Random walks in octants, and related structures

We analyze a Markovian random walk strategy on undirected regular networks involving power matrix functions of the type where L indicates a ‘simple’ Laplacian matrix. We refer to such walks as ‘fractional random walks’ with admissible interval . We deduce probability-generating functions (network Green’s functions) for the fractional random walk. From these analytical results we establish a generalization of Polya’s recurrence theorem for fractional random walks on d-dimensional infinite lattices: The fractional random walk is transient for dimensions (recurrent for ) of the lattice. As a consequence, for the fractional random walk is transient for all lattice dimensions and in the range for dimensions . Finally, for , Polya’s classical recurrence theorem is recovered, namely the walk is transient only for lattice dimensions . The generalization of Polya’s recurrence theorem remains valid for the class of random walks with Lévy flight asymptotics for long-range steps. We also analyze the mean first passage probabilities, mean residence times, mean first passage times and global mean first passage times (Kemeny constant) for the fractional random walk. For an infinite 1D lattice (infinite ring) we obtain for the transient regime closed form expressions for the fractional lattice Green’s function matrix containing the escape and ever passage probabilities. The ever passage probabilities (fractional lattice Green’s functions) in the transient regime fulfil Riesz potential power law decay asymptotic behavior for nodes far from the departure node. The non-locality of the fractional random walk is generated by the non-diagonality of the fractional Laplacian matrix with Lévy-type heavy tailed inverse power law decay for the probability of long-range moves. This non-local and asymptotic behavior of the fractional random walk introduces small-world properties with the emergence of Lévy flights on large (infinite) lattices.

Vicious walkers, flows and directed percolation

Chapter 5 is an overview of Dyck, Motzkin and partially directed path models of polymer entropy. Adsorbing paths, the Temperley method and factorisation methods are discussed, including the Bethe ansatz and models of random walks in the square lattice. The chapter also discusses solution of directed path models by the kernel method. In addition, staircase polygons and directed path models in layered environments and in wedges are reviewed.

Polysaccharides, such as cellulose, are often processed by dissolution in solvent mixtures, e.g. an ionic liquid (IL) combined with a dipolar aprotic co-solvent (CS) that the polymer does not dissolve in. A multi-walker, discrete-time, discrete-space 1-dimensional random walk can be applied to model solvation of a polymer in a multi-component solvent mixture. The number of IL pairs in a solvent mixture and the number of solvent shells formable, x, is associated with n, the model time-step, and N, the number of random walkers. The mean number of distinct sites visited is proportional to the amount of polymer soluble in a solution. By also fitting a polynomial regression model to the data, we can associate the random walk terms with chemical interactions between components and probe where the system deviates from a 1-D random walk. The 'frustration' between solvents shells is given as ln x in the random walk model and as a negative IL:IL interaction term in the regression model. This frustration appears in regime II of the random walk model (high volume fractions of IL) where walkers interfere with each other, and the system tends to its limiting behaviour. In the low concentration regime, (regime I) the solvent shells do not interact, and the system depends only on IL and CS terms. In both models (and both regimes), the system is almost entirely controlled by the volume available to solvation shells, and thus is a counting/space-filling problem, where the molar volume of the CS is important. Small deviations are observed when there is an IL-CS interaction. The use of two models, built on separate approaches, confirm these findings, demonstrating that this is a real effect and offering a route to identifying such systems. Specifically, the majority of CSs - such as dimethylformide - follow the random walk model, whilst 1-methylimidazole, dimethyl sulfoxide, 1,3-dimethyl-2-imidazolidinone and tetramethylurea offer a CS-mediated improvement and propylene carbonate results in a CS-mediated hindrance. It is shown here that systems, which are very complex at a molecular level, may, nonetheless, be effectively modelled as a simple random walk in phase-space. The 1-D random walk model allows prediction of the ability of solvent mixtures to dissolve cellulose based on only two dissolution measurements (one in neat IL) and molar volume.

