Subdiffusive random walk. Dans le cas ou #7B-G est l'arbre d'un processus de branchement critique, conditionne par la non-extinction, si h (x) denote la distance sur #7B-G de x a la racine de l'arbre alors la distribution de n Nov 9, 2007 · The asymptotic mean number of distinct sites visited by a subdiffusive continuous time random walker in two dimensions seems not to have been explicitly calculated anywhere in the literature. Stat. Nichols and Bruce Ian Henry and Christopher N. Mech Funding Information: A Z acknowledges support of the INSPIRE fellowship from DST India and the Physics Computing Facility lab at UCSD. The governing Abstract. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk (Xn) in random environment on a regular tree, which is closely related to Mandelbrot’s (C. Further details can be found in the reviews [12], [13], [14]. =. Majumdar; Published 30 January 2023; Physics Download scientific diagram | Lagrangian propagators for a subdiffusive random walk with μ = 2, α = 1/2, and K α = 1/2. Motivated by trap models of slow dynamics, we consider a model in which the trap depth increases logarithmically with distance from Nov 5, 2020 · The existence of a nonequilibrium stationary state and finite mean first arrival time is proved and the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power-law tails. A class of discrete time random walks has recently been introduced to provide a stochastic process based numerical scheme for solving fractional order partial differential equations, including the fractional subdiffusi… . [Phys. 5) from an ensemble average over N = 10 8 different particle trajectories. H <. Continuous-time RWs have Jul 2, 2015 · Systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power-law scaling of the mean square displacement. Comput. The distribution of the waiting time between the reset events is represented as a sum of an Jan 30, 2023 · A sluggish random walk with subdiffusive spread. The first part of the paper is devoted to proving an almost sure analogue of H. AU - Evans, Martin R. , 2008). This formulation is exemplified by means of an advection-diffusion and a jump-diffusion scheme. H. We provide two forms of the waiting time density Apr 1, 2016 · In the paper [28], [29], researchers applied continuous-time random walks in finance and economics. Dirichlet environments are weakly elliptic and the walk can be slowdowned by local traps whose strength are Langevin formulation of a subdiffusive continuous time random walk in physical time Andrea Cairoli and Adrian Baule∗ arXiv:1501. Acad. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to Abstract. Stochastic trajectories of a CTRW can be described in terms of the subordination of a normal diffusive process by an inverse Lévy-stable process. January 2022. for a static domain. If the waiting-time distribution has long tails, such that the ensemble average waiting time ⟨ t ⟩ pause diverges, then the overall motion becomes subdiffusive. Apr 9, 2024 · This paper studies random walks in random Dirichlet environments, which can be trapped and delayed by local irregularities. However, this model inherently cannot describe subdiffusive cell movement, i. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. Theorem A (Lyons and Pemantle [11]) With P-probability one, the walk (Xn) is recurrent or transient, according to whether p ≤ 1 b or p > 1 b. ( 15 ) which Oct 26, 2012 · Abstract. IntroductionThe understanding of field line random walk (FLRW) is a central part of turbulence theory and space-science. Here we develop a Monte Carlo method for Jan 1, 2022 · Scaling limit of the subdiffusive random walk on a Galton–Watson tree in random environment. AU - Majumdar, Satya N. Dirichlet environments are weakly elliptic and the walk can be slowdowned by local traps whose strength are Feb 4, 2023 · Random walks are usually characterized by the spatial territory they cover, described by the number of sites visited at a given time. AU - Allen, Rosalind J. from publication: Continuous Time Random Walk in Apr 6, 2015 · Main subdiffusive models. Evans and Satya N. a random walk in any of these random environments is subdiffusive in any dimension d < ~. It proves a limit theorem describing the fluctuations when the walk is ballistic (moves linearly in time) but subdiffusive (has fluctuations larger than diffusive). The asymptotic behaviour of hitting Sep 13, 2012 · random walk mod el for anomalou s subdiffusive transport of cells. Continuous-time random walks of a particle that is randomly reset to an initial position are considered. The generalized method of images approach | In this paper