Simple Random Walk Research Articles

Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree

This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen the known asymptotics regarding the on-diagonal part of the averaged heat kernel and the expected distance travelled by the associated simple random walk.

A bio-inspired localization-free stochastic coverage algorithm with verified reachability

Dependence on constant availability to an external localization service is often unreliable and infeasible in mobile robots. In this paper, we take inspiration from a continuous fish motion model, the persistent turning Walker (PTW), to devise a strategy which is able to achieve 2D and 3D coverage in an unknown environment in the absence of a localization service, such as a global positioning system (GPS). This is achieved by converting the continuous-time dynamical system into a discrete-time Markov chain which is then shown to exhibit strongly connected properties that are verifiable through numerical methods. The aforementioned proposed framework can also be used to study the continuous-time dynamics of other biological systems and evaluate their properties. The performance of the PTW model is also compared with two existing random search strategies, simple random walks (SRW) and correlated random walks (CRW) by using analytical bounds, simulation results, and statistical tests. The simulation results show that the proposed PTW algorithm covers a given search-space at a faster rate compared to the CRW and SRW models. Hence, the PTW may be effectively used as a coverage strategy by mobile robots in underwater or underground environments where the availability of a GPS cannot be guaranteed at all times.

Estimating Vehicle Speed from Traffic Count and Occupancy Data

Automatic vehicle detectors are now common on road systems across the world. Many of these detectors are based on single inductive loops, from which data on traffic volumes (i.e. vehicle counts) and occupancy (i.e. proportion of time during which the loop is occupied) are available for 20 or 30 second observational periods. However, for the purposes of traffic management it is frequently useful to have data on (mean) vehicle speeds, but this is not directly available from single loop detectors. While detector occupancy is related in a simple fashion to vehicle speed and length, the latter variable is not measured on the vehicles that pass.In this paper a new method for speed estimation from traffic count and occupancy data is proposed. By assuming a simple random walk model for successive vehicle speeds an MCMC approach to speed estimation can be applied, in which missing vehicle lengths are sampled from an exogenous data set. Unlike earlier estimation methods, measurement error in occupancy data is explicitly modelled. The proposed methodology is applied to traffic flow data from Interstate 5 near Seattle, during a weekday morning. The efficacy of the estimation scheme is examined by comparing the estimates with independently collected vehicle speed data. The results are encouraging.

Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials

In this paper we analyze the large genus asymptotics for intersection numbers between $\psi$-classes, also called correlators, on the moduli space of stable curves. Our proofs proceed through a combinatorial analysis of the recursive relations (Virasoro constraints) that uniquely determine these correlators, together with a comparison between the coefficients in these relations with the jump probabilities of a certain asymmetric simple random walk. As an application of this result, we provide the large genus limits for Masur-Veech volumes and area Siegel-Veech constants associated with principal strata in the moduli space of quadratic differentials. These confirm predictions of Delecroix-Goujard-Zograf-Zorich from 2019.

Convergence of loop-erased random walk in the natural parameterization

Loop-erased random walk (LERW) is one of the most well-studied critical lattice models. It is the self-avoiding random walk that one gets after erasing the loops from a simple random walk in order or, alternatively, by considering the branches in a spanning tree chosen uniformly at random. We prove that planar LERW parameterized by renormalized length converges in the lattice-size scaling limit to SLE2 parameterized by 5∕4-dimensional Minkowski content. In doing this, we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theorem, several of our results about LERW are of independent interest. We give, for example, two-point estimates, estimates on maximal content, and a “separation lemma” for two-sided LERW.

Joint position and clock tracking of wireless nodes

In this paper we consider the problem of joint position and clock tracking of a mobile wireless node by a set of reference nodes. Imperfections of the mobile clock are characterized by its skew and offset, which are assumed to change with time according to simple random walk models.We put forth a measurement protocol, similar to that used in two-way ranging, and apply extended and unscented Kalman filters to estimate the position and the velocity of the mobile, and the skew and offset of its clock. We analyze the performance of the algorithms by means of extensive simulations, where the mobile’s velocity is assumed to follow a random walk. Simulation results are compared to the Cramér–Rao bound for a simplified model of a mobile with constant velocity.We show that estimation errors are largely independent of the mean values of the offset and the skew, but they increase with the mean speed. We also study how estimation errors are influenced by other factors such as the number of reference nodes.We believe these results to be of relevance, specially, in indoor positioning applications.

