Simple Random Walk Research Articles

A multivariate extension of the Erdős–Taylor theorem

AbstractThe Erdős–Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if $$\textsf{L}_N$$ L N is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then $$\tfrac{\pi }{\log N} \,\textsf{L}_N$$ π log N L N converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if $$\textsf{L}_N^{(1,2)}=\sum _{n=1}^N \mathbb {1}_{\{S_n^{(1)}= S_n^{(2)}\}}$$ L N ( 1 , 2 ) = ∑ n = 1 N 1 { S n ( 1 ) = S n ( 2 ) } , then $$\tfrac{\pi }{\log N}\, \textsf{L}^{(1,2)}_N$$ π log N L N ( 1 , 2 ) converges in distribution to an exponential random variable of parameter one. We prove that for every $$h \geqslant 3$$ h ⩾ 3 , the family $$ \big \{ \frac{\pi }{\log N} \,\textsf{L}_N^{(i,j)} \big \}_{1\leqslant i<j\leqslant h}$$ { π log N L N ( i , j ) } 1 ⩽ i < j ⩽ h , of logarithmically rescaled, two-body collision local times between h independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdős–Taylor theorem. We also discuss connections to directed polymers in random environments.

Weighted Jump in Random Walk graph sampling

Graph sampling is a challenging problem in network analysis due to the complex structures of networks. Currently, a series of graph sampling algorithms based on random walks achieve good results in graph sampling tasks. However, the existing methods often reduce the conductance of graphs, causing the sampler to stay in the same node for a long time. This results in undersampling. In this paper, we propose a novel Weighted Jump Random Walk (WJRW) algorithm to generate representative samples. We design a parameter in the WJRW algorithm that can adjust the proportions of random walk and random jump in every step. According to the issue of repeated sample nodes in the Generalized Maximum Degree (GMD) method and the problem of large deviations in the Simple Random Walk (SRW), WJRW addresses the weaknesses of the GMD method and enhances the diffusion of a random walker on graphs, leading to a more representative sample. Then, WJRW addresses the issue of large deviations in SRW and enhances the efficiency of the unbiased estimator. By generating smoother stationary distributions. Numerical experiments with extensive real-world networks verify that our method achieves higher accuracy than classical and state-of-the-art methods in estimating distribution estimation.

The GHP Scaling Limit of Uniform Spanning Trees in High Dimensions

We show that the Brownian continuum random tree is the Gromov–Hausdorff–Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the d-dimensional torus Znd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_n^d$$\\end{document} with d>4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d>4$$\\end{document}, the hypercube {0,1}n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{0,1\\}^n$$\\end{document}, and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.

Mixing time of random walk on dynamical random cluster

AbstractWe study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate $$\mu $$ μ between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order $$n^2/\mu $$ n 2 / μ . In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.

The GHP scaling limit of uniform spanning trees of dense graphs

AbstractWe consider dense graph sequences that converge to a connected graphon and prove that the GHP scaling limit of their uniform spanning trees (USTs) is Aldous' Brownian CRT. Furthermore, we are able to extract the precise scaling constant from the limiting graphon. As an example, we can apply this to the scaling limit of the USTs of the Erdös–Rényi sequence for any fixed , and sequences of dense expanders. A consequence of GHP convergence is that several associated quantities of the spanning trees also converge, such as the height, diameter and law of a simple random walk.

Explosion and non-explosion for the continuous-time frog model

We consider the continuous-time frog model on Z. At time t=0, there are η(x) particles at x∈Z, each of which is represented by a random variable. In particular, (η(x))x∈Z is a collection of independent random variables with a common distribution μ, μ(Z+)=1, Z+≔N∪{0}, N={1,2,3,…}. The particles at the origin are active, all other ones being assumed as dormant, or sleeping, hence not active. Active particles perform a simple symmetric continuous-time random walk in Z (that is, a random walk with exp(1)-distributed jump times and jumps −1 and 1, each with probability 1/2), independently of all other particles. Sleeping particles stay still until the first arrival of an active particle to their location; upon arrival they become active and start their own simple random walks. Different sets of conditions are given ensuring explosion, respectively non-explosion, of the continuous-time frog model. Our results show in particular that if μ is the distribution of eYlnY with a non-negative random variable Y satisfying EY<∞, then a.s. no explosion occurs. On the other hand, if a∈(0,1) and μ is the distribution of eX, where P{X≥t}=t−a, t≥1, then explosion occurs a.s. The proof relies on a certain type of comparison to a percolation model which we call totally asymmetric discrete inhomogeneous Boolean percolation.

Some martingale properties of the simple random walk and its maximum process

Martingales related to simple random walks and their maximum processes are investigated, and characterizations of those martingales are obtained. As applications, derivation of the Kennedy martingale, proofs of the corresponding Doob inequalities, and a solution to the Skorokhod embedding problem are presented.

