arXiv subjects

Alberto Bassanoni

Publications and source records attributed to Alberto Bassanoni.

4 recordsLinked to original sources

arXiv2026

Beyond the Big Jump: A Perturbative Approach to Stretched-Exponential Processes

The problem of sums of independent, identically distributed random variables with stretched-exponential tails exhibits a dynamical phase transition and has recently reemerged in the context of active transport and condensation phenomena. We develop a perturbative expansion for the distribution of the sum that systematically extends the Big Jump Principle beyond its asymptotic regime. The expansion yields explicit higher order corrections that describe moderate deviations, bridging the gap between typical Gaussian fluctuations and the far-tail behavior dominated by single big jump events. In this sense, our approach is complementary to the classical Edgeworth expansion, which provides corrections to the Gaussian core, whereas we construct systematic corrections to the big jump regime. The leading terms reveal the scaling structure governing the crossover between typical and condensed fluctuations, in agreement with large deviation predictions but without relying on its asymptotic limit. We further extend the framework to continuous-time random walks (CTRWs), where stretched-exponential jump statistics combined with stochastic renewal times generate nontrivial propagators through subordination. This setting is particularly relevant for transport processes with non-Gaussian displacement statistics, where super-exponential or Laplace-like tails emerge from the interplay of rare large jumps and temporal fluctuations. All analytical predictions are supported by numerical simulations.

cond-mat.stat-mech↗

arXiv2025

Rare Events and Redundancy in Random Walkers Target Search in a Finite Domain

Finding a target in a complex environment is a fundamental challenge across natural systems, from chemical reactions to sperm cells reaching an egg. A powerful strategy to reduce search times is redundancy: deploying many independent searchers increases the probability that at least one succeeds, particularly when success is driven by rare events. When the underlying stochastic motion features broadly distributed step lengths, rare long relocations dominate the dynamics, making redundancy especially effective. Here, we investigate the statistics of extreme events for the mean first passage time in a system of $N$ independent walkers performing power-law distributed jumps with finite velocity, where target-reaching events are governed by single large fluctuations. We show that the mean first passage time of the fastest walker scales as $\langle T_N \rangle \sim 1/N$, representing a dramatic speed-up compared to classical Brownian motion, and saturates at the minimum value $X/v$. We further extend the model to include random velocity. For fixed $N$, we identify a crossover, governed by a critical tail exponent $\alpha_c$, separating a regime dominated by a single large fluctuation (big jump) from a regime characterised by Gaussian extreme-value statistics arising from finite sampling effects. From these results, we derive a scaling law that links the number of walkers $N$ to the size $X$ of the search region. Our results demonstrate how redundancy, combined with rare-event statistics, can efficiently organise target-search processes in complex biological environments. As a prototypical example, we consider mammalian fertilization and derive, within a coarse-grained description, a cross-species scaling relation between the number of spermatozoa and the typical uterine size.

cond-mat.stat-mech↗

arXiv2025

Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes

Even in a simple stochastic process, the study of the full distribution of time integrated observables can be a difficult task. This is the case of a much-studied process such as the Ornstein-Uhlenbeck process where, recently, anomalous dynamical scaling of large deviations of time integrated functionals has been highlighted. Using the mapping of a continuous stochastic process to a continuous time random walk via the "excursions technique'', we introduce a comprehensive formalism that enables the calculation of the complete distribution of the time-integrated observable $A = \int_0^T v^n(t) dt$, where $n$ is a positive integer and $v(t)$ is the random velocity of a particle following Ornstein-Uhlenbeck dynamics. We reveal an interesting connection between the anomalous rate function associated with the observable $A$ and the statistics of the area under the first-passage functional during an excursion. The rate function of the latter, analyzed here for the first time, exhibits anomalous scaling behavior and a dynamical phase transition, both of which are explored in detail. The case of the anomalous scaling of large deviations, originally associated to the presence of an instantonic solution in the weak noise regime of a path integral approach, is here produced by a so called "big jump effect'', in which the contribution to rare events is dominated by the largest excursion. Our approach, which is quite general for continuous stochastic processes, allows to associate a physical meaning to the anomalous scaling of large deviations, through the big jump principle.

cond-mat.stat-mech↗

arXiv2024

Rare Events in Extreme Value Statistics of Jump Processes with Power Tails

We study rare events in the extreme value statistics of stochastic symmetric jump processes with power tails in the distributions of the jumps, using the big-jump principle. The principle states that in the presence of stochastic processes with power tails statistics, if at a certain time a physical quantity takes on a value much larger than its typical value, this large fluctuation is realised through a single macroscopic jump that exceeds the typical scale of the process by several orders of magnitude. In particular, our estimation focuses on the asymptotic behaviour of the tail of the probability distribution of maxima, a fundamental quantity in a wide class of stochastic models used in chemistry to estimate reaction thresholds, in climatology for earthquake risk assessment, in finance for portfolio management, and in ecology for the collective behaviour of species. We determine the analytical form of the probability distribution of rare events in the extreme value statistics of three jump processes with power tails; L\'evy flights, L\'evy walks and the L\'evy-Lorentz gas. For the L\'evy flights, we re-obtain through the big-jump approach recent analytical results, extending their validity. For the L\'evy-Lorentz gas we show that the topology of the disordered lattice along which the walker moves induces memory effects in its dynamics, which influences the extreme value statistics. Our results are confirmed by extensive numerical simulations.

cond-mat.stat-mech↗