arXiv subjects

Luca Giuggioli

Publications and source records attributed to Luca Giuggioli.

At least 19 recordsLinked to original sources

arXiv2025

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

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.

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arXiv2025

Dynamics and steady states of tight-binding chains in presence of isolated defects

Reduced transport and localization in isolated quantum systems are typically attributed to spatially-extended disorder, but may also emerge from the influence of a few controllable defects. We show here how a single defect profoundly reshapes wave-function spreading on a finite and periodic tight-binding lattice. Adapting the defect technique from classical random-walk studies, we obtain exact time-resolved site-occupation probabilities and several observables of interest. Even a single defect induces remarkable nonlinear effects, including non-monotonic suppression of transport, enhanced localization at distant sites, and strong sensitivity to the initial particle position at long times. These results demonstrate that minimal perturbations can generate nontrivial long-time transport signatures, giving rise to a microscopic defect-driven mechanism of quantum localization. Although the main results presented pertain to a single isolated defect, we show that the developed formalism may naturally extend to multiple as well as to a wider class of defects.

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arXiv2025

Predicting First-Passage Dynamics in Disordered Systems Exactly: Application to Sparse Networks

Quantifying how spatial disorder affects the movement of a diffusing particle or agent is fundamental to target search studies. When diffusion occurs on a network, that is on a highly disordered environment, we lack the mathematical tools to calculate exactly the temporal characteristics of search processes, instead relying on estimates provided by stochastic simulations. To close this knowledge gap we devise a general methodology to represent analytically the movement and search dynamics of a diffusing random walk on sparse graphs. We show its utility by uncovering the existence of a bi-modality regime in the time-dependence of the first-passage probability to hit a target node in a small-world network. By identifying the network features that give rise to the bi-modal regime, we challenge long-held beliefs on how the statistics of the so-called direct, intermediate, and indirect trajectories influence the shape of the resulting first-passage and first-absorption probabilities and the interpretation of their mean values. Overall these findings show that temporal features in first-passage studies can be utilised to unearth novel transport paradigms in spatially heterogeneous environments.

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arXiv2024

Collective Foraging and Behavioural Syndromes in Ants: First-Passage Statistics with Heterogeneous Walkers on a Honeycomb Lattice

Behavioral heterogeneities in animals, also known as syndromes, play a crucial role in understanding how natural populations flexibly adapt to environmental changes. In ant species like \textit{Aphaenogaster senilis}, two key roles in collective foraging are commonly recognised: scouts, who discover food patches, and recruits, who exploit these patches and transport food back to the nest. These roles involve distinct movement patterns and exploratory behaviours. In this chapter, we develop a correlated random walk model on a bounded honeycomb lattice to interpret and replicate empirical observations of foraging ants in an enclosed arena with honeycomb tiling. We do so by extending the theory of first-passage processes for $\mathcal{N}$ random walkers when individuals belong to a heterogeneous population. We apply this theory to examine how individual behavioural heterogeneity in ants affects collective foraging efficiency, focusing on first-passage time statistics for nest-to-patch and patch-to-patch movements. With the combined use of the mathematical model and the controlled experimental setup we evaluate (i) the impact of distinct movement strategies by scouts and recruits on finding food items and (ii) whether ants practice strict central place foraging or utilise previously discovered patches as starting points for further exploration.

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arXiv2024

Local Time Statistics and Permeable Barrier Crossing: from Poisson to Birth-Death Diffusion Equations

Barrier crossing is a widespread phenomenon across natural and engineering systems. While an abundant cross-disciplinary literature on the topic has emerged over the years, the stochastic underpinnings of the process are yet to be linked quantitatively to easily measurable observables. We bridge this gap by developing a microscopic representation of Brownian motion in the presence of permeable barriers that allows to treat barriers with constant asymmetric permeabilities. Our approach relies upon reflected Brownian motion and on the crossing events being Poisson processes subordinated by the local time of the underlying motion at the barrier. Within this paradigm we derive the exact expression for the distribution of the number of crossings, and find an experimentally measurable statistical definition of permeability. We employ Feynman-Kac theory to derive and solve a set of governing birth-death diffusion equations and extend them to the case when barrier permeability is constant and asymmetric. As an application we study a system of infinite, identical and periodically placed asymmetric barriers for which we derive analytically effective transport parameters. This periodic arrangement induces an effective drift at long times whose magnitude depends on the difference in the permeability on either side of the barrier as well as on their absolute values. As the asymmetric permeabilities act akin to localised 'ratchet' potentials that break spatial symmetry and detailed balance, the proposed arrangement of asymmetric barriers provides an example of a noise-induced drift without the need to time-modulate any external force or create temporal correlations on the motion of a diffusing particle. By placing only one asymmetric barrier in a periodic domain we also show the emergence of a non-equilibrium steady state.