The first part of this chapter introduces the random walk initially in a form with stochastic shocks generated by a random variable with a binomial distribution. In the simplest version of this process the random variable has two equally likely outcomes resulting in a symmetric binomial random walk. The idea is simple enough and the terminology is due to a problem posed by Pearson (see Hughes, 1995, p.53), although the concept dates from much earlier, originating in games of chance. Starting from the origin, at regular intervals a walker takes equally spaced steps either to the left (north) or to the right (south), with equal probability. The walker’s progress can be plotted in two dimensions by recording the distance from the origin on the vertical axis and the elapsed time on the horizontal axis; for example, one step to the north followed by one step to the south returns the walker to the origin. Such a graph will look like a series of equally sized steps, see Figure 1.1; perhaps surprisingly, the resulting path does not generally oscillate around zero, the theoretical mean of the process. This lack of ‘mean reversion’ is one of the key characteristics of a random walk. In this form the process has its origins in gambling, where the gambler gains or loses an equal amount at each gamble, with equal probability, and intuition might suggest that the gambler is not systematically losing or winning. The ‘distance from the origin’ corresponds to the gambler’s win/lose tally, which is one-dimensional and can be represented on the vertical axis.

The probability distribution of the number s of distinct sites visited up to time t by a random walk on the fully-connected lattice with N sites is first obtained by solving the eigenvalue problem associated with the discrete master equation. Then, using generating function techniques, we compute the joint probability distribution of s and r, where r is the number of sites visited only once up to time t. Mean values, variances and covariance are deduced from the generating functions and their finite-size-scaling behaviour is studied. Introducing properly centered and scaled variables u and v for r and s and working in the scaling limit (, with w = t/N fixed) the joint probability density of u and v is shown to be a bivariate Gaussian density. It follows that the fluctuations of r and s around their mean values in a finite-size system are Gaussian in the scaling limit. The same type of finite-size scaling is expected to hold on periodic lattices above the critical dimension .

We analyze growth and diversification of U.S. dairy farms by examining longitudinal changes in ten size cohorts and new entrants through three successive censuses. Gibrat's law (random walk) and mean reversion hypotheses of growth are tested and rejected. Growth rates are bimodal with the largest farm cohort growing the fastest. All cohorts become more diversified over time, and smaller farms diversify most rapidly. New entrants are generally large, and they diversify more rapidly than incumbents. These data suggest that scale economies persist even for the largest cohort of dairy farms and that scale economies dominate scope economies for large farms.

We describe a simple algorithm for approximating the empirical entropy of a stream of m values in a single pass, using O(e-2 log(δ-1) log m) words of space. Our algorithm is based upon a novel extension of a method introduced by Alon, Matias, and Szegedy [1]. We show a space lower bound of Ω(e-2 / log(e-1)), meaning that our algorithm is near-optimal in terms of its dependency on e. This improves over previous work on this problem [8, 13, 17, 5]. We show that generalizing to kth order entropy requires close to linear space for all k ≥ 1, and give additive approximations using our algorithm. Lastly, we show how to compute a multiplicative approximation to the entropy of a random walk on an undirected graph.

We investigate the statistics of three kinds of records associated with planar random walks, namely diagonal, simultaneous and radial records. The mean numbers of these records grow as universal power laws of time, with respective exponents 1/4, 1/3 and 1/2. The study of diagonal and simultaneous records relies on the underlying renewal structure of the successive hitting times and locations of translated copies of a fixed target. In this sense, this work represents a two-dimensional extension of the analysis made by Feller of ladder points, i.e. records for one-dimensional random walks. This approach yields a variety of analytical asymptotic results, including the full statistics of the numbers of diagonal and simultaneous records, the joint law of the epoch and location of the current diagonal record and the angular distribution of the current simultaneous record. The sequence of radial records cannot be constructed in terms of a renewal process. In spite of this, their mean number is shown to grow with a super-universal square-root law for isotropic random walks in any spatial dimension. Their full distribution is also obtained. Higher-dimensional diagonal and simultaneous records are also briefly discussed.