A soft Lasso model for the motion of a ball falling in the non-Newtonian fluid

From the mesoscopic point of view, a new concept of soft matching is proposed innovatively. Then a soft Lasso’s approach to learn the soft dynamical equation for the physical mechanical relationship is proposed, too. Furthermore, a discrete iterative algorithm combining the Newton–Stokes term and the soft Lasso’s term is developed to simulate the motion of a ball falling in non-Newtonian fluids. The theory is validated by numerical examples and shows satisfactory results, which exhibit the chaotic phenomena, occasional sudden accelerations and continual random oscillations. These behaviors will maintain for a long time. Furthermore, the pattern of the motion is independent of the initial values as the solution to the Newton–Stokes equation. We find the soft relation in physics to characterize the chaotic phenomena based on the learning method, and improve the method of adding a simple random walk or normal distribution.

Communities in C. elegans connectome through the prism of non-backtracking walks

The fundamental relationship between the mesoscopic structure of neuronal circuits and organismic functions they subserve is one of the major challenges in contemporary neuroscience. Formation of structurally connected modules of neurons enacts the conversion from single-cell firing to large-scale behaviour of an organism, highlighting the importance of their accurate profiling in the data. While connectomes are typically characterized by significant sparsity of neuronal connections, recent advances in network theory and machine learning have revealed fundamental limitations of traditionally used community detection approaches in cases where the network is sparse. Here we studied the optimal community structure in the structural connectome of Caenorhabditis elegans, for which we exploited a non-conventional approach that is based on non-backtracking random walks, virtually eliminating the sparsity issue. In full agreement with the previous asymptotic results, we demonstrated that non-backtracking walks resolve the ground truth annotation into clusters on stochastic block models (SBM) with the size and density of the connectome better than the spectral methods related to simple random walks. Based on the cluster detectability threshold, we determined that the optimal number of modules in a recently mapped connectome of C. elegans is 10, which precisely corresponds to the number of isolated eigenvalues in the spectrum of the non-backtracking flow matrix. The discovered communities have a clear interpretation in terms of their functional role, which allows one to discern three structural compartments in the worm: the Worm Brain (WB), the Worm Movement Controller (WMC), and the Worm Information Flow Connector (WIFC). Broadly, our work provides a robust network-based framework to reveal mesoscopic structures in sparse connectomic datasets, paving way to further investigation of connectome mechanisms for different functions.

On Random Walks with Geometric Lifetimes

We consider the following stochastic process. Initially, there is a single particle placed at the origin of Z . This particle moves as a discrete-time simple random walk on Z , which takes one step to the right with probability p. Furthermore, the particle has a probability of death ( 1 − α ) before each step. Whenever this particle returns to the origin, it gives birth to a new particle, which performs the same dynamics. For which values of the pair ( p , α ) does this model survive with positive probability? Can we obtain the survival probability as a function of ( p , α ) ? By answering these questions, we review some fundamental concepts and tools in probability, such as the return times of a simple random walk on the integers, probability generating functions, branching processes, and coupling.

Relations between scaling exponents in unimodular random graphs

We investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents: \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} d_{w} &= d_{f} + \ ilde{\\zeta }, \\\\ d_{s} &= 2 d_{f}/d_{w}, \\end{aligned}$$ \\end{document} where dw is the walk dimension, df is the fractal dimension, ds is the spectral dimension, and tilde{zeta } is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if df and tilde{zeta } geqslant 0 exist, then dw and ds exist, and the aforementioned equalities hold. Moreover, our primary new estimate d_{w} geqslant d_{f} + tilde{zeta } is established for all tilde{zeta } in mathbb{R}.For the uniform infinite planar triangulation (UIPT), this yields the consequence dw=4 using df=4 (Angel in Geom. Funct. Anal. 13(5):935–974, 2003) and tilde{zeta }=0 (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2020 and (Ding and Gwynne in Commun. Math. Phys. 374(3):1877–1934, 2020)). The conclusion dw=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that dw=df for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since df>2.

On the average hitting times of [formula omitted]

The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks on Cay(ZN,{±1,±2}) was given by Y. Doi et al.. Y. Doi et al. give a simple formula for the HT’s of simple random walks on Cay(ZN,{±1,±2}) by using an elementary method. In this paper, using an elementary method also used by Y. Doi et al., we give a simple formula for HT’s of simple random walks on Cay(ZN,{+1,+2}).

Scale-free percolation mixing time

Assign to each vertex of the one-dimensional torus i.i.d. weights with a heavy-tail of index τ−1>0. Connect then each couple of vertices with probability roughly proportional to the product of their weights and that decays polynomially with exponent α>0 in their distance. The resulting graph is called scale-free percolation. The goal of this work is to study the mixing time of the simple random walk on this structure. We depict a rich phase diagram in α and τ. In particular we prove that the presence of hubs can speed up the mixing of the chain. We use different techniques for each phase, the most interesting of which is a bootstrap procedure to reduce the model from a phase where the degrees have bounded averages to a setting with unbounded averages.