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arXiv2024

Persistent and anti-Persistent Motion in Bounded and Unbounded Space: Resolution of the First-Passage Problem

The presence of temporal correlations in random movement trajectories is a widespread phenomenon across biological, chemical and physical systems. The ubiquity of persistent and anti-persistent motion in many natural and synthetic systems has led to a large literature on the modelling of temporally correlated movement paths. Despite the substantial body of work, little progress has been made to determine the dynamical properties of various transport related quantities, including the first-passage or first-hitting probability to one or multiple absorbing targets when space is bounded. To bridge this knowledge gap we generalise the renewal theory of first-passage and splitting probabilities to correlated discrete variables. We do so in arbitrary dimensions on a lattice for the so-called correlated or persistent random walk, the one step non-Markovian extension of the simple lattice random walk in bounded and unbounded space. We focus on bounded domains and consider both persistent and anti-persistent motion in hypercubic lattices as well as the hexagonal lattice. The discrete formalism allows us to extend the notion of the first-passage to that of the directional first-passage, whereby the walker must reach the target from a prescribed direction for a hitting event to occur. As an application to spatio-temporal observations of correlated moving cells that may be either repelled or attracted to hard surfaces, we compare the first-passage statistics to a target within a reflecting domain depending on whether an interaction with the reflective interface invokes a reversal of the movement direction or not. With strong persistence we observe multi-modality in the first-passage distribution in the former case, which instead is greatly suppressed in the latter.

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arXiv2023

Multi-target search in bounded and heterogeneous environments: a lattice random walk perspective

For more than a century lattice random walks have been employed ubiquitously, both as a theoretical laboratory to develop intuition about more complex stochastic processes and as a tool to interpret a vast array of empirical observations. Recent advances in lattice random walk theory in bounded and heterogeneous environments have opened up opportunities to cope with the finely resolved spatio-temporal nature of modern movement data. We review such advances and their formalisms to represent analytically the walker spatio-temporal dynamics in arbitrary dimensions and geometries. As new findings, we derive the exact spatio-temporal representation of biased walks in a periodic hexagon, we use the discrete Feynman-Kac equation to describe a walker's interaction with a radiation boundary, and we unearth a disorder indifference phenomenon. To demonstrate the power of the formalism we uncover the appearance of multiple first-passage peaks with biased walkers in a periodic hexagon, we display the dependence of the first-transmission probability on the proximity transfer efficiency between two resetting walkers in a one-dimensional periodic lattice, we present an example of spatial disorder in a two-dimensional square lattice that strongly affects the splitting probabilities to either of two targets, and we study the first-reaction dynamics to a single lattice site in an unbounded one-dimensional lattice.

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arXiv2023

Misconceptions about quantifying animal encounter and interaction processes

Quantifying animal interactions is crucial for understanding various ecological processes, including social community structures, predator-prey dynamics, spreading of pathogens and information. Despite the ubiquity of interaction processes among animals and the advancements in tracking technologies enabling simultaneous monitoring of multiple individuals, a common theoretical framework to analyse movement data is still lacking. The diverse mechanisms governing how organisms perceive the proximity of others have led to species-specific theoretical approaches, hindering a common currency with which to evaluate and compare findings across taxa. We propose a general framework, borrowing tools from statistical physics, specifically from the theory of reaction diffusion processes. While some of these tools have been employed to predict pathogen transmission events, they have not yet pervaded the movement ecology literature. Using both continuous and discrete variables, we demonstrate the suitability of our framework to study interaction processes. Defining interactions as the transfer of information between individuals, we show that the probability of information transfer for the first time is equivalent to the first-passage probability of reacting in a multi-target environment. As interaction events reduce to encounter events for perfectly efficient information transfer, we compare our formalism to a recent approach that takes the joint occupation probability of two animals over a region of interaction as a measure of the encounter probability, rather than the first-encounter probability. We show the discrepancy between the two approaches by comparing analytically their predictions with continuous variables, while with discrete variables we quantify their difference over time. We conclude by pointing to some of the open problems that reaction diffusion formalism might be able to tackle.