we study subdiffusion in a system with a thin membrane. The first part of the Jul 15, 2019 · The limit of the random walk, with a force derived from a logarithmic potential, defines a stochastic process that is a fractional generalization of geometric Brownian motion. R. Allen, +1 author S. Under incomplete We study a one-dimensional sluggish random walk with space-dependent transition probabilities between nearest-neighbour lattice sites. Fractional Brownian motion and the random walk on a random walk model (RWRW) [10], which describes the random motion of a tagged particle on a random path, fall in the first class. Two situations are considered. We study a one-dimensional sluggish random walk with space-dependent transition probabilities. 1 2. Jan 27, 2015 · Systems living in complex non equilibrated environments often exhibit subdiffusion characterized by a sublinear power-law scaling of the mean square displacement. Nov 2017 Pierre Rousselin Jun 11, 2020 · We study continuous-time random walks (CTRW) with power-law distribution of waiting times under resetting which brings the walker back to the origin, with a power-law distribution of times between the resetting events. In this context, we analyze anomalous diffusion and dynamics in a one-dimensional random walk, arising from the effects of the memory in a model proposed by Kumar et al. We investigate the effects of Markovian resetting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed Dec 5, 2008 · We formulate the generalized master equation for a class of continuous-time random walks in the presence of a prescribed deterministic evolution between successive transitions. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk (X n ) in random environment on a regular tree, which is closely related to Mandelbrot’s (C. The resetting process is considered as a renewal process with power-law distribution of waiting times between the Feb 28, 2024 · We also consider that, starting from the origin (x 0 = 0, t 0 = 0), seismicity undergoes a random walk in time and space, where each new event site is the new position of the random walk that Feb 8, 2016 · Within the T cell zone of lymph nodes, naive T cells display features of a diffusive (Brownian-type) or subdiffusive random walk 9,17,18,19,20 but also show speed fluctuations, alternating between II. 2010 Mathematics Subject Classification. 1) Jan 15, 2020 · Non-Markovian processes are considerably more difficult to study numerically. Allen and Martin R. 35, 1978–1997 (2007) Random walks in random environments, Galton–Watson trees, conductance. Angstmann}, journal={J. In the first model the whole process is reset to the initial state, whereas in the Mar 15, 2006 · We are interested in the random walk in random environment on an infinite tree. Markov random walk model. Paris 278, 289 Jan 27, 2015 · One of the most common models to describe such subdiffusive dynamics is the continuous time random walk (CTRW). At the beginning, the T1 - A sluggish random walk with subdiffusive spread. The method presented here provides the general form of the Green's function which is also valid for various kinds of diffusion (normal diffusion May 1, 2009 · The random walk of energetic charged particles in turbulent magnetic fields is investigated. Probab. This appears to be an instance of a single random walk model leading to all three forms of behavior by simply changing parameter values. Annales de l'I. 1214/21-AOP1535. First let us consider a Markov model for random cell movement along one-dimensional lattice such that all steps are of equal length 1. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and Random walks with strongly inhomogeneous rates and singular diffusions: Convergence, localization and aging in one dimension (τi) = ∞ and X is subdiffusive LIMIT THEOREMS FOR RANDOM WALKS ON A STRIP IN SUBDIFFUSIVE REGIMES D. jcp. This leads to a random walk which has symmetric transition probabilities that decrease with distance $|k|$ from the May 1, 2009 · 1. May 19, 2015 · Request PDF | Subdiffusive random walk in a membrane system. The particle waits at some position for a random time, governed by the waiting time probability density function, before Dec 26, 2019 · Continuous-time random walks of a particle that is randomly reset to an initial position are considered. Sci. e. May 14, 2019 · We analyze two models of subdiffusion with stochastic resetting. Aug 25, 2016 · We consider a random walk on a Galton-Watson tree in random environment, in the subdiffusive case. Stochastic trajectories of a CTRW can be described mathematically in terms of a Nov 16, 2017 · Subdiffusive discrete time random walks via Monte Carlo and subordination. A subdiffusive behaviour of recurrent random walk 523 We recall a recurrence/transience criterion from Lyons and Pemantle [11] (Theorem 1 and Proposition 2). We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of … Jan 1, 2023 · Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk (X n ) in random environment on a regular tree, which is closely related to Mandelbrot’s (C. Webb et al. Lyons and Pemantle [11] give a precise recurrence/transience criterion. 