Fluctuations of random walks in critical random environments

First passage on disordered intervals

We derive unexpected first-passage properties for nearest-neighbor hopping on finite intervals with disordered hopping rates, including (a) a highly variable spatial dependence of the first-passage time, (b) huge disparities in first-passage times for different realizations of hopping rates, (c) significant discrepancies between the first moment and the square root of the second moment of the first-passage time, and (d) bimodal first-passage time distributions. Our approach relies on the backward equation, in conjunction with probability generating functions, to obtain all moments, as well as the distribution of first-passage times. Our approach is simpler than previous approaches based on the forward equation, in which computing the mth moment of the first-passage time requires all preceding moments.

Topological Properties, Spectra Analysis, and Consensus Problems for a Class of Network Models Based on m-Fission Operation

The network model, especially the network model constructed by network operation, is an important tool to study complex networks. Many topological and dynamic properties of complex networks can be studied in this way. In this article, the m -fission operation is constructed based on the phenomenon of node splitting in the network, which is quite common in complex networks. Many network models, including dual Sierpinski gaskets, are built based on this operation, but it has never been systematically studied. Then, the topological and dynamic properties of the m -fission operation and the corresponding iterative fission network model are studied, and the influence of the operation on the network structure is revealed. Among them, the topological properties of the network include diameter, degree distribution, clustering coefficient, average distance, and modularity. By studying these properties, it can be concluded that the iterative fission network is a fractal homogeneous network with high clustering and high community characteristics. Since the dynamic properties are closely related to the spectrum of the Laplacian matrix corresponding to the network, the iterative relation of the spectrum in operation is studied, and the complete solution of the spectrum of the iterative fission network is obtained. Based on the above results, we calculate the analytical expressions of the characteristic quantities related to the dynamic properties on the network, including the Kirchhoff index and the average of hitting times. Finally, due to the close connection between the network model and the system, we further analyzed the consensus problems on the system corresponding to the network, including convergence rate, delay robustness, first-order noise coherence, and second-order noise coherence.

Shifted Brownian Fluctuation Game

This article analyzes the behavior of a Brownian fluctuation process under a mixed strategic game setup. A variant of a compound Brownian motion has been newly proposed, which is called the Shifted Brownian Fluctuation Process to predict the turning points of a stochastic process. This compound process evolves until it reaches one step prior to the turning point. The Shifted Brownian Fluctuation Game has been constructed based on this new process to find the optimal moment of actions. Analytically tractable results are obtained by using the fluctuation theory and the mixed strategy game theory. The joint functional of the Shifted Brownian Fluctuation Process is targeted for transformation of the first passage time and its index. These results enable us to predict the moment of a turning point and the moment of actions to obtain the optimal payoffs of a game. This research adapts the theoretical framework to implement an autonomous trader for value assets including stocks and cybercurrencies.

Ubiquity of anomalous transport in porous media: Numerical evidence, continuous time random walk modelling, and hydrodynamic interpretation

Anomalous transport in porous media is commonly believed to be induced by the highly complex pore space geometry. However, this phenomenon is also observed in porous media with rather simple pore structure. In order to answer how ubiquitous can anomalous transport be in porous media, we in this work systematically investigate the solute transport process in a simple porous medium model with minimal structural randomness. The porosities we consider range widely from 0.30 up to 0.85, and we find by lattice Boltzmann simulations that the solute transport process can be anomalous in all cases at high Péclet numbers. We use the continuous time random walk theory to quantitatively explain the observed scaling relations of the process. A plausible hydrodynamic origin of anomalous transport in simple porous media is proposed as a complement to its widely accepted geometric origin in complex porous media. Our results, together with previous findings, provide evidence that anomalous transport is indeed ubiquitous in porous media. Consequently, attentions should be paid when modelling solute transport by the classical advection-diffusion equation, which could lead to systematic error.

Exact results for the first-passage properties in a class of fractal networks

In this work, we consider a class of recursively grown fractal networks G_n(t) whose topology is controlled by two integer parameters, t and n. We first analyse the structural properties of G_n(t) (including fractal dimension, modularity, and clustering coefficient), and then we move to its transport properties. The latter are studied in terms of first-passage quantities (including the mean trapping time, the global mean first-passage time, and Kemeny's constant), and we highlight that their asymptotic behavior is controlled by the network's size and diameter. Remarkably, if we tune n (or, analogously, t) while keeping the network size fixed, as n increases (t decreases) the network gets more and more clustered and modular while its diameter is reduced, implying, ultimately, a better transport performance. The connection between this class of networks and models for polymer architectures is also discussed.

A general phenomenological relation for the subdiffusive exponent of anomalous diffusion in disordered media

This work numerically investigates the diffusion of finite inert tracer particles in different types of fixed gels. The mean square displacement (MSD) of the tracers reveals a transition to subdiffusive motion MSD ∼ tα as soon as the accessible volume fraction p in the gel decreases from unity. Individual tracer dynamics reveals two types of particles in the gels: mobile tracers cross the system through percolating pores following subdiffusive dynamics MSDmob ∼ tαmob, while a fraction ptrap(p) of the particles remain trapped in finite pores. Below the void percolation threshold p < pc all the particles get trapped and α → 0. By separately studying both populations we find a simple phenomenological law for the mobile tracers αmob(p) ≈ a ln p + c where c ≈ 1 and a ∼ 0.2 depends on the gel type. On the other hand, a cluster-analysis of the gel accessible volume reveals a power law for the trapping probability ptrap ∼ (p/pc)-γ, with γ ≃ 2.9. This yields a prediction for the ensemble averaged subdiffusion exponent α = αmob(1 - ptrap). Our predictions are successfully validated against the different gels studied here and against numerical and experimental results in the literature (silica gels, polyacrylamide gels, flexible F-actin networks and in different random obstacles). Notably, the parameter a ∼ 0.2 presents small differences amongst all these cases, indicating the robustness of the proposed relation.