Dimensional Random Walk Research Articles

The Recurrence and Transience of Random Walks

This article reviews the research history and application fields of random walks and abstracts random walks into a time-homogeneous Markov chain to study their recurrent and transient properties. For one-dimensional and two-dimensional random walks, the likelihood of returning in steps and the probability of returning for the first time in steps of each state are first introduced, along with their relationship. Then the Stirling's formula is given, which is utilized to estimate the probability of returning in n steps, and the convergence and divergence of infinite series is used to prove that random walk in one dimension is recurrent. Random walk in two dimension has similar properties, which is a bit more complicated by using the relation between polynomials. Higher dimensional random walks need to consider special states first, and then generalize them to other states. Finally, this paper concluded that one-dimensional and two-dimensional random walks are recurrent, and random walks with dimensions higher than two are transient.

Visible lattice points along curves in higher dimensional random walks

Let k≥2 be an integer, and α=(α1,⋯,αk) with 0<α1,⋯,αk<1 and α1+⋯+αk=1, this paper concerns the generalized visibility of lattice points visited by an α-random walk in Zk. Specifically, for b=(b1,⋯,bk)∈Nk with gcd⁡(b1,⋯,bk)=1, we focus on the lattice points in the walk which are b-visible from a set of N watch-points, and we show that, almost surely, the proportion of b-visible steps is ∏p(1−N/ps(b)), where p runs over all primes and s(b):=b1+⋯+bk. This paper generalizes our previous work for the visible lattice points in higher dimensional random walks.

Moments of the 2D Directed Polymer in the Subcritical Regime and a Generalisation of the Erdös–Taylor Theorem

We compute the limit of the moments of the partition function Z_{N}^{beta _N} of the directed polymer in dimension d=2 in the subcritical regime, i.e. when the inverse temperature is scaled as beta _N sim hat{beta } sqrt{tfrac{pi }{log N}} for hat{beta } in (0,1). In particular, we establish that for every h in {mathbb {R}}, lim _{N rightarrow infty } {{mathbb {E}}} big [big (Z_{N}^{beta _N}big )^h big ]=big (frac{1}{1-hat{beta }^2}big )^{frac{h(h-1)}{2}}. We also identify the limit of the moments of the averaged field tfrac{sqrt{log N}}{N} sum _{x in {mathbb {Z}}^2} varphi (tfrac{x}{sqrt{N}})big (Z_{N}^{beta _N} (x)-1 big ), for varphi in C_c({mathbb {R}}^2), as those of a gaussian free field. As a byproduct, we identify the limiting probability distribution of the total pairwise collisions between h independent, two dimensional random walks starting at the origin. In particular, we derive that πlogN∑1≤i<j≤hLN(i,j)→N→∞(d)Γ(h(h-1)2,1),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\pi }{\\log N}\\sum _{1 \\le i<j\\le h} {\ extsf{L}}_N^{(i,j)}\\xrightarrow [N \\rightarrow \\infty ]{(d)} \\Gamma \\big ( \ frac{h(h-1)}{2},1\\big ) , \\end{aligned}$$\\end{document}where {{textsf{L}}}^{(i,j)}_N denotes the collision local time by time N between copies i, j and Gamma denotes the Gamma distribution. This generalises a classical result of Erdös and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960).

High dimensional expanders and coset geometries

High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are abundant.In this work, we construct new families of bounded degree high dimensional expanders obeying the local spectral expansion property. This property has a number of important consequences, including geometric overlapping, fast mixing of high dimensional random walks, agreement testing and agreement expansion. Our construction also yields new families of expander graphs which are close to the Ramanujan bound, i.e., their spectral gap is close to optimal.The construction is quite elementary and it is presented in a self contained manner; This is in contrary to the highly involved previously known construction of the Ramanujan complexes. The construction is also very symmetric (such symmetry properties are not known for Ramanujan complexes) ; The symmetry of the construction could be used, for example, in order to obtain good symmetric LDPC codes that were previously based on Ramanujan graphs.The main tool that we use for is the theory of coset geometries. Coset geometries arose as a tool for studying finite simple groups. Here, we show that coset geometries arise in a very natural manner for groups of elementary matrices over any finitely generated algebra over a commutative unital ring. In other words, we show that such groups act simply transitively on the top dimensional face of a pure, partite, clique complex.

Some recent advances in random walks and random environments

Recent contributions to random walks in random environments and related topics are presented. We focus on non parametric estimation for one dimensional random walks in random environment and on the Dirichlet distribution on decomposable graphs.

