Biased lattice random walks (BLRW) are used to model random motion with drift in a variety of empirical situations in engineering and natural systems such as phototaxis, chemotaxis or gravitaxis. When motion is also affected by the presence of external borders resulting from natural barriers or experimental apparatuses, modelling biased random movement in confinement becomes necessary. To study these scenarios, confined BLRW models have been employed but so far only through computational techniques due to the lack of an analytic framework. Here, we lay the groundwork for such an analytical approach by deriving the Green's functions, or propagators, for the confined BLRW in arbitrary dimensions and arbitrary boundary conditions. By using these propagators we construct explicitly the time dependent first-passage probability in one dimension for reflecting and periodic domains, while in higher dimensions we are able to find its generating function. The latter is used to find the mean first-passage passage time for a $d$-dimensional box, $d$-dimensional torus or a combination of both. We show the appearance of surprising characteristics such as the presence of saddles in the spatio-temporal dynamics of the propagator with reflecting boundaries, bimodal features in the first-passage probability in periodic domains and the minimisation of the mean first-return time for a bias of intermediate strength in rectangular domains. Furthermore, we quantify how in a multi-target environment with the presence of a bias shorter mean first-passage times can be achieved by placing fewer targets close to boundaries in contrast to many targets away from them.
The model of self-avoiding lattice walks and the asymptotic analysis of power-series have been two of the major research themes of Tony Guttmann. In this paper we bring the two together and perform a new analysis of the generating functions for the number of square lattice self-avoiding walks and some of their metric properties such as the mean-square end-to-end distance. The critical point $x_c$ for self-avoiding walks is known to a high degree of accuracy and we utilise this knowledge to undertake a new numerical analysis of the series using biased differential approximants. The new method i…
We study random walks on large random graphs that are biased towards a randomly chosen but fixed target node. We show that a critical bias strength b_c exists such that most walks find the target within a finite time when b>b_c. For b<b_c, a finite fraction of walks drifts off to infinity before hitting the target. The phase transition at b=b_c is second order, but finite size behavior is complex and does not obey the usual finite size scaling ansatz. By extending rigorous results for biased walks on Galton-Watson trees, we give the exact analytical value for b_c and verify it by large scale s…
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.
We analyse a biased random walk on a 1D lattice with unequal step lengths. Such a walk was recently shown to undergo a phase transition from a state containing a single connected cluster of visited sites to one with several clusters of visited sites (fragments) separated by unvisited sites at a critical probability p_c, [PRL 99, 180602 (2007)]. The behaviour of rho(l), the probability of formation of fragments of length l is analysed. An exact expression for the generating function of rho(l) at the critical point is derived. We prove that the asymptotic behaviour is of the form rho(l) ~ 3/[l(l…
We study the first passage time (FPT) problem for biased continuous time random walks. Using the recently formulated framework of fractional Fokker-Planck equations, we obtain the Laplace transform of the FPT density function when the bias is constant. When the bias depends linearly on the position, the full FPT density function is derived in terms of Hermite polynomials and generalized Mittag-Leffler functions.
We present a new approach of topology biased random walks for undirected networks. We focus on a one parameter family of biases and by using a formal analogy with perturbation theory in quantum mechanics we investigate the features of biased random walks. This analogy is extended through the use of parametric equations of motion (PEM) to study the features of random walks {\em vs.} parameter values. Furthermore, we show an analysis of the spectral gap maximum associated to the value of the second eigenvalue of the transition matrix related to the relaxation rate to the stationary state. Applic…
We consider the impact of inertia on biased Brownian motion of point particles in a two-dimensional channel with sinusoidally varying width. If the time scales of the problem separate, the adiabatic elimination of the transverse degrees of freedom leads to an effective description for the motion along the channel given by the potential of mean force. The possibility of such description is intimately connected with equipartition. Numerical simulations show that in the presence of external bias the equipartition may break down leading to non-monotonic dependence of mobility on external force and…
Flory-Huggins theory is a mean field theory for modelling the free energy of dense polymer solutions and polymer melts. In this paper we use Flory-Huggins theory as a model of a dense two dimensional self-avoiding walk confined to a square in the square lattice. The theory describes the free energy of the walk well, and we estimate the Flory interaction parameter of the walk to be $χ_{saw} = 0.32(1)$.
