Rigorous renormalisation group and disordered systems

Journal of Statistical Physics, 2008

We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in ε = 4 − d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d = 4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d = 4, which can be used as a further test of the conjecture.

The Annals of Probability, 2017

We consider a random walk in random environment in the low disorder regime on Z d , that is, the probability that the random walk jumps from a site x to a nearest neighboring site x + e is given by p(e) + εξ(x, e), where p(e) is deterministic, {{ξ(x, e) : |e| 1 = 1} : x ∈ Z d } are i.i.d. and ε > 0 is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in ε for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in ε for the invariant measure of random perturbations of the simple symmetric random walk in dimensions d = 2.

Physical Review A, 2019

We consider a quantum particle (walker) on a line who coherently chooses to jump to the left or right depending on the result of toss of a quantum coin. The lengths of the jumps are considered to be independent and identically distributed quenched Poisson random variables. We find that the spread of the walker is significantly inhibited, whereby it resides in the near-origin region, with respect to the case when there is no disorder. The scaling exponent of the quenched-averaged dispersion of the walker is sub-ballistic but super-diffusive. We also show that the features are universal to a class of sub-and super-Poissonian distributed quenched randomized jumps.

Journal of Computational and Applied Mathematics, 2008

We consider a nearest neighbor, symmetric random walk on a homogeneous, ergodic random lattice Z d . The jump rates of the walk are independent, identically Bernoulli distributed random variables indexed by the bonds of the lattice. A standard result from the homogenization theory, see [A. De Masi, P.A. Ferrari, S. Goldstein, W.D. Wick, An invariance principle for reversible Markov processes, Applications to random walks in random environments, J. Statist. Phys. 55(3/4) (1989) 787-855], asserts that the scaled trajectory of the particle satisfies the functional central limit theorem. The covariance matrix of the limiting normal distribution is called the effective diffusivity of the walk. We use the duality structure corresponding to the product Bernoulli measure to construct a numerical scheme that approximates this parameter when d 3. The estimates of the convergence rates are also provided. © 2007 Published by Elsevier B.V. MSC: Primary 65C35; 82C41; Secondary 65Z05 Keywords: Eandom walk on a random lattice; Corrector; Duality ( ) t converge weakly, in P-probability with respect to , to the law of a Gaussian

2002

We study the behavior of self-avoiding walks ͑SAWs͒ on square and cubic lattices in the presence of strong disorder. We simulate the disorder by assigning random energy ⑀ taken from a probability distribution P(⑀) to each site ͑or bond͒ of the lattice. We study the strong disorder limit for an extremely broad range of energies with P(⑀)ϰ1/⑀. For each configuration of disorder, we find by exact enumeration the optimal SAW of fixed length N and fixed origin that minimizes the sum of the energies of the visited sites ͑or bonds͒. We find the fractal dimension of the optimal path to be d opt ϭ1.52Ϯ0.10 in two dimensions ͑2D͒ and d opt ϭ1.82Ϯ0.08 in 3D. Our results imply that SAWs in strong disorder with fixed N are much more compact than SAWs in disordered media with a uniform distribution of energies, optimal paths in strong disorder with fixed end-toend distance R, and SAWs on a percolation cluster. Our results are also consistent with the possibility that SAWs in strong disorder belong to the same universality class as the maximal SAW on a percolation cluster at criticality, for which we calculate the fractal dimension d max ϭ1.64Ϯ0.02 for 2D and d max ϭ1.87Ϯ0.05 for 3D, values very close to the fractal dimensions of the percolation backbone in 2D and 3D.

2018

We consider a quantum particle (walker) on a line who coherently chooses to jump to the left or right depending on the result of toss of a quantum coin. The lengths of the jumps are considered to be independent and identically distributed quenched Poisson random variables. We find that the spread of the walker is significantly inhibited, whereby it resides in the near-origin region, with respect to the case when there is no disorder. The scaling exponent of the quenched-averaged dispersion of the walker is sub-ballistic but super-diffusive. We also show that the features are universal to a class of sub-and super-Poissonian distributed quenched randomized jumps.

Journal of Mathematical Physics, 1978

We use the exact renormalization group equations to determine the asymptotic behavior of long selfavoiding random walks on some pseudolattices. The lattices considered are the truncated 3-simplex, the truncated 4-simplex, and the modified rectangular lattices. The total number of random walks C", the number of polygons P" of perimeter n, and the mean square end to end distance (RA) are assumed to be asymptotically proportional to JL"n Y-I, JL"n a-3 , and n 2 v respectively for large n, where n is the total length of the walk. The exact values of the connectivity constant IL• and the critical exponents A, a, v are determined for the three lattices. We give an example of two lattice systems that have the same effective nonintegral dimensionality 3/2 but different values of the critical exponents y, a, and v.

Physical Review A, 1984

This paper introduces a position-space renormalization-group approach for nonequilibrium systems and applies the method to a driven stochastic one-dimensional gas with open boundaries. The dynamics are characterized by three parameters: the probability ␣ that a particle will flow into the chain to the leftmost site, the probability ␤ that a particle will flow out from the rightmost site, and the probability p that a particle will jump to the right if the site to the right is empty. The renormalization-group procedure is conducted within the space of these transition probabilities, which are relevant to the system's dynamics. The method yields a critical point at ␣ c ϭ␤ c ϭ1/2, in agreement with the exact values, and the critical exponent ϭ2.71, as compared with the exact value ϭ2.00.

Journal de Physique, 1988

Random walks provide a simple conventional model to describe various transport processes, for example propagation of heat or diffusion of matter through a medium. However, in many practical cases the medium is highly irregular due to defects, impurities, fluctuations etc., and it is natural to model this as random environment. In the random walks context, such models are referred to as Random Walks in Random Environments (RWRE). This is a relatively new chapter in applied probability and physics of disordered systems, initiated in the 1970s. Early interest was motivated by some problems in biology, crystallography and metal physics, but later applications have spread through numerous areas. After 30 years of extensive work, RWRE remain a very active area of research, which has already led to many surprising discoveries. The goal of this article is to give a brief introduction to the beautiful area of RWRE. The principal model to be discussed is a random walk with nearest-neighbor ju...