Two-dimensional Random Walk Research Articles

Establishing the kinetics of ballistic-to-diffusive transition using directional statistics.

We establish the kinetics of ballistic-to-diffusive (BD) transition observed in two-dimensional random walk using directional statistics. Directional correlation is parameterized using the walker's turning angle distribution, which follows the commonly adopted wrapped Cauchy distribution (WCD) function. During the BD transition, the concentration factor (ρ) governing the WCD shape is observed to decrease from its initial value. We next analytically derive the relationship between effective ρ and time, which essentially quantifies the BD transition rate. The prediction of our kinetic expression agrees well with the empirical datasets obtained from correlated random walk simulation. We further connect our formulation with the conventionally used scaling relationship between the walker's mean-square displacement and time.

Recurrence of multidimensional persistent random walks. Fourier and series criteria

The recurrence and transience of persistent random walks built from variable length Markov chains are investigated. It turns out that these stochastic processes can be seen as Lévy walks for which the persistence times depend on some internal Markov chain: they admit Markov random walk skeletons. A recurrence versus transience dichotomy is highlighted. Assuming the positive recurrence of the driving chain, a sufficient Fourier criterion for the recurrence, close to the usual Chung–Fuchs one, is given and a series criterion is derived. The key tool is the Nagaev–Guivarc’h method. Finally, we focus on particular two-dimensional persistent random walks, including directionally reinforced random walks, for which necessary and sufficient Fourier and series criteria are obtained. Inspired by (Adv. Math. 208 (2007) 680–698), we produce a genuine counterexample to the conjecture of (Adv. Math. 117 (1996) 239–252). As for the one-dimensional case studied in (J. Theoret. Probab. 31 (2018) 232–243), it is easier for a persistent random walk than its skeleton to be recurrent. However, such examples are much more difficult to exhibit in the higher dimensional context. These results are based on a surprisingly novel – to our knowledge – upper bound for the Lévy concentration function associated with symmetric distributions.

Mapping of the Bak, Tang, and Wiesenfeld sandpile model on a two-dimensional Ising-correlated percolation lattice to the two-dimensional self-avoiding random walk.

The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T_{c} the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D_{f}^{SAW}=4/3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R_{rms}∼t^{3/4} (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T=T_{c} the model has some aspects compatible with the 2D BTW model (e.g., the 1/log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T=T_{c}. In the off-critical temperatures in the close vicinity of T_{c} the exponents show some additional power-law behaviors in terms of T-T_{c} with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L^{1/2}, which is different from the regular 2D BTW model.

Relative complexity of random walks in random scenery in the absence of a weak invariance principle for the local times

We answer a question of Aaronson about the relative complexity of Random Walks in Random Sceneries driven by either aperiodic two-dimensional random walks, two-dimensional Simple Random walk, or by aperiodic random walks in the domain of attraction of the Cauchy distribution. A key step is proving that the range of the random walk satisfies the Fölner property almost surely.

First passage percolation on the exponential of two-dimensional branching random walk

We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with Gaussian increments indexed over a two-dimensional box $V_N$ of side length $N$, and we study the first passage percolation where each vertex is assigned weight $e^{\gamma \mathcal R^N_z}$ for $\gamma>0$. We show that for $\gamma>0$ sufficiently small but fixed, the expected FPP distance between the left and right boundaries is at most $O(N^{1 - \gamma^2/10})$.

Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map II: local estimates and empty reduced word exponent

We continue our study of the inventory accumulation introduced by Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various local estimates for the inventory accumulation model, i.e., estimates for the precise number of symbols of a given type in a reduced word sampled from the model. Using our estimates, we obtain the scaling limit of the associated two-dimensional random walk conditioned on the event that it stays in the first quadrant for one unit of time and ends up at a particular position in the interior of the first quadrant. We also obtain the exponent for the probability that a word of length $2n$ sampled from the inventory accumulation model corresponds to an empty reduced word, which is equivalent to an asymptotic formula for the partition function of the critical FK planar map model. The estimates of this paper will be used in a subsequent paper to obtain the scaling limit of the lattice walk associated with a finite-volume FK planar map.