A random walk model that accounts for space occupation and movements of a large herbivore

Animal movement has been identified as a key feature in understanding animal behavior, distribution and habitat use and foraging strategies among others. Large datasets of invididual locations often remain unused or used only in part due to the lack of practical models that can directly infer the desired features from raw GPS locations and the complexity of existing approaches. Some of them being disputed for their lack of biological justifications in their design. We propose a simple model of individual movement with explicit parameters, based on a two-dimensional biased and correlated random walk with three forces related to advection (correlation), attraction (bias) and immobility of the animal. These forces can be directly estimated using individual data. We demonstrate the approach by using GPS data of 5 red deer with a high frequency sampling. The results show that a simple random walk template can account for the spatial complexity of wild animals. The practical design of the model is also verified for detecting spatial feature abnormalities and for providing estimates of density and abundance of wild animals. Integrating even more additional features of animal movement, such as individuals’ interactions or environmental repellents, could help to better understand the spatial behavior of wild animals.

Some criteria for circle packing types and combinatorial Gauss-Bonnet Theorem

We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)} = \infty, \] where $k_n$ is the degree excess sequence defined by \[ k_n = \sum_{v \in B_n} (\mbox{deg}\, v - 6) \] for combinatorial balls $B_n$ of radius $n$ and centered at a fixed vertex. It is also shown that the simple random walk on a disk triangulation graph is recurrent if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)+\sum_{j=0}^{n} (k_j +6)} = \infty. \] These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998.

Two-type annihilating systems on the complete and star graph

Red and blue particles are placed in equal proportion throughout either the complete or star graph and iteratively sampled to take simple random walk steps. Mutual annihilation occurs when particles with different colors meet. We compare the time it takes to extinguish every particle to the analogous time in the (simple to analyze) one-type setting. Additionally, we study the effect of asymmetric particle speeds.

Random Walks with Invariant Loop Probabilities: Stereographic Random Walks.

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.

Run-and-Tumble Motion: The Role of Reversibility

We study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

Limit law for the cover time of a random walk on a binary tree

Let Tn denote the binary tree of depth n augmented by an extra edge connected to its root. Let Cn denote the cover time of Tn by simple random walk. We prove that the sequence of random variables Cn 2−(n+1)−mn, where mn is an explicit constant, converges in distribution as n→∞, and identify the limit.

The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity

We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to γ-Liouville quantum gravity (γ-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and γ ranges from 0 to 2 as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling SLEκ for κ=16/γ2 (in the annealed sense) and that simple random walk on the embedded map converges to Brownian motion (in the quenched sense). This work constitutes the first proof that a discrete conformal embedding of a random planar map converges to LQG. Many more such statements have been conjectured. Since the mated-CRT map can be viewed as a coarse-grained approximation to other random planar maps (the UIPT, tree-weighted maps, bipolar-oriented maps, etc.), our results indicate a potential approach for proving that embeddings of these maps converge to LQG as well. To prove the main result, we establish several (independently interesting) theorems about LQG surfaces decorated by space-filling SLE. There is a natural way to use the SLE curve to divide the plane into “cells” corresponding to vertices of the mated-CRT map. We study the law of the shape of the origin-containing cell, in particular proving moments for the ratio of its squared diameter to its area. We also give bounds on the degree of the origin-containing cell and establish a form of ergodicity for the entire configuration. Ultimately, we use these properties to show (with the help of a general theorem proved in a separate paper) that random walk on these cells converges to a time change of Brownian motion, which in turn leads to the Tutte embedding result.

A reverse Aldous–Broder algorithm

The Aldous–Broder algorithm provides a way of sampling a uniformly random spanning tree for finite connected graphs using simple random walk. Namely, start a simple random walk on a connected graph and stop at the cover time. The tree formed by all the first-entrance edges has the law of a uniform spanning tree. Here we show that the tree formed by all the last-exit edges also has the law of a uniform spanning tree. This answers a question of Tom Hayes and Cris Moore from 2010. The proof relies on a bijection that is related to the BEST theorem in graph theory. We also give other applications of our results, including new proofs of the reversibility of loop-erased random walk, of the Aldous–Broder algorithm itself, and of Wilson’s algorithm.

The spectral gap of sparse random digraphs

The second largest eigenvalue of a transition matrix P has connections with many properties of the underlying Markov chain, and especially its convergence rate towards the stationary distribution. In this paper, we give an asymptotic upper bound for the second eigenvalue when P is the transition matrix of the simple random walk over a random directed graph with given degree sequence. This is the first result concerning the asymptotic behavior of the spectral gap for sparse non-reversible Markov chains with an unknown stationary distribution. An immediate consequence of our result is a proof of the Alon conjecture for directed regular graphs.

