Tethering effects on first-passage variables of lattice random walks in linear and quadratic focal point potentials

TL;DR

The paper analyzes lattice random walks subjected to focal-point confinement in 1D, focusing on a V-shaped potential and an elastic U-shaped potential to understand how deterministic forces influence first-passage dynamics and spreading. It develops exact analytical frameworks: the V-shaped case yields an explicit propagator and first-passage results on unbounded, semi-bounded, and fully bounded domains via a defect method; the U-shaped case is solved in a fully bounded domain using Krawtchouk polynomials. Key findings include a logarithmic long-time growth of the mean number of distinct sites visited under V-tethering, nontrivial mean first-passage times with bias-dependent minima, and distinct steady-state shapes under resetting for the two potentials. The work provides rigorous tools for diffusion in heterogeneous, confining environments, with implications for search and transport in disordered media and for understanding resetting-induced regimes on lattices.

Abstract

Diffusion in a confining potential offers a minimal setting to understand the interplay between random motion and deterministic forces driving a particle towards a focal point or potential minimum. In continuous space and time, two extensively studied examples are Brownian motion in a linear (V-shaped) or a quadratic (U-shaped) potential. The deterministic bias towards the minimum is represented, respectively, by a constant force for the former and by an elastic restoring force that increases proportionally with distance for the latter. Surprisingly, unlike Brownian walks, random walks under focal point potentials in discrete space and time have received little attention. Here, we bridge this gap by analysing the dynamics of lattice random walkers in the presence of a V-shaped potential, both in a finite and an infinite spatial domain, and a finite U-shaped potential. For the V-potential in unbounded space, we find the generating function of the occupation probability and analyse the time dependence of the mean number of distinct sites visited, demonstrating that its long-time growth is logarithmic. We also study the first-passage probability and show that its mean may display a minimum as a function of bias strength, depending on the location of the initial and target sites relative to the focal point. Qualitatively similar dependencies in the first-passage probability and its mean appear for the finite U-potential. As a comparative analysis to the U-potential, we construct the bounded V-potential and superimpose in both cases a resetting process, in which the walker returns at random times to a site distinct from the focal point with some probability. We quantify the different effects of resetting on the steady-state probability and the first-passage dynamics in the two cases, and show a motion-limited regime emerges even for relatively moderate resetting probabilities.

Figures (11)

• Figure 1: Schematic diagram of transition probabilities for a walker under the V--shaped potential centred at the focal site $n_c$ in unbounded space. The red (dotted) and green (dashed) arrows denote the constant probabilities $q(1-g)/2$ and $q(1+g)/2$, respectively, and thus make the potential symmetric about $n_c$.
• Figure 2: Propagator $Q(n, t|n_0)$ for dynamics \ref{['eq:deb_vpot_master_compact']} under a V--potential centred at site $n_c=10$ in unbounded spatial domain. The walker starts from site $n_0=5$ with $q=0.5$ and $g=0.3$. The points are obtained from $5\times10^5$ stochastic realizations of dynamics \ref{['eq:deb_vpot_master_compact']}. The broken lines are obtained by numerically inverting Eq. \ref{['eq:GF_inZ_VPot_gen_sol']}, while the continuous line in green denotes the steady-state given in Eq. \ref{['eq:VPot_ss']}.
• Figure 3: First-passage walker statistics in a V--potential centred at site $n_c=10$ in unbounded space. The walker starts from site $n_0=8$ and reaches a target at $n$ with $q=0.5$. In the plots, the points are obtained from $5\times10^5$ stochastic realizations of dynamics \ref{['eq:deb_vpot_master_compact']}, while the lines denote analytical results. Top left: First-passage probability $F(n, t|n_0)$ as a function of time $t$ for different targets $n$ with $g=0.3$. The dashed lines in black are obtained by numerically inverting abate_numerical_1992 Eq. \ref{['eq:FPT_inZ_VPot']}. Top right: Mean first-passage time $\expval{T}$ from Eq. \ref{['eq:MFPT_Vpot']}. Bottom left: Variance of the first-passage time from Eq. \ref{['eq:VarFPT_Vpot']}. Bottom right: The signal-to-noise ratio obtained using the relation $\mathrm{SNR}=\expval{T}^2/\mathrm{Var}(T)$. Note that for $g=0$ the dynamics reduce to that of a symmetric random walk, leading to diverging mean and variance for all target sites.
• Figure 4: Average number of distinct sites visited $\expval{N(t)}$ by a walker moving within the V--potential with different values of $g$ in unbounded domain. The walker starts from the centre of the potential $(n_0=n_c=0)$ with diffusivity $q=0.5$. The left panel shows $\expval{N(t)}$ vs $t$, while the right panel shows the scaling law. Points are obtained from $2000$ stochastic realizations. The lines in the left panel are obtained by numerically inverting Eq. \ref{['eq:MNDSV']}, where the infinite spatial sum is truncated to the range $-100 \le n \le 100$, while the dotted straight line in the right panel is shown as a guide to the eye.
• Figure 5: Schematic diagram of transition probabilities for a walker under the U--shaped potential centred at the focal site $R$. The green (dashed) and red (dotted) arrows denote space-dependent probabilities $a_n$ and $b_n$, respectively, given by $a_n = q(R+|n-R|)/(2R)$ and $b_n = q(R-|n-R|)/(2R)$. Note that one has $a_{R+n}=a_{R-n}$ and $b_{R+n}=b_{R-n}$, which make the potential symmetric about $R$. The probability strength of the green (dashed) arrows weakens as we approach $R$, while that of the red (dotted) arrows weakens as we move away from $R$. Imposing reflecting boundary conditions at sites 0 and $2R$ (which are not shown here) yields the Master equation \ref{['eq:Master_Upot']}.