In this paper, we study nonlocal random walk strategies generated with the fractional Laplacian matrix of directed networks. We present a general approach to analyzing these strategies by defining the dynamics as a discrete-time Markovian process with transition probabilities between nodes expressed in terms of powers of the Laplacian matrix. We analyze the elements of the transition matrices and their respective eigenvalues and eigenvectors, the mean first passage times and global times to characterize the random walk strategies. We apply this approach to the study of particular local and non…
Diffusion of point-like non interacting particles in a two-dimensional (2D) channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. We apply our recurrence mapping procedure, which enables us to derive an effective one-dimensional (1D) evolution equation, governing the 1D density of the particles in the channel. In the limit of stationary flow, we arrive at an extended Fick-Jacobs equation, corrected by an effective diffusion coefficient D(x), depending on the longitudinal coordinate x. Our result is an approximate formula for D…
We study the biased random walk process in random uncorrelated networks with arbitrary degree distributions. In our model, the bias is defined by the preferential transition probability, which, in recent years, has been commonly used to study efficiency of different routing protocols in communication networks. We derive exact expressions for the stationary occupation probability, and for the mean transit time between two nodes. The effect of the cyclic search on transit times is also explored. Results presented in this paper give the basis for theoretical treatment of the transport-related pro…
Sood and Grassberger studied in [Phys. Rev. Lett. 99, 098701 (2007)] random walks on random graphs that are biased towards a fixed target point. They put forward a critical bias strength b_c such that a random walker on an infinite graph eventually reaches the target with probability 1 when b>b_c, while a finite fraction of walks drift off to infinity for b<b_c. They rely on rigorous results obtained for biased walks on Galton-Watson (GW) trees to calculate b_c, and give arguments indicating that this result should also hold for random graphs such as Erdos-Renyi (ER) graphs and Molloy-Reed (MR…
A power-law distance-dependent biased random walk model with a tuning parameter ($σ$) is introduced in which finite mean first passage times are realizable if $σ$ is less than a critical value $σ_c$. We perform numerical simulations in $1$-dimension to obtain $σ_c \sim 1.14$. The three-dimensional version of this model is related to the phenomenon of chemotaxis. Diffusiophoretic theory supplemented with coarse-grained simulations establish the connection with the specific value of $σ= 2$ as a consequence of in-built solvent diffusion. A variant of the one-dimensional power-law model is found t…
We study biased random walkers on lattices with randomly dispersed static traps in one, two and three dimensions. As the external bias is increased from zero the system undergoes a phase transition, most clearly manifested in the asymptotic drift velocity of survivors which is zero below a critical bias. This transition is second-order in one dimension but of first order in higher dimensions. The model can be mapped to a stretched polymer with attractive interaction between monomers, and this phase transition would then describe sudden unfolding of the polymer when the stretching force exceeds…
Characterizing the occupation statistics of a radiation flow through confined geometries is key to such technological issues as nuclear reactor design and medical diagnosis. This amounts to assessing the distribution of the travelled length $\ell$ and the number of collisions $n$ performed by the underlying stochastic transport process, for which remarkably simple Cauchy-like formulas were established in the case of branching Pearson random walks with exponentially distributed jumps. In this Letter, we show that such formulas strikingly carry over to the much broader class of branching process…
Random walk on discrete lattice models is important to understand various types of transport processes. The extreme events, defined as exceedences of the flux of walkers above a prescribed threshold, have been studied recently in the context of complex networks. This was motivated by the occurrence of rare events such as traffic jams, floods, and power black-outs which take place on networks. In this work, we study extreme events in a generalized random walk model in which the walk is preferentially biased by the network topology. The walkers preferentially choose to hop toward the hubs or sma…
Motivated by the diffusion-reaction kinetics on interstellar dust grains, we study a first-passage problem of mortal random walkers in a confined two-dimensional geometry. We provide an exact expression for the encounter probability of two walkers, which is evaluated in limiting cases and checked against extensive kinetic Monte Carlo simulations. We analyze the continuum limit which is approached very slowly, with corrections that vanish logarithmically with the lattice size. We then examine the influence of the shape of the lattice on the first-passage probability, where we focus on the aspec…
When random walks on a square lattice are biased horizontally to move solely to the right, the probability distribution of their algebraic area can be exactly obtained. We explicitly map this biased classical random system on a non hermitian Hofstadter-like quantum model where a charged particle on a square lattice coupled to a perpendicular magnetic field hopps only to the right. In the commensurate case when the magnetic flux per unit cell is rational, an exact solution of the quantum model is obtained. Periodicity on the lattice allows to relate traces of the Nth power of the Hamiltonian to…
We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process `{\it space-time Mittag-Leffler process}'. We derive explicit formulae for the state pro…
Extreme events have low occurrence probabilities and display pronounced deviation from their average behaviour, such as earthquakes or power blackouts. Such extreme events occurring on the nodes of a complex network have been extensively studied earlier through the modelling framework of unbiased random walks. They reveal that the occurrence probability for extreme events on nodes of a network has a strong dependence on the nodal properties. Apart from these, a recent work has shown the independence of extreme events on edges from those occurring on nodes. Hence, in this work, we propose a mor…
In this paper we present a computation of the mean first-passage times both for a random walk in a discrete bounded lattice, between a starting site and a target site, and for a Brownian motion in a bounded domain, where the target is a sphere. In both cases, we also discuss the case of two targets, including splitting probabilities, and conditional mean first-passage times. In addition, we study the higher-order moments and the full distribution of the first-passage time. These results significantly extend our earlier contribution [Phys. Rev. Lett. 95, 260601].
Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we provide a novel mechanism for the enhancement of diffusion in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance, i.e., ergodicity is preserved. Ho…
We study the dynamics of a deterministic walk confined in a narrow two-dimensional space randomly filled with point-like targets. At each step, the walker visits the nearest target not previously visited. Complex dynamics is observed at some intermediate values of the domain width, when, while drifting, the walk performs long intermittent backward excursions. As the width is increased, evidence of a transition from ballistic motion to a weakly non-ergodic regime is shown, characterized by sudden inversions of the drift velocity with a probability slowly decaying with time, as $1/t$ at leading …
The spatial coverage produced by a single discrete-time random walk, with asymmetric jump probability $p\neq 1/2$ and non-uniform steps, moving on an infinite one-dimensional lattice is investigated. Analytical calculations are complemented with Monte Carlo simulations. We show that, for appropriate step sizes, the model displays a critical phenomenon, at $p=p_c$. Its scaling properties as well as the main features of the fragmented coverage occurring in the vicinity of the critical point are shown. In particular, in the limit $p\to p_c$, the distribution of fragment lengths is scale-free, wit…
We consider a random walk in confined geometry, starting from a site and eventually reaching a target site. We calculate analytically the distribution of the occupation time on a third site, before reaching the target site. The obtained distribution is exact, and completely explicit in the case or parallepipedic confining domains. We discuss implications of these results in two different fields: The mean first passage time for the random trap model is computed in dimensions greater than 1, and is shown to display a non-trivial dependence with the source and target positions ; The probability o…
We study biased, diffusive transport of Brownian particles through narrow, spatially periodic structures in which the motion is constrained in lateral directions. The problem is analyzed under the perspective of the Fick-Jacobs equation which accounts for the effect of the lateral confinement by introducing an entropic barrier in a one dimensional diffusion. The validity of this approximation, being based on the assumption of an instantaneous equilibration of the particle distribution in the cross-section of the structure, is analyzed by comparing the different time scales that characterize th…
Exponential, and not Gaussian, decay of probability density functions was studied by Laplace in the context of his analysis of errors. Such Laplace propagators for the diffusive motion of single particles in disordered media were recently observed in numerous experimental systems. What will happen to this universality when an external driving force is applied? Using the ubiquitous continuous time random walk with bias, and the Crooks relation in conjunction with large deviations theory, we derive two properties of the positional probability density function $P_F(x,t)$ that hold for a wide spec…
The mean-squared displacement (MSD) is an averaged quantity widely used to assess anomalous diffusion. In many cases, such as molecular motors with finite processivity, dynamics of the system of interest produce trajectories of varying duration. Here we explore the effects of finite processivity on different measures of the MSD. We do so by investigating a deceptively simple dynamical system: a one-dimensional random walk (with equidistant jump lengths, symmetric move probabilities, and constant step duration) with an origin-directed detachment bias. By tuning the time dependence of the detach…
We study analytically the dynamics and the micro-structural changes of a host medium caused by a driven tracer particle moving in a confined, quiescent molecular crowding environment. Imitating typical settings of active micro-rheology experiments, we consider here a minimal model comprising a geometrically confined lattice system -- a two-dimensional strip-like or a three-dimensional capillary-like -- populated by two types of hard-core particles with stochastic dynamics -- a tracer particle driven by a constant external force and bath particles moving completely at random. Resorting to a dec…
The continuous time random walk (CTRW) model exhibits a non-ergodic phase when the average waiting time diverges. Using an analytical approach for the non-biased and the uniformly biased CTRWs, and numerical simulations for the CTRW in a potential field, we obtain the non-ergodic properties of the random walk which show strong deviations from Boltzmann--Gibbs theory. We derive the distribution function of occupation times in a bounded region of space which, in the ergodic phase recovers the Boltzmann--Gibbs theory, while in the non-ergodic phase yields a generalized non-ergodic statistical law…
The principle characteristics of biased greedy random walks (BGRWs) on two-dimensional lattices with real-valued quenched disorder on the lattice edges are studied. Here, the disorder allows for negative edge-weights. In previous studies, considering the negative-weight percolation (NWP) problem, this was shown to change the universality class of the existing, static percolation transition. In the presented study, four different types of BGRWs and an algorithm based on the ant colony optimization (ACO) heuristic were considered. Regarding the BGRWs, the precise configurations of the lattice wa…