Correction to: Exact value of the resistance exponent for four dimensional random walk trace

We correct a proof in the article ‘D. Shiraishi, Exact value of the resistance exponent for four dimensional random walk trace, Probab. Theory and Related Fields 153 (2012), no. 1–2, 191–232’.

Localization of Defects Via Residence Time Measures

We show that residence time measure can be used to identify the geometrical and transmission properties of a defect along a path. The model we study is based on a one--dimensional simple random walk. The sites of the lattice are regular, i.e., the jumping probabilities are the same in each site, except for a site, called \emph{defect}, where the jumping probabilities are different. At each side of the lattice an absorbing site is present. We show that by measuring the fraction of particles crossing the channel and/or the typical time they need to cross it, it is possible to identify the main features of the lattice and of the defect site, namely, the jumping probabilities at regular and at the defect sites and the position of the defect in the lattice.

Study on the existence of tracking model for a d-dimensional random walk transportation radionuclide particle in a fractured medium

This paper presents a novel tracking model to detect a [Formula: see text]-dimensional random walk radionuclides and toxic chemicals in a fractured medium with minimum time and maximum probability. This model improves the physical and chemical properties of a fractured medium for a long time. It depends on a coordinated search technique with multiple sensors. The radionuclide particle has a linear flows on one direction from [Formula: see text]-directions to spread. We consider this particle dispersed through a cylinder path in a very small time. The tracking or searching process in each cylinder path will be done by two sensors which start from any point in the fractured medium (space). The particle’s initial position is a vector of random variables with a known probability distribution. More than presenting the analytical solution which shows the existence of a finite tracking technique, we derive some auxiliary results related to optimal tracking technique. Besides that, we derive this optimal tracking technique which minimizes the expected value of the first meeting time between one of the sensors and the radionuclide or the toxic chemical particle. The effectiveness of this model appears in some numerical results.

Time-Lagged Independent Component Analysis of Random Walks and Protein Dynamics.

Time-lagged independent component analysis (tICA) is a widely used dimension reduction method for the analysis of molecular dynamics (MD) trajectories and has proven particularly useful for the construction of protein dynamics Markov models. It identifies those “slow” collective degrees of freedom onto which the projections of a given trajectory show maximal autocorrelation for a given lag time. Here we ask how much information on the actual protein dynamics and, in particular, the free energy landscape that governs these dynamics the tICA-projections of MD-trajectories contain, as opposed to noise due to the inherently stochastic nature of each trajectory. To answer this question, we have analyzed the tICA-projections of high dimensional random walks using a combination of analytical and numerical methods. We find that the projections resemble cosine functions and strongly depend on the lag time, exhibiting strikingly complex behavior. In particular, and contrary to previous studies of principal component projections, the projections change noncontinuously with increasing lag time. The tICA-projections of selected 1 μs protein trajectories and those of random walks are strikingly similar, particularly for larger proteins, suggesting that these trajectories contain only little information on the energy landscape that governs the actual protein dynamics. Further the tICA-projections of random walks show clusters very similar to those observed for the protein trajectories, suggesting that clusters in the tICA-projections of protein trajectories do not necessarily reflect local minima in the free energy landscape. We also conclude that, in addition to the previous finding that certain ensemble properties of nonconverged protein trajectories resemble those of random walks; this is also true for their time correlations.

A note on the intersections of two random walks in two dimensions

In this note we prove a large deviation result for the intersection of ranges of two independent random walks in dimension two. This complements the study of Phetpradap from 2011, where the intersection in dimension three and above was studied.

The two regimes of moderate deviations for the range of a transient walk

We obtain sharp upper and lower bounds for the downward moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, which strengthen the approach developed in Asselah and Schapira (Sci Éc Norm Supér 50(4):755–786, 2017).

