property of the probability density function (PDF) of the process, viz., the probability of occupying the origin at time $t$, $P(t)$. We test our results for two different models of lattice random walks with spatially inhomogeneous transition probabilities, one of which of non-Markovian nature, and find good agreement with theory. We also show that the distributions depend only on the occupation probability of the origin by comparing them for the two systems: when $P(t)$ show the same long-time behavior, each observable follows indeed the same distribution.

cond-mat.stat-mech↗

arXiv2018

Non-homogeneous persistent random walks and averaged environment for the Lévy-Lorentz gas

We consider transport properties for a non-homogeneous persistent random walk, that may be viewed as a mean-field version of the Lévy-Lorentz gas, namely a 1-d model characterized by a fat polynomial tail of the distribution of scatterers' distance, with parameter $α$. By varying the value of $α$ we have a transition from normal transport to superdiffusion, which we characterize by appropriate continuum limits.

cond-mat.stat-mech↗