60J80, 60G50, 60F25, 60F15. Dirichlet environments are weakly elliptic and the walk can be slowdowned by local traps whose strength are Using two random walk models in a system with a thin membrane we find the Green’s functions describing various kinds of diffusion in this system; the membrane is treated here as a thin, partially permeable wall. Apr 9, 2024 · We consider random walk in Dirichlet random environment in ${\\mathbf{Z}^d, d\\ge 3}$, which corresponds to the case where the environment is constructed from i. P. In geometric Brownian motion the expectation of the logarithm of the position of the particle scales linearly with time. We analyze two models of subdiffusion with stochastic resetting. This theoretical result is Sep 24, 2015 · DOI: 10. Kesten’s subdiffusivity theorem for the random walk on the incipient infinite cluster and the invasion percolation cluster using ideas of M. 1. In this paper we develop a coupled continuous-time random walk model in which the waiting time is power-law coupled with the local environmental diffusion coefficient. The asymptotic mean number of distinct sites visited by a subdiffusive continuous-time random walker in two dimensions seems not to have been explicitly calculated anywhere in the literature. . [6] (for a somewhat different but simpler model, see [7]). Authors: Loïc de Raphélis. Stochastic trajectories of a CTRW can be described mathematically in terms of a subordination of a normal diffusive May 14, 2019 · A general version of random walk namely the mesoscopic continuous time random walks with heavy tailed jump distributions (such as Lévy flights) were studied in [49, 50]. Magdziarz applied the CTRW into the option pricing problem [30]. Allen, Martin R. We define the probability p(k,t) = Pr{X(t) = k} (1. 1016/j. we obtain the subdiffusive dynamics, [8,9]. The key element of the CTRW is the subordination of random processes [21]. 06. On considere deux cas de marche aleatoire {X n } n≥0 sur un graphe aleatoire #7B-G. d. : A subdiffusive behavior of recurrent random walk in random environment on a regular tree. We derive "quenched" subdiffusive lower bounds for the exit time tau (n) from a box of size n for the simple random walk on the planar invasion percolation cluster. Motivated by trap models of slow dynamics, we consider a model in which the trap depth increases logarithmically with distance from the origin. that describes the non-homogeneous in space subdiffusive transport of cells. Probabilités et statistiques (1986) Volume: 22, Issue: 4, page 425-487 May 2, 2019 · Scaling limit of the subdiffusive random walk on a Galton-Watson tree in random environment. 58, 1100 (1987)] with respect to the squared displacements. Lett. Evans, Satya N. Each of them consists of two parts: subdiffusion based on the continuous-time random walk scheme and independent resetting events generated uniformly in time according to the Poisson point process. Majumdar. Stochastic trajectories of a CTRW can be described in terms of the subordination of a normal diffusive process by an Physical Review Link Manager Jun 19, 2018 · The persistent random walk (PRW) model accurately describes cell migration on two-dimensional (2D) substrates. 1 Introduction The strong links between electric networks and reversible random walks on graphs have emerged during the second half of the last century and were popularized in the seminal book [9]. : Slow movement of random walk in random environment on a regular tree. The main aim of the paper is to incorporate the nonlinear kinetic term into non-Markovian transport equations described by a continuous Jan 6, 2007 · We are interested in the random walk in random environment on an infinite tree. transition probabilities at each vertex with a Dirichlet distribution with parameters $(α_i)_{1 \\le i \\le 2d}$. We study the asymptotic behaviour of occupation times of a transient random walk in a quenched random environment on a strip in a sub-di usive regime. This strategy was used in the papers of Kesten, Kozlov, Spitzer [19] in the case of random walks in random environment on Z, and Basdevant and