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arXiv2023

Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier

The Arcsine laws of Brownian motion are a collection of results describing three different statistical quantities of one-dimensional Brownian motion: the time at which the process reaches its maximum position, the total time the process spends in the positive half-space and the time at which the process crosses the origin for the last time. Remarkably the cumulative probabilities of these three observables all follows the same distribution, the Arcsine distribution. But in real systems, space is often heterogeneous, and these laws are likely to hold no longer. In this paper we explore such a scenario and study how the presence of a spatial heterogeneity alters these Arcsine laws. Specifically we consider the case of a thin permeable barrier, which is often employed to represent diffusion impeding heterogeneities in physical and biological systems such as multilayer electrodes, electrical gap junctions, cell membranes and fragmentation in the landscape for dispersing animals. Using the Feynman-Kac formalism and path decomposition techniques we are able to find the exact time-dependence of the probability distribution of the three statistical quantities of interest. We show that a permeable barrier has a large impact on these distributions at short times, but this impact is less influential as time becomes long. In particular, the presence of a barrier means that the three distributions are no longer identical with symmetry about their means being broken. We also study a closely related statistical quantity, namely, the distribution of the maximum displacement of a Brownian particle and show that it deviates significantly from the usual half-Gaussian form.

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arXiv2023

Exact Spatio-Temporal Dynamics of Lattice Random Walks in Hexagonal and Honeycomb Domains

A variety of transport processes in natural and man-made systems are intrinsically random. To model their stochasticity, lattice random walks have been employed for a long time, mainly by considering Cartesian lattices. However, in many applications in bounded space the geometry of the domain may have profound effects on the dynamics and ought to be accounted for. We consider here the cases of the six-neighbour (hexagonal) and three-neighbour (honeycomb) lattice, which are utilised in models ranging from adatoms diffusing in metals and excitations diffusing on single-walled carbon nanotubes to animal foraging strategy and the formation of territories in scent-marking organisms. In these and other examples, the main theoretical tool to study the dynamics of lattice random walks in hexagonal geometries has been via simulations. Analytic representations have in most cases been inaccessible, in particular in bounded hexagons, given the complicated zig-zag boundary conditions that a walker is subject to. Here we generalise the method of images to hexagonal geometries and obtain closed-form expressions for the occupation probability, the so-called propagator, for lattice random walks both on hexagonal and honeycomb lattices with periodic, reflective and absorbing boundary conditions. In the periodic case, we identify two possible choices of image placement and their corresponding propagators. Using them, we construct the exact propagators for the other boundary conditions, and we derive transport related statistical quantities such as first passage probabilities to one or multiple targets and their means, elucidating the effect of the boundary condition on transport properties.

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arXiv2023

Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach

We derive, through subordination techniques, a generalized Feynman-Kac equation in the form of a time fractional Schrodinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman-Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman-Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.

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arXiv2022

Dynamics of lattice random walk within regions composed of different media and interfaces

We study the lattice random walk dynamics in a heterogeneous space of two media separated by an interface and having different diffusivity and bias. Depending on the position of the interface, there exist two exclusive ways to model the dynamics: (1) Type A dynamics whereby the interface is placed between two lattice points, and (2) Type B dynamics whereby the interface is placed on a lattice point. For both types, we obtain exact results for the one-dimensional generating function of the Green's function or propagator for the composite system in unbounded domain as well as domains confined with reflecting, absorbing, and mixed boundaries. For the case with reflecting confinement in the absence of bias, the steady-state probability shows a step-like behavior for the Type A dynamics, while it is uniform for the Type B dynamics. We also derive explicit expressions for the first-passage probability and the mean first-passage time, and compare the hitting time dependence to a single target. Finally, considering the continuous-space continuous-time limit of the propagator, we obtain the boundary conditions at the interface. At the interface, while the flux is the same, the probability density is discontinuous for Type A and is continuous for Type B. For the latter we derive a generalized version of the so-called leather boundary condition in the appropriate limit.

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arXiv2022

Discrete space-time resetting model: Application to first-passage and transmission statistics

We consider the dynamics of lattice random walks with resetting. The walker moving randomly on a lattice of arbitrary dimensions resets at every time step to a given site with a constant probability $r$. We construct a discrete renewal equation and present closed-form expressions for different quantities of the resetting dynamics in terms of the underlying reset-free propagator or Green's function. We apply our formalism to the biased random walk dynamics in one-dimensional unbounded space and show how one recovers in the continuous limits results for diffusion with resetting. The resetting dynamics of biased random walker in one-dimensional domain bounded with periodic and reflecting boundaries is also analyzed. Depending on the bias the first-passage probability in periodic domain shows multi-fold non-monotonicity as $r$ is varied. Finally, we apply our formalism to study the transmission dynamics of two lattice walkers with resetting in one-dimensional domain bounded by periodic and reflecting boundaries. The probability of a definite transmission between the walkers shows non-monotonic behavior as the resetting probabilities are varied.

