Open Quantum Random Walks with Decoherence on Coins with n Degrees of Freedom

In this paper we define a new type of decoherent quantum random walk with parameter 0≤p≤1, which becomes a unitary quantum random walk (UQRW) when p=0 and an open quantum random walk (OQRW) when p=1, respectively. We call this process a partially open quantum random walk (POQRW). We study the limiting distribution of a POQRW on Z ¹ subject to decoherence on coins with n degrees of freedom. The limiting distribution of the POQRW converges to a convex combination of normal distributions, under an eigenvalue condition. A Perron-Frobenius type of theorem is established to determine whether or not a POQRW satisfies the eigenvalue condition. Moreover, we explicitly compute the limiting distributions of characteristic equations of the position probability functions when n=2 and 3.     

Microscopic Structure of Shocks and Antishocks in the ASEP Conditioned on Low Current

We study the time evolution of the ASEP on a one-dimensional torus with L sites, conditioned on an atypically low current up to a finite time t. For a certain one-parameter family of initial measures with a shock we prove that the shock position performs a biased random walk on the torus and that the measure seen from the shock position remains invariant. We compute explicitly the transition rates of the random walk. For the large scale behavior this result suggests that there is an atypically low current such that the optimal density profile that realizes this current is a hyperbolic tangent with a traveling shock discontinuity. For an atypically low local current across a single bond of the torus we prove that a product measure with a shock at an arbitrary position and an antishock at the conditioned bond remains a convex combination of such measures at all times which implies that the antishock remains microscopically stable under the locally conditioned dynamics. We compute the coefficients of the convex combinations.     

Discrete random walk models for symmetric Lévy–Feller diffusion processes

We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index α (0<α⩽2), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous Markovian stochastic processes which we refer to as Lévy–Feller diffusion processes.     

Two-state random walk model of diffusion. 2. Oscillatory diffusion

Continuing our study of interrupted diffusion, we consider the problem of a particle executing a random walk interspersed with localized oscillations during its halts (e.g., at lattice sites). Earlier approaches proceedvia approximation schemes for the solution of the Fokker-Planck equation for diffusion in a periodic potential. In contrast, we visualize a two-state random walk in velocity space with the particle alternating between a state of flight and one of localized oscillation. Using simple, physically plausible inputs for the primary quantities characterising the random walk, we employ the powerful continuous-time random walk formalism to derive convenient and tractable closed-form expressions for all the objects of interest: the velocity autocorrelation, generalized diffusion constant, dynamic mobility, mean square displacement, dynamic structure factor (in the Gaussian approximation), etc. The interplay of the three characteristic times in the problem (the mean residence and flight times, and the period of the ‘local mode’) is elucidated. The emergence of a number of striking features of oscillatory diffusion (e.g., the local mode peak in the dynamic mobility and structure factor, and the transition between the oscillatory and diffusive regimes) is demonstrated.     

Subdiffusivity of a Random Walk Among a Poisson System of Moving Traps on $\mathbb {Z}$

We consider a random walk among a Poisson system of moving traps on \(\mathbb {Z}\). In earlier work (Drewitz et al. Springer Proc. Math. 11, 119-158 2012), the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time t in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk’s range.     

Analysis of price diffusion in financial markets using PUCK model

Based on the new type of random walk process called the potentials of unbalanced complex kinetics (PUCK) model, we theoretically show that the price diffusion in large scales is amplified 2(2+b)^-1 times, where b is the coefficient of quadratic term of the potential. In short time scales the price diffusion depends on the size M of the super moving average. Both numerical simulations and real data analysis of Yen-Dollar rates are consistent with theoretical analysis.     

The melting phenomenon in random-walk model of DNA

The melting phenomenon in a double-stranded homopolypeptide is considered. The relative distance between the corresponding monomers of two polymer chains is modeled by the two-dimensional random walk on the square lattice. Returns of the random walk to the origin describe the formation of hydrogen bonds between complementary units. To take into account the two competing interactions of monomers inside the chains, we obtain a completely denatured state at finite temperature T _c .     

Pólya number and first return of bursty random walk: Rigorous solutions

The recurrence properties of random walks can be characterized by Pólya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we investigate Pólya number and first return for bursty random walk on a line, in which the walk has different step size and moving probabilities. Using the concept of the Catalan number, we obtain exact results for first return probability, the average first return time and Pólya number for the first time. We show that Pólya number displays two different functional behavior when the walk deviates from the recurrent point. By utilizing the Lagrange inversion formula, we interpret our findings by transferring Pólya number to the closed-form solutions of an inverse function. We also calculate Pólya number using another approach, which corroborates our results and conclusions. Finally, we consider the recurrence properties and Pólya number of two variations of the bursty random walk model.     

Dynamical correlations for vicious random walk with a wall

A one-dimensional system of nonintersecting Brownian particles is constructed as the diffusion scaling limit of Fisher's vicious random walk model. N Brownian particles start from the origin at time t=0 and undergo mutually avoiding motion until a finite time t=T. Dynamical correlation functions among the walkers are exactly evaluated in the case with a wall at the origin. Taking an asymptotic limit N→∞, we observe discontinuous transitions in the dynamical correlations. It is further shown that the vicious walk model with a wall is equivalent to a parametric random matrix model describing the crossover between the Bogoliubov–deGennes universality classes.     