Singh [4],[5],[6] in the case of multi-excited Jan 20, 2022 · The TSRW model is a typical two-state renewal process alternating between the continuous-time random walk state and the Lévy walk state, in both of which the sojourn time distributions follow a Jun 10, 2013 · Request PDF | Subdiffusive continuous time random walks and weak ergodicity breaking analyzed with the distribution of generalized diffusivities | We propose a new tool for analyzing data from Sep 1, 2013 · We derive quenched subdiffusive lower bounds for the exit time τ (n) from a box of size n for the simple random walk on the planar invasion percolation cluster. We explore the dynamics of a tuneable box-trapped Bose gas under strong periodic forcing in the presence of weak disorder. Furthermore, the model … Nov 9, 2023 · In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, two of them diffusive and one of them normal diffusive. We present an explicit derivation for two cases in all Apr 6, 2015 · Random walks are non-stationary, since the ensemble average 〈 x 2 (t) 〉 is explicitly time dependent. We attach to each vertex v of this tree a random variable X (v) and define \ (S (v) = \Sigma _ {w \varepsilon \pi (0 A sluggish random walk with subdiffusive spread @inproceedings{Zodage2023ASR, title={A sluggish random walk with subdiffusive spread}, author={Aniket Zodage and Rosalind J. Majumdar}, year={2023} } Aniket Zodage, R. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk $(X\\_n)$ in random environment on a regular tree, which is closely related to Mandelbrot [13]'s multiplicative cascade. In absence of interparticle interactions, the interplay of the drive and disorder results in an isotropic nonthermal momentum distribution Aug 25, 2016 · We consider a random walk on a Galton-Watson tree in random environment, in the subdiffusive case. This refines existing results on the subballistic regime. 1007/S00440-016-0739-8 Corpus ID: 119322143; Scaling limit of the recurrent biased random walk on a Galton–Watson tree @article{Adkon2015ScalingLO, title={Scaling limit of the recurrent biased random walk on a Galton–Watson tree}, author={Elie A{\"i}d{\'e}kon and Lo{\"i}c de Raph{\'e}lis}, journal={Probability Theory and Related Fields}, year={2015}, volume={169}, pages={643-666 subdiffusive model. The models differ in the assumptions concerning how the particle is stopped or reflected by the membrane when the particle’s attempts to pass through it fail. Symbols represent simulations results with 10 6 realisations. For example, for the PDF to find a particle (with continuous distribution of Apr 19, 2024 · semble of continuous time random walks (CTRWs), characterized by power law waiting time densities and nearest neighbour, or Gaussian, step length densities [1]. 20, 125–136, 1992) give a precise recurrence/transience criterion. Oct 21, 2015 · We use two random walk models which differ in assumptions as to whether the particle is stopped or reflected by the membrane when the particle's attempt to pass through the membrane fails. The idea of the proof is to look at the local times of the random walk. , 2006, Shalchi and Kourakis, 2007c, Weinhorst et al. The Annals of Probability 50 (1) DOI: 10. We obtain the scaling limits of the local times and the quenched local probability for the biased walk in the subdiffusive case. Starting with a M arkov model involving a structured probability density function, we derive the non-local in time master equation DOI: 10. The models we will consider are a special case of what we have called [9] random walk on a random hillside. A class of discrete time random walks has recently been introduced to provide a stochastic process based numerical scheme for solving fractional order partial differential equations, including the fractional subdiffusion equation. Dirichlet environments are weakly elliptic and the walk can be slowdowned by local traps whose strength are Subdiffusivity of random walk on the 2D invasion percolation cluster @article{Damron2012SubdiffusivityOR, title={Subdiffusivity of random walk on the 2D invasion percolation cluster}, author={Michael Damron and Jack Hanson and Philippe Sosoe}, journal={Stochastic Processes and their Applications}, year={2012}, volume={123}, pages={3588-3621 May 19, 2015 · Using two random walk models in a system with a thin membrane we find the Green’s functions describing various kinds of diffusion in this system; the membrane is treated here as a thin, partially permeable wall. Lyons and Pemantle (Ann. Based on this master equation, we also derive reaction-diffusion equations for subdiffusive chemical species, using a mean the convergence of the height of the random walk under a second moment assumption. 