On properties of non-markovian random walk in one dimension

We study a strongly Non-Markovian variant of random walk in which the probability of visiting a given site i is a function f of number of previous visits v(i) to the site. If the probability is inversely proportional to number of visits to the site, say f(i)=1/(v(i)+1)α the probability distribution of visited sites tends to be flat for α>0 compared to simple random walk. For f(i)=exp(-v(i)), we observe a distribution with two peaks. The origin is no longer the most probable site. The probability is maximum at site k(t) which increases in time. For f(i)=exp(-v(i)) and for α>0 the properties do not change as the walk ages. However, for α<0, the properties are similar to simple random walk asymptotically. We study lattice covering time for these functions. The lattice covering time scales as Nz, with z=2, for α ≤ 0, z>2 for a >0 and z<2 for f(i)=exp(-v(i)).

Limit Theorems for the One-Dimensional Random Walk with Random Resetting to the Maximum

The first part of this paper is devoted to study a model of one-dimensional random walk with memory to the maximum position described as follows. At each step the walker resets to the rightmost visited site with probability $$r \in (0,1)$$ and moves as the simple random walk with remaining probability. Using the approach of renewal theory, we prove the laws of large numbers and the central limit theorems for the random walk. These results reprove and significantly enhance the analysis of the mean value and variance of the process established in Majumdar et al. (Phys Rev E 92:052126, 2015). In the second part, we expand the analysis to the situation where the memory of the walker decreases over time by assuming that at the step n the resetting probability is $$r_n = \min \{rn^{-a}, \tfrac{1}{2}\}$$ with r, a positive parameters. For this model, we first establish the asymptotic behavior of the mean values of $$X_n$$ -the current position and $$M_n$$ -the maximum position of the random walk. As a consequence, we observe an interesting phase transition of the ratio $${{\mathbb {E}}}[X_n]/{{\mathbb {E}}}[M_n]$$ when a varies. Precisely, it converges to 1 in the subcritical phase $$a\in (0,1)$$ , to a constant $$c\in (0,1)$$ in the critical phase $$a=1$$ , and to 0 in the supercritical phase $$a>1$$ . Finally, when $$a>1$$ , we show that the model behaves closely to the simple random walk in the sense that $$\frac{X_n}{\sqrt{n}} \overset{(d)}{\longrightarrow } {\mathcal {N}}(0,1)$$ and $$\frac{M_n}{\sqrt{n}} \overset{(d)}{\longrightarrow } \max _{0 \le t \le 1} B_t$$ , where $${\mathcal {N}}(0,1)$$ is the standard normal distribution and $$(B_t)_{t\ge 0}$$ is the standard Brownian motion.

Lower bounds for in-network computation of arbitrary functions

In this paper, we provide a family of bounds for the rate at which the functions of many inputs can be computed in-network on general topologies. Going beyond simple symmetric functions where the output is invariant to the permutation of the operands, e.g., average, parity, we describe an algorithm that is analyzed to provide throughput bounds (both lower and upper) for the general functions. In particular, we analyze our algorithm when the function to be computed is given as a binary tree schema. Our lower bounds depend on schema parameters like the number of operands and graph parameters like the second largest eigenvalue of the transition matrix of simple random walk on network graph, the maximum and minimum degree of any node in the network. The lower bounding technique that we have used is based on the network flows and can capture general multi-commodity flow settings. Our proposed algorithm uses the well-known simple random walk on a network as its basic primitive for routing. We show that the lower bound obtained on the rate of computation is tight for the complete network topology, the hypercube and the star topology. We also present an upper bound on the expected latency of any data operand in terms of the height of schema, well-studied random walk parameter like the hitting time, and the relative distance from the critical data rate. For the computation time of symmetric functions on the random geometric graph under the gossip model, our approach achieves an order-optimal \(\widetilde{O}(n)\) time despite enforcing a binary tree schema for function computation. In general, Big-O notation represents an upper bound and \(\widetilde{O}\) hides \(\text {poly}\log n\) factors.

Mixing time of PageRank surfers on sparse random digraphs

AbstractWe consider the generalized PageRank walk on a digraph G, with refresh probability and resampling distribution . We analyze convergence to stationarity when G is a large sparse random digraph with given degree sequences, in the limit of vanishing . We identify three scenarios: when is much smaller than the inverse of the mixing time of G the relaxation to equilibrium is dominated by the simple random walk and displays a cutoff behavior; when is much larger than the inverse of the mixing time of G on the contrary one has pure exponential decay with rate ; when is comparable to the inverse of the mixing time of G there is a mixed behavior interpolating between cutoff and exponential decay. This trichotomy is shown to hold uniformly in the starting point and uniformly in the resampling distribution .

The number of zeros in Elephant random walks with delays

In the simple random walk the steps are independent, whereas in the elephant random walk (ERW), which was introduced by Schütz and Trimper (2004), the next step always depends on the whole path so far. In an earlier paper we investigated elephant random walks when the elephant has a restricted memory. Inspired by a suggestion by Bercu et al. (2019) we extended our results to the case when delays are allowed. In this paper we examine how the number of delays (that possibly stop the process) increases as time goes by.