Far tails of the biased CTRW model under the short time limit

TL;DR

This work analyzes the far-tail statistics of biased continuous-time random walks in the short-time limit, focusing on both Gaussian and discrete displacements. By combining analytic waiting-time PDFs near zero with large-deviation and saddle-point methods, it derives exponential, bias-dependent tails for the position distribution and introduces explicit rate-function forms for both position and time. The study extends to a binomial random walk and general waiting-time PDFs, revealing how bias shifts and amplifies rare-event statistics, and it establishes Einstein-like relations in the tail regime. The results clarify how short-time, rare fluctuations in CTRWs are governed by near-origin waiting-time behavior rather than the heavy tails of waiting-time distributions, with implications for interpreting experiments in crowded or heterogeneous media and for understanding rapid, bias-driven transport. Overall, the paper provides a concrete, technically detailed framework to quantify rare fluctuations in biased CTRWs via saddle-point analyses and rate-function descriptions.

Abstract

It has been observed in numerous experiments, simulations, and various theoretical treatments that the spreading of particles can be modeled by the continuous-time random walk. We consider two well-known cases, i.e., Gaussian displacements and discrete displacements, to compute the position distribution and demonstrate the emergence of exponential decay in the far tails when a bias is introduced. We further analyze the temporal rate function and the positional rate function to examine the convergence of the theoretical predictions. For Gaussian displacements, we further discuss the relationship between the position distributions with and without bias in different asymptotic limits.

Figures (9)

• Figure 1: Plot of the ratio of $N^+$ and $N^-$ versus $x$ with the waiting time following exponential distribution $\phi(\tau)=\exp(-\tau)$. The symbols illustrate the exact result from Eq. \ref{['AT19SEC2EQ1201']}. We find $P_N(x,t)=Q_t(N)f(x|N)$ then choose the maximum of $P_N(x,t)$ for a given $x$, finally, we record the corresponding $N^+$ or $N^-$ for positive or negative $x$, respectively. Here we choose $\sigma=0.1$, $a=0.01$, and $t=2$.
• Figure 2: The positional distribution exhibits asymmetric exponential tails for different biases, considering various waiting time PDFs, such as exponential distribution $\phi(\tau)=\exp(-\tau)$ ('$\square$'), Dagum distribution $\phi(\tau)=1/(1+\tau)^2$ ('$\circ$') and a special form of Beta distribution $\phi(\tau)=6\tau(1-\tau)$ ('$\ast$'). The symbols represent simulations generated from $10^9$ trajectories and the solid lines depict theoretical predictions based on Eq. \ref{['AT19SEC3fsds101']}. Note that Throughout we did not discuss the non-moving particles. In our setting, we choose $\sigma=1$ and $t=0.5$ for different $a$.
• Figure 3: Plot of $\ln(\frac{P(x,t)_{a\neq 0}}{P(x,t)_{a=0}})$ versus the position $x$ for different variables $a$ where waiting times are drawn from Dagum distribution $\phi(\tau)=1/(1+\tau)^2$. The red solid lines correspond to the theoretical result given by Eq. \ref{['AT19SEC3fsis1001']}, which is valid for all types of biases. The corresponding asymptotic behavior, as described by Eq. \ref{['AT19SEC3fss1006']}, is shown by the dash-dotted lines for a weak bias. The linear relation, given by Eq. \ref{['AT19SEC3fsis1004']}, is represented by the dashed lines. The symbols are simulations generated from $10^8$ realizations with $\sigma=1$. The figure shows that as we decrease $a$, the linear relationship Eq. \ref{['AT19SEC3fsis1004']} becomes readily detected.
• Figure 4: Rate function $\mathcal{I}_x(l)$ plotted for different times $t$. Here, the waiting time follows the Erlang distribution $\phi(\tau)=\tau^2\exp(-\tau)/2$ with the mean $3$. In that sense $A=2$, $C_2=1/2$ and $C_{3}=-C_2$. The red solid line is the theoretical prediction \ref{['AT19SEC3fsds102']} without fitting. The exact results plot $\ln(P(x,t)/(-|x|))$, where $P(x,t)$ is estimated from Eq. \ref{['AT19SEC2EQ1201']} using the infinite terms Wang2020Large. In our setting $a=1$ and $\sigma=1$. Though the observation time $t$ is not very long, the convergence to the exact result is fast.
• Figure 5: Decay of the positional distribution, where waiting times are drawn from the exponential distribution $\phi(\tau)=\exp(-\tau)$ and the displacements are generated from Eq. \ref{['AT19APP4eq102']}. The symbols describe the exact result for various $p$, obtained from Eq. \ref{['AT19SEC2EQ1201']}, and the solid lines are the corresponding theoretical prediction Eq. \ref{['AT19APP1eqbwe3001']}. The observation time is $t=2$.