06680v2 [cond-mat. Apr 9, 2024 · We consider random walk in Dirichlet random environment in ${\\mathbf{Z}^d, d\\ge 3}$, which corresponds to the case where the environment is constructed from i. g. AU - Zodage, Aniket. LANGEVIN APPROACH TO SUBDIFFUSIVE CTRW MODEL In this Section we recall briefly the subdiffusive jump CTRW model [20] and the corresponding Langevin approach [12], [13], [14]. One of the most common models to describe such subdiffusive dynamics is the continuous-time random walk (CTRW). He introduced the subdiffusive geometric Brownian motion (SGBM) as the model of asset prices, which means the time t in the right side of Eq. Ann. Very few models have been solved exactly. N1 - 17 pages, revised version accepted for J. We show that the Jun 19, 2018 · The persistent random walk (PRW) model accurately describes cell migration on two-dimensional (2D) substrates. However, one may consider the increments of the process at a given time lag τ, and consider the ensemble averages of the MSD from the initial position at time t, 〈 x 2 (τ | t) 〉 = 〈 [x (τ + t) − x (t)] 2 〉. Special focus is placed on transport across the mean magnetic field, which had been found to be Apr 13, 2023 · Observation of subdiffusive dynamic scaling in a driven and disordered Bose gas. Dec 1, 2000 · The continuous time random walk (CTRW) in a homogeneous velocity field and in arbitrary force fields is studied. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. 1: (Color online) Distribution of generalized diffusivities for a subdiffusive continuous time random walk (α = 0. Aizenman, A Jul 1, 2017 · It is shown that anomalous subdiffusion has a significant impact on the shape of the stationary state as a sum of an arbitrary number of exponentials. random walk (CTRW) and the corresponding Jan 1, 2009 · We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger et al. 1. Rev. We consider the branching tree T (n) of the first ( n +1) generations of a critical branching process, conditioned on survival till time β n for some fixed β>0 or on extinction occurring at time k n with k n /n→β. Sci In the present work we revisit the problem of the behavior of a subdiffusive continuous time random walk (CTRW) under resetting. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck- Smoluchowski equations with the corresponding memory kernels. Aug 1, 2010 · Clearly, (15) demonstrates that the random walk with self-reinforcement (5) and Mittag-Leffler distributed rest times (6) is subdiffusive in the long time limit. For > 0 there is a drift towards the origin while for < 0 there is a drift a way Apr 1, 2008 · Following continuous-time random-walk dynamics, we consider particle diffusion and trapping on fractals, and study the interplay of the spatial (alpha) and temporal (gamma) stochastic aspects. The power law waiting time density models the phenomenon of trapping whereby the longer a particle remains at a site the more likely it is to con-tinue waiting. 07061, Aug 2016. Michael Damron, Jack Hanson, Philippe Sosoe. These results are a consequence of a sharp estimate on the return time, whose analysis is driven by a family of concave recursive equations on trees. Sep 3, 2016 · Hu, Y. stat-mech] 29 Jan 2015 School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, E1 4NS, UK (Dated: January 31, 2015) Systems living in complex non equilibrated environments often exhibit subdiffusion characterized by a Nov 1, 2018 · Here we develop a Monte Carlo method for simulating discrete time random walks with Sibuya power law waiting times, providing another approximate solution of the fractional subdiffusion equation Subdiffusive behavior of random walk on a random cluster Harry Kesten. The first part of the paper is devoted to proving an almost sure Subdiffusive behavior of random walk on a random cluster. The CTRW is a stochastic process that tracks the position of a particle in time. Paris 278, 289–292 An explicit expression for the speed of a propagating front in the case of subdiffusive transport is found and applied to the problem of front propagation in the reaction-transport systems with Kolmogorov-Petrovskii-Piskunov kinetics and anomalous diffusion. Aniket Zodage, Rosalind J. The proof requires an additional assumption on Jul 15, 2019 · Brownian motion can be derived as the diffusion limit of a continuous time random walk (CTRW) with a Markovian exponential waiting time density. We calculate the qth order moments in the unbiased In a continuous time random walk (CTRW), a diffusing particle experiences random pause events