Asymptotic Behaviour of a Random Walk Killed on a Finite Set

We study asymptotic behavior, for large time $n$, of the transition probability of a two-dimensional random walk killed when entering into a non-empty finite subset $A$. We show that it behaves like $4 \tilde u_A(x) \tilde u_{-A}(-y) (\lg n)^{-2} p^n(y- x)$ for large $n$, uniformly in the parabolic regime $|x|\vee |y| =O(\sqrt n)$, where $p^n(y-x)$ is the transition kernel of the random walk (without killing) and $\tilde u_A$ is the unique harmonic function in the 'exterior of $A$' satisfying the boundary condition $\tilde u_A(x) \sim \lg |x|$ at infinity.

Multispecies random walk simulations in radial symmetry: model concept, benchmark, and application to HTO, 22 Na and 36 Cl diffusion in clay

Abstract Modelling of radionuclide transport in clay can be simplified and accelerated by exploiting radial symmetry of laboratory experiments or nuclear waste repositories design. Consequently, the multispecies reactive transport code MCOTAC has been extended to radial symmetry, exploiting the advantages of its random walk transport description. Random walk in radial r -symmetry is mimicked by two-dimensional (2D) random walk projected to a radial coordinate for geochemical calculations. This guarantees fast 2D( x , y ) transport calculations, together with complex geochemical calculations in 1D( r ) only. The new model concept has been benchmarked for simple geochemical systems and applied to laboratory diffusion experiments with HTO, 22 Na and 36 Cl in Opalinus Clay using higher spatial resolution for modelling than reported in the literature. This makes it possible to distinguish between the transport properties of the sample and the filters, and yields more careful determination of radionuclide transport parameters: for example, there was a difference of a factor of 2 between the diffusion coefficient and log K d sorption coefficient in the case of 22 Na compared to the former best-fit analysis.

Statistics of two-dimensional random walks, the cyclic sieving phenomenon and the Hofstadter model

We focus on the algebraic area probability distribution of planar random walks on a square lattice with and l2 steps right, left, up and down. We aim, in particular, at the algebraic area generating function evaluated at a root of unity, when both and multiples of q. In the simple case of staircase walks, a geometrical interpretation of in terms of the cyclic sieving phenomenon is illustrated. Then, an expression for which is relevant to the Stembridge case, is proposed. Finally, the related problem of evaluating the nth moments of the Hofstadter Hamiltonian in the commensurate case is addressed.

The Hitting Distribution of a Line Segment for Two-Dimensional Random Walks

Asymptotic estimates of the hitting distribution of a long segment on the real axis for two dimensional random walks on ${\bf Z}^2$ of zero mean and finite variances are obtained: some are general and exhibit its apparent similarity to the corresponding Brownian density, while others are so detailed as to involve certain characteristics of the random walk.

Modulation effect with global ambiguity in 2-dimensional random walk

Here, we developed a two-dimensional multi-agent random walk algorithm. In our algorithm, agents interact with each other and change their directional rules by locally detecting moving directions of other agents. In addition to that, we introduced modulation effects in which agents controlled rule-intervals depending on number of local other agents. We show that modulation effects which introduce global ambiguity play a crucial role in establishing optimal random walk by checking the slope value (μ), depending on the density of agents. We set a modulation-added model and non-modulation model. The latter is control model. In case of non-modulation model, the slope values (μ) highly depend on the density of agents. However, in case of modulation-added model, the slope values (μ) are flexible and independent of the density of agents.

HITTING TIME OF A HALF-LINE BY A TWO-DIMENSIONAL NON-SYMMETRIC RANDOM WALK

We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line (−∞, 0] × {0} before time n. Let X = (X1, X2) be the increment of the two-dimensional random walk. For an aperiodic random walk with moment conditions E [X2] = 0 and E [|X1|δ] < ∞, E [|X2|2+δ] < ∞ for some δ ∈ (0, 1), we obtain an asymptotic estimate (as n →∞) of this probability by assuming the behavior of the characteristic function of X near zero.

Maximum and minimum of local times for two-dimensional random walk

We obtain the leading orders of the maximum and the minimum of local times for the simple random walk on the two-dimensional torus at time proportional to the cover time. We also estimate the number of points with large (or small) values of the local times. These are analogues of estimates on the two-dimensional Gaussian free fields by Bolthausen, Deuschel, and Giacomin [Ann. Probab., 29 (2001)] and Daviaud [Ann. Probab., 34 (2006)], but we have different exponents from the case of the Gaussian free field.

