Self avoiding walk trees and laces

A new graph theoretical method is introduced to analyze a data set of objects described by a dissimilarity matrix d ij . This method is based on the generation of a series of random walks in the data set. We define a random walk in a data set by moving in each time step from one object to another one at random. In order that the random walk depends on the pattern of the data, a restriction depending on the previous moves is imposed during its generation, so that the random walk is attracted by clusters formed of similar objects. We define a hierarchical set of graphs consisting of all connections of a series of random walks at a certain time step to detect the structure of the data and to derive an similarity measure between the objects. In an example an application of the method is shown.

Static step-step correlations for restricted and self-avoiding random walks (SARW) on quadratic and simple cubic lattices are studied with the help of Monte Carlo simulation technique. For the SARW ofN steps our results, for largeN tend to the theoretically predicted values. An analysis of the correlations for SARW in terms of the individual restrictions indicate that its value between steps at a distancer have a dominant contribution coming from the restriction prohibiting polygon closures of sides2r (r≧2). Our results also show that the contribution of successive restrictions to the mean-square end-to-end distance of a SARW decay with the same exponent as that of the correlations for the SARW.

Level Touchings in a Restricted Random Walk

We consider an ensemble of restricted discrete random walks in 2+1 dimensions. The restriction on the walks is such as to given particles an intrinsic angular momentum. The walks are embedded in a field which affects the mean free path of the walks. We show that the dynamics of the walks is such that second-order effects are described by a discrete form of Schrodinger's equation for particles in a potential field. This provides a classical context of the equation which is independent of its quantum context.

A weighted Markov chain technique is used to find the generating functions for several restricted one-dimensional random walks. Examples with simple restrictions concern the number of penetrations of a penetrable barrier. Examples with compound restrictions include the number of full crossings of the origin. A typical asymptotic evaluation is carried out.

A randomm‐ary seach tree is constructed from a random permutation of 1,…,n. A law of large numbers is obtained for the heightHnof these trees by applying the theory of branching random walks. in particular, it is shown thatHn/logn→γin probability asn→∞ where γ = γ(m) is a constant depending uponmonly. Interestingly, asm→∞, γ(m) is asymptotic to 1/logm, the coefficient of lognin the asymptotic expression for the height of the completem‐ary search tree. This proves that for largem, randomm‐ary search trees behave virtually like completem‐ary trees.

Much recent work in satisficing planning has aimed at striking a balance between coverage - solving as many problems as possible - and plan quality. Current planners achieve near perfect coverage on the latest IPC benchmarks. It is therefore natural to investigate their scaling behavior on more difficult instances. Among state of the art planners, LAMA (Richter, Helmert, and Westphal 2008) is able to generate high quality plans, but its coverage drops off rapidly with increasing prob- lem complexity. The Arvand planner (Nakhost and Müller 2009) scales to much harder instances but generates lower quality plans. This paper introduces a new algorithm, Monte Carlo Random Walk-based Local Tree Search (MRW-LTS), which uses random walks to selectively build local search trees. Experiments demonstrate that MRW-LTS combines a scaling behavior that is better than LAMA’s with a plan quality that is better than Arvand’s.

Large deviation rates for supercritical and critical branching processes.- How fast does a general branching random walk spread?.- Boltzmann-Gibbs weights in the branching random walk.- Stochastic monotonicity and branching processes.- Multilevel multitype branching models of an information system.- On the shape of the wavefront of branching random walk.- Limiting distributions in branching processes with two types of particles.- Depth-first search of random trees, and Poisson point processes.- Towards dependence in general branching processes.- A criterion of boundedness of discrete branching random walk.- Quasistationarity in a branching model of division-within-division.- Population and density dependent branching processes.- Directed polymers in random media and spin glass models on trees..- A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes.- On two measures defined on the boundary of a branching tree.- Which critically branching populations persist?.- A simple path to Biggins' martingale convergence for branching random walk.- Unsolved problems concerning random walks on trees.- Branching processes with local dependencies.- Sharpness of second moment criteria for branching and tree-indexed processes.- On the recognition and structure of probability generating functions.- Record values of a family of branching processes.- Limit skeleton for critical crump-mode-Jagers branching processes.- Markov cascades.- Limit theorems for branching processes with random migration stopped at zero.

In this paper we develop a diagrammatic technique for determining the exact recursive relations for the partition functions of the Ising model in a field, situated on finitely ramified deterministic fractal lattices. Applying this method on the members of the two-dimensional Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski gasket type of fractal lattices, which are characterized by generators of side length b, we first recover the known exact-space renormalization-group results for b=2 and 3, in the case of zero field, and for b=2 when H\ensuremath{\ne}0. Then we obtain new results for all b up to b=15, for H=0, and up to b=8 for H\ensuremath{\ne}0. These results enable us to initiate the study of the crossover of thermodynamic properties of the Ising model caused by changing of the underlying fractal lattices towards the Euclidean (triangular) lattice. Accordingly, we calculate the temperature dependence of the specific heat, for b\ensuremath{\le}15, and susceptibility for b\ensuremath{\le}8, and compare these functions with the known results for the triangular lattice. This comparison demonstrates the difference between the standard thermodynamic limit and the fractal-to-Euclidean crossover behavior.