where it waits a time t before re-engaging in its motion. These models can be essentially di-vided into two classes: the models with stationary incre-ments, and those intrinsically with nonstationary ones. Within the extended CTRW scheme, anomalous transport properties due to long-tailed … Expand We present a random walk model that exhibits asymptotic subdiffusive, diffusive, or superdiffusive behavior in different parameter regimes. It is, moreover, positive recurrent if p < 1 b. 2018. The second model of subdiffusion is the continuous-time. Mar 15, 2006 · We are interested in the random walk in random environment on an infinite tree. Thus, for. Under complete resetting, the CTRW after the resetting event starts anew, with a new waiting time, independent of the prehistory. Jan 30, 2023 · particle undergoes a biased random walk in which the parameter ∈ [− 1, 1] controls the strength of the bias. GOLDSHEID Abstract. , migration paths in which the root mean square displacement increases more slowly than the square root of the time interval. Fields 138, 521–549 (2007) Article MATH Google Scholar Hu, Y. , Shi, Z. i. One of the most common models to describe such subdiffusive dynamics is the continuous time random walk (CTRW). transition probabilities at each vertex with a Dirichlet distribution with parameters $(\\alpha_i)_{1 \\le i \\le 2d}$. We have investigated the moments for this process. We show that the Aug 4, 2015 · We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. Having established the relevant background and basic definitions regarding random walks and anomalous diffusion, we proceed to discuss three main classes of physical scenarios that may lead to subdiffusive behavior, and their corresponding mathematical models. This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. We consider a random walk on a Galton–Watson tree in random environment, in the subdiffusive case. We consider random walk in Dirichlet random environment in ${\\mathbf{Z}^d, d\\ge 3}$, which corresponds to the case where the environment is constructed from i. DOLGOPYAT AND I. is the Hurst exponent. with α ∈ ]1,2[; subdiffusive RWs on deterministic Jul 27, 2020 · Kurtosis for a subdiffusive random walk with α = 1/2 and K α = 1/2 on an evolving domain with power-law scale factor , for t 0 = 10 3 and γ = 0, 1/20, 1/10, 3/20, 1/5, and 1/4. arXiv e-prints, page arXiv:1608. We prove, under some general Oct 26, 2012 · Subdiffusivity of random walk on the 2D invasion percolation cluster. For instance, it has been demonstrated in several articles that stochastic wandering of magnetic field lines directly influences transport of charged cosmic rays (see, e. For values of τ in this range ( τ = 10 3 ) the distribution is practically independent of the time lag τ and agrees very well with eq. Theory Relat. Oct 1, 2009 · Fractional Brownian motion with Hurst index less then 1/2 and continuous-time random walk with heavy tailed waiting times (and the corresponding fractional Fokker-Planck equation) are two One of the most common models to describe such subdiffusive dynamics is the continuous-time random walk (CTRW). 044 Corpus ID: 52915367; Subdiffusive discrete time random walks via Monte Carlo and subordination @article{Nichols2017SubdiffusiveDT, title={Subdiffusive discrete time random walks via Monte Carlo and subordination}, author={James A. The distribution of the waiting time between the reset events is represented as a sum of an arbitrary number of exponentials. We prove the convergence of the renormalised height function of the walk towards the continuous Jun 10, 2013 · Fig. Here, we propose an equivalent Langevin formulation of a force-free CTRW without Anomalous diffusive behaviors are observed in highly inhomogeneous but relatively stable environments such as intracellular media and are increasingly attracting attention. In these systems one starts with a random * This author partially supported by NSF grant DMS 83-1080 We consider random walk in Dirichlet random environment in ${\\mathbf{Z}^d, d\\ge 3}$, which corresponds to the case where the environment is constructed from i. This number has been calculated for other dimensions for only one specific asymptotic behavior of the waiting time distribution between steps. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable L\'evy process, jointly with the convergence of the renormalised range of the walk towards the real tree coded by the latter continuous-time Consider a class of null-recurrent randomly biased walks on a supercritical Galton–Watson tree. Numerical solutions are represented by solid lines. re zc xz rh oz wz ib hj re hy