Coherent random walks in free space

Two-dimensional continuous time quantum random walks (CTQRW) are physical processes where quantum particles simultaneously evolve in different permissible directions within discrete graphs. In order to force the quantum walkers (QWs) to evolve in such a fashion, one generally requires periodic potentials. Here, we demonstrate that two-dimensional CTQRW can be generated in free space by properly tailoring the initial wave functions. We analytically show that within a certain spatial region the arising probability distribution quantitatively resembles the probability pattern exhibited by a QW traversing a periodic lattice potential. These theoretical predictions were experimentally verified using classical laser light, appropriately shaped by a spatial light modulator. Expanding the presented results to the case of multiple walkers may open new possibilities in quantum information technology using bulk optics.

Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime

We present an analysis of the relaxation dynamics of finite-size topological qubits in contact with a thermal bath. Using a continuous-time Monte Carlo method, we explicitly compute the low-temperature nonequilibrium dynamics of the toric code on finite lattices. In contrast to the size-independent bound predicted for the toric code in the thermodynamic limit, we identify a low-temperature regime on finite lattices below a size-dependent crossover temperature with nontrivial finite-size and temperature scaling of the relaxation time. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial two-dimensional classical random walks. The transition out of this low-temperature regime defines a dynamical finite-size crossover temperature that scales inversely with the log of the system size, in agreement with a crossover temperature defined from equilibrium properties. We find that both the finite-size and finite-temperature scaling are stronger in the low-temperature regime than above the crossover temperature. Since this finite-temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of memory lifetimes of possible physical realizations of topological qubits.

Atomic mechanisms and diffusion anisotropy of Cu tetramers on Cu(111)

The surface diffusion of compact Cu tetramers on Cu(111) has been studied at the atomic scale by means of molecular dynamics simulations using embedded atom interatomic potentials. The Cu clusters diffuse by several different mechanisms; all of them have the common trait of involving concerted displacements of at least some of the atoms forming the island. The anisotropy in the rhombus shape of the Cu tetramers and the different activation energies of the translational and rotational jumps result in highly anisotropic diffusion, with a transition from practically one-dimensional motion at low temperature to nearly isotropical two-dimensional random walk at high temperature. The use of molecular dynamics allows us to also determine the preexponential factors for each jump mechanism, from which analysis we can obtain some insight into their basic nature and the relationships between them.

Optimal Multicast Capacity and DelayTradeoffs in MANETs

In this paper, we give a global perspective of multicast capacity and delay analysis in Mobile Ad Hoc Networks (MANETs). Specifically, we consider four node mobility models: (1) two-dimensional i.i.d. mobility, (2) two-dimensional hybrid random walk, (3) one-dimensional i.i.d. mobility, and (4) one-dimensional hybrid random walk. Two mobility time-scales are investigated in this paper: (i) fast mobility where node mobility is at the same time-scale as data transmissions and (ii) slow mobility where node mobility is assumed to occur at a much slower time-scale than data transmissions. Given a delay constraint <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$D$</tex></formula> , we first characterize the optimal multicast capacity for each of the eight types of mobility models, and then we develop a scheme that can achieve a capacity-delay tradeoff close to the upper bound up to a logarithmic factor. In addition, we also study heterogeneous networks with infrastructure support.

The area distribution of two-dimensional random walks and non-Hermitian Hofstadter quantum mechanics

When random walks on a square lattice are biased horizontally to move solely to the right, the probability distribution of their algebraic area can be obtained exactly (Mashkevich and Ouvry 2009 J. Stat. Phys. 137 71). We explicitly map this biased classical random system onto a non-Hermitian Hofstadter-like quantum model where a charged particle on a square lattice coupled to a perpendicular magnetic field hops only to the right. For the commensurate case, when the magnetic flux per unit cell is rational, an exact solution of the quantum model is obtained. The periodicity of the lattice allows one to relate traces of the Nth power of the Hamiltonian to probability distribution generating functions of biased walks of length N.

On the upper bound in the large deviation principle for sums of random vectors

We consider the random walk generated by a sequence of independent identically distributed random vectors. The known upper bound for normalized sums in the large deviation principle was established under the assumption that the Laplace-Stieltjes transform of the distribution of the walk jumps exists in a neighborhood of zero. In the present article, we prove that, for a twodimensional random walk, this bound holds without any additional assumptions.

Martingale defocusing and transience of a self-interacting random walk

Suppose that (X;Y;Z) is a random walk in Z 3 that moves in the following way: on the rst visit to a vertex only Z changes by 1 equally likely, while on later visits to the same vertex (X;Y ) performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.