Random walks on random lattices with traps

A critical examination is presented of the continuous time random walk (CTRW) approximation and of frequency-dependent effective transition rate methods for calculating the configurational average of the Laplace transform of the probabilityPs, t|₀) that a particle performing a random walk will be at site s timet after it reached the site s₀. Some exact results are derived for the form of P(s, t|s₀) at long times, and these indicate that the effective transition rate methods are the better approximation for systems with symmetric effective hopping rates, while the CTRW approximation is better for systems containing traps, i.e., states that are much easier to enter than to leave. The implications of these results for calculations of transient currents and of the ac conductivity for amorphous semiconductors are discussed.     

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.     

Trapping in the Random Conductance Model

We consider random walks on ? ^d among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time 2n. We show that in the situations when the heat kernel exhibits subdiffusive decay—which is known to occur in dimensions d≥4—the walk gets trapped for a time of order n in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.     

Open Quantum Random Walks

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.     

Energy transfer as a random walk with long-range steps

We consider the incoherent energy transport in molecular crystals, where the transfer rates stem from Coulombic and exchange interactions. For substitutionally disordered lattices we present in a first passage model the excitation decay due to trapping by randomly distributed traps; the decay is related to the distribution of the number of distinct sites visited during the timet and is expressible through the cumulants of this distribution. The validity domains of approximate decay laws based on the first few cumulants are also discussed. We exemplify the findings for dipolar transfer rates between randomly distributed molecules on a square lattice, by comparing the random walk on the random system to its CTRW (continuous time random walk) counterpart.     

Conductivity exponent in three-dimensional percolation by diffusion based on molecular trajectory algorithm and blind-ant rules

Diffusion on random systems above and at their percolation threshold in three dimensions is carried out by a molecular trajectory method and a simple lattice random walk method, respectively. The classical regimes of diffusion on percolation near the threshold are observed in our simulations by both methods. Our Monte Carlo simulations by the simple lattice random walk method give the conductivity exponent μ/ν=2.32±0.02 for diffusion on the incipient infinite clusters and μ/ν=2.21±0.03 for diffusion on a percolating lattice above the threshold. However, while diffusion is performed by the molecular trajectory algorithm either on the incipient infinite clusters or on a percolating lattice above the threshold, the result is found to be μ/ν=2.26±0.02. In addition, it takes less time step for diffusion based on the molecular trajectory algorithm to reach the asymptotic limit comparing with the simple lattice random walk.     

Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

We investigate statistics of occupation times for an over-damped Brownian particle in an external force field, using a backward Fokker–Planck equation introduced by Majumdar and Comtet. For an arbitrary potential field the distribution of occupation times is expressed in terms of solutions of the corresponding first passage time problem. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker–Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding cases, rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.     

Improving Network Representation Learning via Dynamic Random Walk,Self-Attention and Vertex Attributes-Driven Laplacian Space Optimization

Network data analysis is a crucial method for mining complicated object interactions. In recent years, random walk and neural-language-model-based network representation learning (NRL) approaches have been widely used for network data analysis. However, these NRL approaches suffer from the following deficiencies: firstly, because the random walk procedure is based on symmetric node similarity and fixed probability distribution, the sampled vertices’ sequences may lose local community structure information; secondly, because the feature extraction capacity of the shallow neural language model is limited, they can only extract the local structural features of networks; and thirdly, these approaches require specially designed mechanisms for different downstream tasks to integrate vertex attributes of various types. We conducted an in-depth investigation to address the aforementioned issues and propose a novel general NRL framework called dynamic structure and vertex attribute fusion network embedding, which firstly defines an asymmetric similarity and h-hop dynamic random walk strategy to guide the random walk process to preserve the network’s local community structure in walked vertex sequences. Next, we train a self-attention-based sequence prediction model on the walked vertex sequences to simultaneously learn the vertices’ local and global structural features. Finally, we introduce an attributes-driven Laplacian space optimization to converge the process of structural feature extraction and attribute feature extraction. The proposed approach is exhaustively evaluated by means of node visualization and classification on multiple benchmark datasets, and achieves superior results compared to baseline approaches.     

A computer simulation study of ionic conductivity in polymer electrolytes

In this paper we present a computer simulation study of ionic conductivity in solid polymeric electrolytes. The multiphase nature of the material is taken into account. The polymer is represented by a regular lattice whose sites represent either crystalline or amorphous regions with the charge carrier performing a random walk. Different waiting times are assigned to sites corresponding to the different phases. A random walk (RW) is used to calculate the conductivity through the Nernst-Einstein relation. Our walk algorithm takes into account the reorganization of the different phases over time scales comparable to time scales for the conduction process. This is a characteristic feature of the polymer network. The qualitative nature of the variation of conductivity with salt concentration agrees with the experimental values for PEO-NH₄I and PEO-NH₄SCN. The average jump distance estimated from our work is consistent with the reported bond lengths for such polymers.     

Stochastic Modeling of Indoor Air Temperature

Temperature is one of the main parameters describing thermal comfort and indoor air quality. In this paper we propose an approach, based on a modification of the continuous time random walk, to model the indoor air temperature. We perform a statistical analysis of the recorded time series, that allows us to point out the main statistical properties of the recorded variable. The obtained conclusions about the nature of the process lead to a continuous time random walk, that in contrast to the classical approach, models time dependence of the jumps distribution. Moreover, we show that the waiting times can be modeled by a tempered stable distribution, which yields a subdiffusive behavior in short times and diffusive behavior in longer times. Finally, by conducting a simulation study we illustrate possible applications of the presented approach in the thermal comfort monitoring and forecasting.