Random walks on square-lattice percolation clusters are simulated for interaction ranges spanning one to five nearest-neighbor bonds (R=1 , . . . , 5). The relative hopping probability is given by exp(-\ensuremath{\alpha}r), where r is the number of bonds traversed in one hop and \ensuremath{\alpha} is a parameter (0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}10). The fractal exponent for the random walks is universal. For R=2 (and R=1) we obtain a spectral dimension of ${d}_{s}$=1.31\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the Alexander-Orbach conjecture (1.333), and in even better agreement with the Aharony-Stauffer conjecture (1.309). Our results are based on the relation ${d}_{s}$=(91/43)f, where ${S}_{n}$\ensuremath{\sim}${N}^{f}$ describes the mean number (${S}_{N}$) of distinct sites visited in N steps for walks originating on all clusters. While the asymptotic limit of f is closely approached after 5000 nominal time steps for \ensuremath{\alpha}=0, much longer times (>50 000 steps) are required for \ensuremath{\alpha}\ensuremath{\gg}0. We also observe fractal-to-Euclidean crossovers above criticality; again, this crossover takes much longer for \ensuremath{\alpha}\ensuremath{\gg}0.

Monte Carlo simulation of the electronic excitation transport on the freely-jointed polymer chain

A random walk model with a negative binomially fluctuating number of steps is considered in the case where the mean of the number fluctuations, , is finite. The asymptotic behaviour of the resultant statistics in the large limit is derived and shown to give the K distribution. The equivalence of this model to the hitherto unrelated doubly stochastic representation of the K distribution is also demonstrated. The convergence to the K distribution of the probability density function generated by a random walk with a finite mean number of steps is examined along with the moments, and the non-Gaussian statistics are shown to be a direct result of discreteness and bunching effects.

Consider a random walk on a finite group G. Suppose that the time taken for the transition probabilities to approach the uniform distribution is small compared with #G. Then the time taken for the random walk to visit every state is approximately R#G·log(#G), where R is the mean number of returns to the initial state in the short term.

We study the number of distinct sites visited by N random walkers after t steps SN(t) under the condition that all the walkers are initially at the origin. We derive asymptotic expressions for the mean number of distinct sites 〈SN(t)〉 in one, two, and three dimensions. We find that 〈SN(t)〉 passes through several growth regimes; at short times 〈SN(t)〉\ensuremath{\sim}td (regime I), for tx\ensuremath{\ll}t\ensuremath{\ll}tx\ensuremath{'} we find that 〈SN(t)〉\ensuremath{\sim}(t ln[N S1(t)/td/2])d/2 (regime II), and for t\ensuremath{\gg}tx\ensuremath{'}, 〈SN(t)〉\ensuremath{\sim}NS1(t) (regime III). The crossover times are tx\ensuremath{\sim}ln N for all dimensions, and tx\ensuremath{'}\ensuremath{\sim}\ensuremath{\infty}, exp N, and N2 for one, two, and three dimensions, respectively. We show that in regimes II and III 〈SN(t)〉 satisfies a scaling relation of the form 〈SN(t)〉\ensuremath{\sim}td/2f(x), with x\ensuremath{\equiv}N〈S1(t)〉/td/2. We also obtain asymptotic results for the complete probability distribution of SN(t) for the one-dimensional case in the limit of large N and t.

The authors study the number of distinct sites visited by a random walker in d=1 after t steps, S(t), in the presence of a trap. They calculate the distribution q(S, t) of S(t) in the limit of large t. They find an unusual crossover in the probability density at S approximately=Sx identical to Dt. For S<<Sx, q(S, t) approximately S-2 and for S>>Sx, q(S, t) approximately St-3/2 exp(-S2/4Dt). Fro this crossover it follows that the mean number of distinct sites visited is (S(t)) approximately In(t).