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Register a new userAnomalous 1D fluctuations of a simple 2D random walk in a large deviation regime
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The following question is the subject of our work: could a two-dimensional random path pushed by some constraints to an improbable large deviation regime, possess extreme statistics with one-dimensional Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though non-universal, since the fluctuations depend on the underlying geometry. We consider in details two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span $left< d(t) right> sim t^{gamma}$ of the trajectory of $t$ steps above the top of the semicircle or the triangle. We show that $gamma = frac{1}{3}$, i.e. $left< d(t) right>$ shares the KPZ statistics for the semicircle, while $gamma=0$ for the triangle. We propose heuristic derivations of scaling exponents $gamma$ for different geometries, justify them by explicit analytic computations and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.
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By constructing a multicanonical Monte Carlo simulation, we obtain the full probability distribution $rho_N(r)$ of the degree assortativity coefficient $r$ on configuration networks of size $N$ by using the multiple histogram reweighting method. We suggest that $rho_N(r)$ obeys a large deviation principle, $rho_N left( r- r_N^* right) asymp {e^{ - {N^xi }Ileft( {r- r_N^* } right)}}$, where the rate function $I$ is convex and possesses its unique minimum at $r=r_N^*$, and $xi$ is an exponent that scales $rho_N$s with $N$. We show that $xi=1$ for Poisson random graphs, and $xigeq1$ for scale-free networks in which $xi$ is a decreasing function of the degree distribution exponent $gamma$. Our results reveal that the fluctuations of $r$ exhibits an anomalous scaling with $N$ in highly heterogeneous networks.
Expanding media are typical in many different fields, e.g. in Biology and Cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties. Here, we focus on such effects when the diffusion process is described by the Continuous Time Random Walk (CTRW) model. For the case where the jump length and the waiting time probability density functions (pdfs) are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Levy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Greens function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. For a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This Big Crunch effect stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained. Our results confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Levy flights, we quantify this effect by means of the so-called Levy horizon.
The ``Brownian bees model describes an ensemble of $N$ independent branching Brownian particles. When a particle branches into two particles, the particle farthest from the origin is eliminated so as to keep a constant number of particles. In the limit of $Nto infty$, the spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation. At long times the particle density approaches a spherically symmetric steady state solution with a compact support. Here we study fluctuations of the ``swarm of bees due to the random character of the branching Brownian motion in the limit of large but finite $N$. We consider a one-dimensional setting and focus on two fluctuating quantities: the swarm center of mass $X(t)$ and the swarm radius $ell(t)$. Linearizing a pertinent Langevin equation around the deterministic steady state solution, we calculate the two-time covariances of $X(t)$ and $ell(t)$. The variance of $X(t)$ directly follows from the covariance of $X(t)$, and it scales as $1/N$ as to be expected from the law of large numbers. The variance of $ell(t)$ behaves differently: it exhibits an anomalous scaling $ln N/N$. This anomaly appears because all spatial scales, including a narrow region near the edges of the swarm where only a few particles are present, give a significant contribution to the variance. We argue that the variance of $ell(t)$ can be obtained from the covariance of $ell(t)$ by introducing a cutoff at the microscopic time $1/N$ where the continuum Langevin description breaks down. Our theoretical predictions are in good agreement with Monte-Carlo simulations of the microscopic model. Generalizations to higher dimensions are briefly discussed.
The probability of a random walker to return to its starting point in dimensions one and two is unity, a theorem first proven by G. Polya. The recurrence probability -- the probability to be found at the origin at a time t, is a power law with a critical exponent d/2 in dimensions d=1,2. We report an experiment that directly measures the Laplace transform of the recurrence probability in one dimension using Electromagnetically Induced Transparency (EIT) of coherent atoms diffusing in a vapor-cell filled with buffer gas. We find a regime where the limiting form of the complex EIT spectrum is universal and only depends on the effective dimensionality in which the random recurrence takes place. In an effective one-dimensional diffusion setting, the measured spectrum exhibits power law dependence over two decades in the frequency domain with a critical exponent of 0.56 close to the expected value 0.5. Possible extensions to more elaborate diffusion schemes are briefly discussed.
We derive properties of the rate function in Varadhans (annealed) large deviation principle for multidimensional, ballistic random walk in random environment, in a certain neighborhood of the zero set of the rate function. Our approach relates the LDP to that of regeneration times and distances. The analysis of the latter is possible due to the i.i.d. structure of regenerations.
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