A lattice model on the rate of in vivo site-specific DNA-protein interactions

We develop a lattice model of site-specific DNA-protein interactions under in vivo conditions where DNA is modelled as a self-avoiding random walk that is embedded in a cubic lattice box resembling the living cell. The protein molecule searches for its cognate site on DNA via a combination of three dimensional (3D) and one dimensional (1D) random walks. Hopping and intersegmental transfers occur depending on the conformational state of DNA. Results show that the search acceleration ratio (= search time in pure 3D route/search time in 3D and 1D routes) asymptotically increases towards a limiting value as the dilution factor of DNA (= volume of the cell/the volume of DNA) tends towards infinity. When the dilution ratio is low, then hopping and intersegmental transfers significantly enhance the search efficiency over pure sliding. At high dilution ratio, hopping does not enhance the search efficiency much since under such situation DNA will be in a relaxed conformation that favors only sliding. In the absence of hopping and intersegmental transfers, there exists an optimum sliding time at which the search acceleration ratio attains a maximum in line with the current theoretical results. However, existence of such optimum sliding length disappears in the presence of hopping. When the DNA is confined in a small volume inside the cell resembling a natural cell system, then there exists an optimum dilution and compression ratios (= total cell volume/volume in which DNA is confined) at which the search acceleration factor attains a maximum especially in the presence of hopping and intersegmental transfers. These optimum values are consistent with the values observed in the Escherichia coli cell system. In the absence of confinement of DNA, position of the specific binding site on the genomic DNA significantly influences the search acceleration. However, such position dependent changes in the search acceleration ratio will be nullified in the presence of hopping and intersegmental transfers especially when the DNA is confined in a small volume that is embedded in an outer cell.

Loop-Erased Random Walk as a Spin System Observable

The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one-point function of loop-erased random walk.

Determining effective conductivities of fractal objects

We develop formulas to approximate the effective thermal conductivity of composite materials consisting of a matrix with nominal dimensionless conductivity Kmat=1 containing inclusions (holes) with a conductivity Khole≤1 laid out in fractal patterns: 2D Sierpinski carpets and 3D Menger sponges. Direct simulation of steady state heat conduction in finite stages of construction of these objects provide benchmark conductivity values. Recent theory on random walks in fractal media provides conductivity estimates for Sierpinski carpets for the case where Khole=0. Recognizing that the estimate in the fractal generators is related to the classic Rayleigh model, we obtain an expression for the conductivity in the more general case 0≤Khole≤1. Comparison with direct simulated values shows a relative error below 6% when the conductivity contrast is high (Khole≈0) and below 1% when the contrast is low. This significantly improves the accuracy of estimates obtained from simple resistor networks, which are interpreted as upper and lower bounds. The extension of this approach to Menger sponges gives a conductivity estimator related to the classic Maxwell conductivity model. This estimator, accounting for the effects of pattern correlations, has relative errors in the same range as the carpet estimator. A conjecture for the random walk dimension in sponges is also derived. The effective conductivities for 2D and 3D patterns, with low volume fractions of the matrix and high conductivity contrast, also represent a significant improvement over the classic formulas for uncorrelated distributions of inclusions.

Mortal random walks on a family of treelike regular fractals with a deep trap

Due to the ubiquitous occurrence of evanescence in many physical, chemical and biological scenarios, mortal random walks that incorporate evanescence explicitly have drawn more and more attention. It has been a hot topic to study mortal random walks on distinct network models. In this paper, we study mortal random walks on T fractal and a family of treelike regular fractals with a trap located at central node (i.e., innermost node). First, with self-similar setting composed of T fractal, initial position of the walker and location of trap, the total trapping probability of the mortal walker reduces to a function of walker’s single-step survival parameter [Formula: see text]. In more detail, the total trapping probability is expressed by the [Formula: see text]th iteration of map (scaling function) of [Formula: see text]. Based on the map, the analytical expression of total trapping probability’s dominant behavior, the mean time to trapping (MFPT) and temporal factor are obtained, which are related to random walk dimension. Last, we extend the analysis to a family of treelike regular fractals. On them, the total trapping probability is still expressed as the [Formula: see text]th iteration of the map scaling [Formula: see text]. Accordingly, dominant behavior of total trapping probability, MFPT and temporal factor are determined analytically. Both analytical results obtained on T fractal and more general treelike regular fractals show that the mean time to trapping and desired random walk dimension can be obtained by tuning the survival probability parameter [Formula: see text]. In summary, the work advances the understanding of mortal random walks on more general deterministic networks.

Fractal dimension of critical curves in the O(n) -symmetric ϕ4 model and crossover exponent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, XY , and Heisenberg models

We calculate the fractal dimension d_{f} of critical curves in the O(n)-symmetric (ϕ[over ⃗]^{2})^{2} theory in d=4-ε dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at n=-2, self-avoiding walks (n=0), Ising lines (n=1), and XY lines (n=2), in agreement with numerical simulations. It can be compared to the fractal dimension d_{f}^{tot} of all lines, i.e., backbone plus the surrounding loops, identical to d_{f}^{tot}=1/ν. The combination ϕ_{c}=d_{f}/d_{f}^{tot}=νd_{f} is the crossover exponent, describing a system with mass anisotropy. Introducing a self-consistent resummation procedure and combining it with analytic results in d=2 allows us to give improved estimates in d=3 for all relevant exponents at 6-loop order.