We have applied the lattice bond enumeration method to the calculation of the steady-state diffusion in a lattice with fixed traps. We show that, to first order in density of traps, our random walk calculations for the effective diffusion constant in lattices with periodically arrayed traps are in exact agreement with calculations carried out previously for randomly arrayed traps embedded in a three-dimensional continuum medium (fluid). Our lattice random walk results are independent of dimension for d > 1, and we conjecture that this is also true for the continuum diffusion model.

We derive a continuous‐time lace expansion for a broad class of self‐interacting continuous‐time random walks. Our expansion applies when the self‐interaction is a sufficiently nice function of the local time of a continuous‐time random walk. As a special case we obtain a continuous‐time lace expansion for a class of spin systems that admit continuous‐time random walk representations.We apply our lace expansion to the n‐component model on when n=1,2, and prove that the critical Green's function is asymptotically a multiple of when and the coupling is weak. As another application of our method, we establish the analogous result for the lattice Edwards model at weak coupling. © 2021 Wiley Periodicals LLC.

The evaluation of the average number S_N(t) of distinct sites visited up to time t by N independent random walkers all starting from the same origin on an Euclidean lattice is addressed. We find that, for the nontrivial time regime and for large N, S_N(t) \approx \hat S_N(t) (1-\Delta), where \hat S_N(t) is the volume of a hypersphere of radius (4Dt \ln N)^{1/2}, \Delta={1/2}\sum_{n=1}^\infty \ln^{-n} N \sum_{m=0}^n s_m^{(n)} \ln^{m} \ln N, d is the dimension of the lattice, and the coefficients s_m^{(n)} depend on the dimension and time. The first three terms of these series are calculated explicitly and the resulting expressions are compared with other approximations and with simulation results for dimensions 1, 2, and 3. Some implications of these results on the geometry of the set of visited sites are discussed.

We studied the critical properties of flexible polymers, modelled by self-avoiding random walks, in good solvents and homogeneous environments. By applying the PERM Monte Carlo simulation method, we generated the polymer chains on the square and the simplecubic lattice of the maximal length of N=2000 steps.We enumerated approximately the number of different polymer chain configurations of length N,and analysed its asymptotic behaviour (for large N), determined by the connectivity constant μ and the entropic critical exponent γ. Also, we studied the behaviour of the set of effective critical exponents 휈푁, governing the end-to-end distance of a polymer chain of length N. We have established that in two dimensions 휈푁monotonically increases with N, whereas in three dimensions itmonotonically decreases when Nincreases. Values of 휈푁, obtained for both spatial dimensions have been extrapolated in the range of very long chains.In the end, we discuss and compare our results to those obtained previously for polymers on Euclidean lattices.

Placing multiple controllers in a large‐scale software defined network (SDN) is an NP‐hard problem. Currently, it is still a challenge to design a solution algorithm with high accuracy, short time consumption, and compliance with engineering standards. To overcome this challenge, we proposed the concept of synthetical delay, and took the controller capacity as a constraint condition, and established a controller placement model. The firefly algorithm (FA) was chosen as the solving algorithm for the controller placement problem. However, the FA has the drawbacks of insufficient search breadth in the early stage and low convergence efficiency in the later stage. To this end, we first introduced the dynamic parameter strategy for the light attractiveness, the light absorption coefficient, and the parameter to control the random walk of the FA. Then, we adopt the particle swarm optimization to improve the initial static parameters of the FA. Experimental results in the real large‐scale SDN topologies show that, compared with the latest controller placement algorithms based on cuckoo optimization, chaotic salp swarm algorithm, Varna‐based Optimization, and Teaching‐Learning‐based Optimization, the controller placement scheme obtained by the improved FA has a lower synthetical delay and more balanced controller load. Simultaneously, the time consumption of the improved FA is acceptable.

Levy flight is a random walk in which the step length r is a random variable with probability distribution proportional to 1/r1+u, where u is the Levy parameter and 0<u< infinity . The authors study the critical behaviour of fully directed Levy flight on Sierpinski carpets using the Monte Carlo method. The obtained critical exponent v/sub /// is independent of the parameter u, but vperpendicular to is found to be dependent on u. This seems to be interesting compared with the directed Levy flight on ordinary Euclidean lattices previously discussed by Hu and one of the authors, for which v/sub /// and vperpendicular to are independent of u. These results indicate that directed Levy flights on different fractals belong to different universality classes.