arXiv subjects

Omer Hamdi

Publications and source records attributed to Omer Hamdi.

3 recordsLinked to original sources

arXiv2026

From Continuous-Time Random Walks to Laplace Tails

During Brownian motion, the displacement is normally distributed, a classical fact aligned with the central limit theorem. However, single particle tracking in complex media such as glasses, living cells, and colloidal suspensions often reveals pronounced exponential decay of the displacement distribution, known as Laplace tails. In a short letter, two of us presented the emergence of Laplace tails in the continuous time random walk (CTRW) framework. Here, a detailed complementary study is presented. By exploring the behavior of $Q_t(n)$, the probability that exactly $n$ renewals occur during time $t$, we develop a rate function-like framework for this quantity, valid for finite $t$. We show that $Q_t(n)$ exhibits exponential tails, which in turn give rise to exponential tails of the positional probability density function $P(x,t)$. Favorable comparison to finite-time numerical simulations and asymptotic large deviation rate functions establishes the validity of our results over a wide temporal range.

cond-mat.stat-mech↗

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↗

arXiv2024

Laplace's first law of errors applied to diffusive motion

In biological, glassy, and active systems, various tracers exhibit Laplace-like, i.e., exponential, spreading of the diffusing packet of particles. The limitations of the central limit theorem in fully capturing the behaviors of such diffusive processes, especially in the tails, have been studied using the continuous time random walk model. For cases when the jump length distribution is super-exponential, e.g., a Gaussian, we use large deviations theory and relate it to the appearance of exponential tails. When the jump length distribution is sub-exponential the packet of spreading particles is described by the big jump principle. We demonstrate the applicability of our approach for finite time, indicating that rare events and the asymptotics of the large deviations rate function can be sampled for large length scales within a reasonably short measurement time.

cond-mat.stat-mech↗