Correlated disordered interactions on Potts models

Abstract

Using a weak-disorder scheme and real-space renormalization-group techniques, we obtain analytical results for the critical behavior of various q-state Potts models with correlated disordered exchange interactions along d1 of d spatial dimensions on hierarchical (Migdal-Kadanoff) lattices. Our results indicate qualitative differences between the cases d − d1 = 1 (for which we find nonphysical random fixed points, suggesting the existence of nonperturbative fixed distributions) and d − d1 > 1 (for which we do find acceptable perturbartive random fixed points), in agreement with previous numerical calculations by Andelman and Aharony. We also rederive a criterion for relevance of correlated disorder, which generalizes the usual Harris criterion.

Aperiodic quantumXXZchains: Renormalization-group results

“…Randomly distributed inhomogeneities lead to ω = 1 /2, and the general Harris criterion 44,46,47 is recovered. The ground-state of the model in Eq.…”

Two-dimensional quantum gravity - a laboratory for fluctuating graphs and quenched connectivity disorder

“…Such correlations for a random-bond model have been considered occasionally [70][71][72][73] and altered relevance criteria have been proposed [70,74]. Luck [74] has considered a class of irregular systems not covered by the random-bond paradigm, namely that of quasi-crystalline or aperiodic structures, and formulated a generalised relevance criterion.…”

Continuous cascades in the wavelet space as models for synthetic turbulence

“…In [11,12] (see also [10]) dM (t) is obtained as the (weak) limit, when → 0 of the measure with density:…”

“…This lack of stationarity and continuous scale invariance is obviously not suited to accounting for natural phenomena. In order to circumvent these problems, continuous extensions of Mandelbrot cascades were proposed by first Barral and Mandelbrot [10] and later by Bacry and Muzy [11,12]. The idea under these constructions is to replace the discrete multiplicative density n i=1 W i = e n i=1 ln W i by e ω (t) where ω (t) is an infinitely divisible noise chosen with a logarithmic correlation function designed to mimic the tree-like (in general a dyadic tree) structure underlying M-cascades [13][14][15].…”

“…An alternative but related approach, was initially proposed with the construction of "the Multifractal Random Walk" (MRW) in [14,17] and that consists of interpreting M (dt) as the local variance of a fractional Brownian motion B H (t). As emphasized in [11,18,19], this amounts to constructing a multifractal process as (the limit of) a stochastic integral like e ω (t) dB H (t). This point of view is inspired from classical stochastic volatility models of financial markets that aim at accounting for the observed bursts in price fluctuations by the random nature of the variance of an underlying Gaussian law.…”