On covering monotonic paths with simple random walk

In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_{1}$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when $d\geq 4$. The main tool is a general combinatorial inequality, that is interesting in its own right.

First-passage properties of mortal random walks: Ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs.

We consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites. The configuration comprising the graph, the starting point of the walk, and the locations of the trap sites is taken to be exactly self-similar as one goes from one generation of the family to the next. Under these circumstances, the total probability that the walker hits a trap is determined exactly as a function of the single-step survival probability q of the mortal walker. On the nth generation graph of the family, this probability is shown to be given by the nth iterate of a certain scaling function or map q→f(q). The properties of the map then determine, in each case, the behavior of the trapping probability, the mean time to trapping, the temporal scaling factor governing the random walk dimension on the graph, and other related properties. The formalism is illustrated for the cases of a linear hierarchical lattice and the Sierpinski graphs in two and three Euclidean dimensions. We find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint. The relevance of this finding for experiments involving travel times of particles in diffusion-decay systems is discussed.

Models of infiltration into homogeneous and fractal porous media with localized sources.

We study a random-walk infiltration (RWI) model, in homogeneous and in fractal media, with small localized sources at their boundaries. In this model, particles released at a source, maintained at a constant density value, execute unbiased random walks over a lattice; a model that represents solute infiltration by diffusion into a medium in contact with a reservoir of fixed concentration. A scaling approach shows that the infiltrated length, area, or volume evolves in time as the number of distinct sites visited by a single random walker in the same medium. This is consistent with numerical simulations of the lattice model and exact and numerical solutions of the corresponding diffusion equation. In a Sierpinski carpet, the infiltrated area is expected to evolve as t^{D_{F}/D_{W}} (Alexander-Orbach relation), where D_{F} is the fractal dimension of the medium and D_{W} is the random-walk dimension; the numerical integration of the diffusion equation supports this result with very good accuracy and improves results of lattice random-walk simulations. In a Menger sponge in which D_{F}>D_{W} (i.e., a fractal with a dimension close to 3), a linear time increase of the infiltrated volume is theoretically predicted and confirmed numerically. Thus, no evidence of fractality can be observed in measurements of infiltrated volumes or masses in media where random walks are not recurrent, although the tracer diffusion is anomalous. We compare our findings with results for a fluid infiltration model in which the pressure head is constant at the source and the front displacement is driven by the local gradient of that head. Exact solutions in two and three dimensions and numerical results in a carpet show that this type of fluid infiltration is in the same universality class of RWI, with an equivalence between the head and the particle concentration. These results set a relation between different infiltration processes with localized sources and the recurrence properties of random walks in the same media.

Localization in Fractonic Random Circuits

We study the spreading of initially-local operators under unitary time evolution in a 1d random quantum circuit model which is constrained to conserve a $U(1)$ charge and its dipole moment, motivated by the quantum dynamics of fracton phases. We discover that charge remains localized at its initial position, providing a crisp example of a non-ergodic dynamical phase of random circuit dynamics. This localization can be understood as a consequence of the return properties of low dimensional random walks, through a mechanism reminiscent of weak localization, but insensitive to dephasing. The charge dynamics is well-described by a system of coupled hydrodynamic equations, which makes several nontrivial predictions in good agreement with numerics. Importantly, these equations also predict localization in 2d fractonic circuits. Immobile fractonic charge emits non-conserved operators, whose spreading is governed by exponents distinct to non-fractonic circuits. Fractonic operators exhibit a short time linear growth of observable entanglement with saturation to an area law, as well as a subthermal volume law for operator entanglement. The entanglement spectrum follows semi-Poisson statistics, similar to eigenstates of MBL systems. The non-ergodic phenomenology persists to initial conditions containing non-zero density of dipolar or fractonic charge. Our work implies that low-dimensional fracton systems preserve forever a memory of their initial conditions in local observables under noisy quantum dynamics, thereby constituting ideal memories. It also implies that 1d and 2d fracton systems should realize true MBL under Hamiltonian dynamics, even in the absence of disorder, with the obstructions to MBL in translation invariant systems and in d>1 being evaded by the nature of the mechanism responsible for localization. We also suggest a possible route to new non-ergodic phases in high dimensions.