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arXiv2022

Particle-Environment Interactions In Arbitrary Dimensions: A Unifying Analytic Framework To Model Diffusion With Inert Spatial Heterogeneities

Abridged abstract: Inert interactions between randomly moving entities and spatial disorder play a crucial role in quantifying the diffusive properties of a system. These interactions affect only the movement of the entities, and examples range from molecules advancing along dendritic spines to anti-predator displacements of animals due to sparse vegetation. Despite the prevalence of such systems, a general framework to model the movement explicitly in the presence of spatial heterogeneities is missing. Here, we tackle this challenge and develop an analytic theory to model inert particle-environment interactions in domains of arbitrary shape and dimensions. We use a discrete space formulation which allows us to model the interactions between an agent and the environment as perturbed dynamics between lattice sites. Interactions from spatial disorder, such as impenetrable and permeable obstacles or regions of increased or decreased diffusivity, as well as many others, can be modelled using our framework. We provide exact expressions for the generating function of the occupation probability of the diffusing particle and related transport quantities such as first-passage, return and exit probabilities and their respective means. We uncover a surprising property, the disorder indifference phenomenon of the mean first-passage time in the presence of a permeable barrier in quasi-1D systems. We demonstrate the widespread applicability of our formalism by considering three examples that span across scales and disciplines. (1) We explore an enhancement strategy of transdermal drug delivery. (2) We associate the disorder with a decision-making process of an animal to study thigmotaxis. (3) We illustrate the use of spatial heterogeneities to model inert interactions between particles by modelling the search for a promoter region on the DNA by transcription factors during gene transcription.

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arXiv2022

Diffusion through permeable interfaces: Fundamental equations and their application to first-passage and local time statistics

The diffusion equation is the primary tool to study the movement dynamics of a free Brownian particle, but when spatial heterogeneities in the form of permeable interfaces are present, no fundamental equation has been derived. Here we obtain such an equation from a microscopic description using a lattice random walk model. The sought after Fokker-Planck description and the corresponding backward Kolmogorov equation are employed to investigate first-passage and local time statistics and gain new insights. Among them a surprising phenomenon, in the case of a semibounded domain, is the appearance of a regime of dependence and independence on the location of the permeable barrier in the mean first-passage time. The new formalism is completely general: it allows to study the dynamics in the presence of multiple permeable barriers as well as reactive heterogeneities in bounded or unbounded domains and under the influence of external forces.

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arXiv2020

Closed-form solutions to the dynamics of confined biased lattice random walks in arbitrary dimensions

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.

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arXiv2020

Anderson-like localization transition of random walks with resetting

We study several lattice random walk models with stochastic resetting to previously visited sites which exhibit a phase transition between an anomalous diffusive regime and a localization regime where diffusion is suppressed. The localized phase settles above a critical resetting rate, or rate of memory use, and the probability density asymptotically adopts in this regime a non-equilibrium steady state similar to that of the well known problem of diffusion with resetting to the origin. The transition occurs because of the presence of a single impurity site where the resetting rate is lower than on other sites, and around which the walker spontaneously localizes. Near criticality, the localization length diverges with a critical exponent that falls in the same class as the self-consistent theory of Anderson localization of waves in random media. The critical dimensions are also the same in both problems. Our study provides analytically tractable examples of localization transitions in path-dependent, reinforced stochastic processes, which can be also useful for understanding spatial learning by living organisms.

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arXiv2019

Comparison of two models of tethered motion

We consider a random walker whose motion is tethered around a focal point. We use two models that exhibit the same spatial dependence in the steady state but widely different dynamics. In one case, the walker is subject to a deterministic bias towards the focal point, while in the other case, it resets its position to the focal point at random times. The deterministic tendency of the biased walker makes the forays away from the focal point more unlikely when compared to the random nature of the returns of the resetting walker. This difference has consequences on the spatio-temporal dynamics at intermediate times. To show the differences in the two models, we analyze their probability distribution and their dynamics in presence and absence of partially or fully absorbing traps. We derive analytically various quantities: (i) mean first-passage times to one target, where we recover results obtained earlier by a different technique, (ii) splitting probabilities to either of two targets as well as survival probabilities when one or either target is partially absorbing. The interplay between confinement, diffusion and absorbing traps produces interesting non-monotonic effects in various quantities, all potentially accessible in experiments. The formalism developed here may have a diverse range of applications, from study of animals roaming within home ranges and of electronic excitations moving in organic crystals to developing efficient search algorithms for locating targets in a crowded environment.

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