Infinite collision property for the three-dimensional uniform spanning tree

Let [Formula: see text] be the three-dimensional uniform spanning tree, whose probability law is denoted by [Formula: see text]. For [Formula: see text]-a.s. realization of [Formula: see text], the recurrence of the simple random walk on [Formula: see text] is proved in Benjamini et al. (2001) [I. Benjamini, R. Lyons, Y. Peres and O. Schramm, Uniform spanning forests, Ann. Probab. 29(1) (2001) 1–65] and it is also demonstrated in Hutchcroft and Peres (2015) [T. Hutchcroft and Y. Peres, Collisions of random walks in reversible random graphs, Electron. Commun. Probab. 20(63) (2015) 1–6] that two independent simple random walks on [Formula: see text] collide infinitely often. In this paper, we will give a quantitative estimate on the number of collisions of two independent simple random walks on [Formula: see text], which provides another proof of the infinite collision property of [Formula: see text].

A Markov Chain Model for Determining the Optimal Time to Move Pregnant Cows to Individual Calving Pens.

The use of individual calving pens in modern farming is widely recognized as a good practice for promoting good animal welfare during parturition. However, determining the optimal time to move a pregnant cow to a calving pen can be a management challenge. Moving cows too early may result in prolonged occupancy of the pen, while moving them too late may increase the risk of calving complications and production-related diseases. In this paper, a simple random walk type Markov Chain Model to predict the optimal time for moving periparturient cows to individual calving pens was proposed. Behavior changes such as lying time, standing time, and rumination time were analyzed using a video monitoring system, and we formulated these changes as the states of a Markov Chain with an absorbing barrier. The model showed that the first time entering an absorbing state was the optimal time for a pregnant cow to be moved to a calving pen. The proposed method was validated through a series of experiments in a real-life dairy farm, showing promising results with high accuracy.

A scaling limit of the parabolic Anderson model with exclusion interaction

AbstractWe consider the (discrete) parabolic Anderson model , , , where the ξ‐field is ‐valued and plays the role of a dynamic random environment, and Δ is the discrete Laplacian. We focus on the case in which ξ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein–Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension upon considering a suitably renormalised version of the above equation, the sequence of solutions converges in law. As a by‐product of our main result we obtain precise asymptotics for the survival probability of a simple random walk that is killed at a scale dependent rate when meeting an exclusion particle. Our proof relies on the discrete theory of regularity structures of  Erhard and Hairer and on novel sharp estimates of joint cumulants of arbitrary large order for the exclusion process. We think that the latter is of independent interest and may find applications elsewhere.

Characterization of random walks on space of unordered trees using efficient metric simulation

The simple random walk on Zp shows two drastically different behaviors depending on the value of p: it is recurrent when p∈{1,2} while it escapes (with a rate increasing with p) as soon as p≥3. This classical example illustrates that the asymptotic properties of a random walk provides some information on the structure of its state space. This paper aims to explore analogous questions on space made up of combinatorial objects with no algebraic structure. We take as a model for this problem the space of unordered unlabeled rooted trees endowed with Zhang edit distance. To this end, it defines the canonical unbiased random walk on the space of trees and provides an efficient algorithm to evaluate its escape rate. Compared to Zhang algorithm, it is incremental and computes the edit distance along the random walk approximately 100 times faster on trees of size 500 on average. The escape rate of the random walk on trees is precisely estimated using intensive numerical simulations, out of reasonable reach without the incremental algorithm.

Interlacement limit of a stopped random walk trace on a torus

AbstractWe consider a simple random walk on $\mathbb{Z}^d$ started at the origin and stopped on its first exit time from $({-}L,L)^d \cap \mathbb{Z}^d$ . Write L in the form $L = m N$ with $m = m(N)$ and N an integer going to infinity in such a way that $L^2 \sim A N^d$ for some real constant $A \gt 0$ . Our main result is that for $d \ge 3$ , the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level $A d \sigma_1$ , where $\sigma_1$ is the exit time of a Brownian motion from the unit cube $({-}1,1)^d$ that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).

Asymptotics of noncolliding q-exchangeable random walks

We consider a process of noncolliding q-exchangeable random walks on making steps 0 (‘straight’) and −1 (‘down’). A single random walk is called q-exchangeable if under an elementary transposition of the neighboring steps the probability of the trajectory is multiplied by a parameter . Our process of m noncolliding q-exchangeable random walks is obtained from the independent q-exchangeable walks via the Doob’s h-transform for a nonnegative eigenfunction h (expressed via the q-Vandermonde product) with the eigenvalue less than 1. The system of m walks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is the q-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncolliding q-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel of q-distributed random lozenge tilings of sawtooth polygons. In the limit as , with γ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.

A note about the maximum local time of a random walk on the square lattice

This paper conducts a numerical analysis of the behavior of the average value of the Maximum Local Time, [Formula: see text], in the Simple Random Walk on the square lattice. It has been established in the literature that the sequence [Formula: see text] converges to [Formula: see text]. The author found numerical evidence that the average value of [Formula: see text] ([Formula: see text]) increases until a certain value of [Formula: see text], denoted as [Formula: see text], after which it decreases and approaches [Formula: see text]. Furthermore, estimates are presented for [Formula: see text] and [Formula: see text].