We study self-avoiding walks on the square lattice restricted to a square box of side $L$ weighted by a length fugacity without restriction of their end points. This models a confined polymer in dilute solution. The model admits a phase transition between an `empty' phase, where the average length of walks are finite and the density inside large boxes goes to zero, to a `dense' phase, where there is a finite positive density. We prove various bounds on the free energy and develop a scaling theory for the phase transition based on the standard theory for unconstrained polymers. We compare this …
We consider the occurrence of record-breaking events in random walks with asymmetric jump distributions. The statistics of records in symmetric random walks was previously analyzed by Majumdar and Ziff and is well understood. Unlike the case of symmetric jump distributions, in the asymmetric case the statistics of records depends on the choice of the jump distribution. We compute the record rate $P_n(c)$, defined as the probability for the $n$th value to be larger than all previous values, for a Gaussian jump distribution with standard deviation $σ$ that is shifted by a constant drift $c$. For…
Microscopic mechanisms of natural processes are frequently understood in terms of random walk models by analyzing local particle transitions. This is because these models properly account for dynamic processes at the molecular level and provide a clear physical picture. Recent theoretical studies made a surprising discovery that in complex systems the symmetry of molecular forward/backward transition times with respect to local bias in the dynamics may be broken and it may take longer to go downhill than uphill. The physical origins of these phenomena remain not fully understood. Here we explo…
First passage under restart has recently emerged as a conceptual framework to study various stochastic processes under restart mechanism. Emanating from the canonical diffusion problem by Evans and Majumdar, restart has been shown to outperform the completion of many first passage processes which otherwise would take longer time to finish. However, most of the studies so far assumed continuous time underlying first passage time processes and moreover considered continuous time resetting restricting out restart processes broken up into synchronized time steps. To bridge this gap, in this paper,…
We compute the average shape of trajectories of some one--dimensional stochastic processes x(t) in the (t,x) plane during an excursion, i.e. between two successive returns to a reference value, finding that it obeys a scaling form. For uncorrelated random walks the average shape is semicircular, independently from the single increments distribution, as long as it is symmetric. Such universality extends to biased random walks and Levy flights, with the exception of a particular class of biased Levy flights. Adding a linear damping term destroys scaling and leads asymptotically to flat excursion…
Efficient techniques to navigate networks with local information are fundamental to sample large-scale online social systems and to retrieve resources in peer-to-peer systems. Biased random walks, i.e. walks whose motion is biased on properties of neighbouring nodes, have been largely exploited to design smart local strategies to explore a network, for instance by constructing maximally mixing trajectories or by allowing an almost uniform sampling of the nodes. Here we introduce and study biased random walks on multiplex networks, graphs where the nodes are related through different types of l…
Polymers in confined spaces lose conformational entropy. This induces a net repulsive entropic force on the walls of the confining space. A model for this phenomenon is a lattice walk between confining walls, and in this paper a model of an adsorbing partially directed walk is used. The walk is placed in a half square lattice $L^2_+$ with boundary $\partial L^2_+$, and confined between two vertical parallel walls, which are vertical lines in the lattice, a distance $w$ apart. The free energy of the walk is determined, as a function of $w$, for walks with endpoints in the confining walls and ad…
We analyze the dynamics of random walks in which the jumping probabilities are periodic {\it time-dependent} functions. In particular, we determine the survival probability of biased walkers who are drifted towards an absorbing boundary. The typical life-time of the walkers is found to decrease with an increment of the oscillation amplitude of the jumping probabilities. We discuss the applicability of the results in the context of complex adaptive systems.
Dynamical features of tagged particles are studied in a one dimensional $A+A \rightarrow kA$ system for $k=0$ and 1, where the particles $A$ have a bias $ε$ $(0 \leq ε\leq 0.5)$ to hop one step in the direction of their nearest neighboring particle. $ε=0$ represents purely diffusive motion and $ε=0.5$ represents purely deterministic motion of the particles. We show that for any $ε$, there is a time scale $t^*$ which demarcates the dynamics of the particles. Below $t^*$, the dynamics are governed by the annihilation of the particles, and the particle motions are highly correlated, while for $t …
We compute the exponent gamma for self-avoiding walks in three dimensions. We get gamma = 1.1575 +- 0.0006 in agreement with renormalization-group predictions. Earlier Monte Carlo and exact-enumeration determinations are now seen to be biased by corrections to scaling.
