Level number variance and spectral compressibility in a critical two-dimensional random matrix model

We study level number variance in a two-dimensional random matrix model characterized by a power-law decay of the matrix elements. The amplitude of the decay is controlled by the parameter b. We find analytically that at small values of b the level number variance behaves linearly, with the compressibility chi between 0 and 1, which is typical for critical systems. For large values of b, we derive that chi=0, as one would normally expect in the metallic phase. Using numerical simulations we determine the critical value of b at which the transition between these two phases occurs.Comment: 6 page

Universal and non-universal features of the multifractality exponents of critical wave-functions

We calculate perturbatively the multifractality spectrum of wave-functions in critical random matrix ensembles in the regime of weak multifractality. We show that in the leading order the spectrum is universal, while the higher order corrections are model-specific. Explicit results for the anomalous dimensions are derived in the power-law and ultrametric random matrix ensembles.Comment: 9 page

Global properties of Stochastic Loewner evolution driven by Levy processes

Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian motion which then produces a trace, a continuous fractal curve connecting the singular points of the motion. If jumps are added to the driving function, the trace branches. In a recent publication [1] we introduced a generalized SLE driven by a superposition of a Brownian motion and a fractal set of jumps (technically a stable L\'evy process). We then discussed the small-scale properties of the resulting L\'evy-SLE growth process. Here we discuss the same model, but focus on the global scaling behavior which ensues as time goes to infinity. This limiting behavior is independent of the Brownian forcing and depends upon only a single parameter, $\alpha$, which defines the shape of the stable L\'evy distribution. We learn about this behavior by studying a Fokker-Planck equation which gives the probability distribution for endpoints of the trace as a function of time. As in the short-time case previously studied, we observe that the properties of this growth process change qualitatively and singularly at $\alpha =1$. We show both analytically and numerically that the growth continues indefinitely in the vertical direction for $\alpha > 1$, goes as $\log t$ for $\alpha = 1$, and saturates for $\alpha< 1$. The probability density has two different scales corresponding to directions along and perpendicular to the boundary. In the former case, the characteristic scale is $X(t) \sim t^{1/\alpha}$. In the latter case the scale is $Y(t) \sim A + B t^{1-1/\alpha}$ for $\alpha \neq 1$, and $Y(t) \sim \ln t$ for $\alpha = 1$. Scaling functions for the probability density are given for various limiting cases.Comment: Published versio

On harmonic measure of critical curves

Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories with central charge $c\leqslant 1$, scaling exponents of harmonic measure have been computed by B. Duplantier [Phys. Rev. Lett. {\bf 84}, 1363 (2000)] by relating the problem to boundary two-dimensional gravity. We present a simple argument that allows us to connect harmonic measure of critical curves to operators obtained by fusion of primary fields, and compute characteristics of fractal geometry by means of regular methods of conformal field theory. The method is not limited to theories with $c\leqslant 1$.Comment: Some more correction

Stochastic Loewner evolution driven by Levy processes

Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then produces a continuous fractal trace. If jumps are added to the driving function, the trace branches. We consider a generalized SLE driven by a superposition of a Brownian motion and a stable Levy process. The situation is defined by the usual SLE parameter, $\kappa$, as well as $\alpha$ which defines the shape of the stable Levy distribution. The resulting behavior is characterized by two descriptors: $p$, the probability that the trace self-intersects, and $\tilde{p}$, the probability that it will approach arbitrarily close to doing so. Using Dynkin's formula, these descriptors are shown to change qualitatively and singularly at critical values of $\kappa$ and $\alpha$. It is reasonable to call such changes ``phase transitions''. These transitions occur as $\kappa$ passes through four (a well-known result) and as $\alpha$ passes through one (a new result). Numerical simulations are then used to explore the associated touching and near-touching events.Comment: Published version, minor typos corrected, added reference

Critical curves in conformally invariant statistical systems

We consider critical curves -- conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. We show how the multifractal spectrum of harmonic measure and other fractal characteristics of critical curves can be obtained.Comment: Published versio

Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal 1/f1/f noise

To understand the sample-to-sample fluctuations in disorder-generated multifractal patterns we investigate analytically as well as numerically the statistics of high values of the simplest model - the ideal periodic $1/f$ Gaussian noise. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of number of points above a given level. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level, results in an important difference between the mean and the typical values of the counting function. This can be further used to determine the typical threshold $x_m$ of extreme values in the pattern which turns out to be given by $x_m^{(typ)}=2-c\ln{\ln{M}}/\ln{M}$ with $c=3/2$. Such observation provides a rather compelling explanation of the mechanism behind universality of $c$. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. In particular, we predict that the typical value of the maximum $p_{max}$ of intensity is to be given by $-\ln{p_{max}} = \alpha_{-}\ln{M} + \frac{3}{2f'(\alpha_{-})}\ln{\ln{M}} + O(1)$, where $f(\alpha)$ is the corresponding singularity spectrum vanishing at $\alpha=\alpha_{-}>0$. For the $1/f$ noise we also derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism.Comment: 28 pages; 7 figures, published version with a few misprints corrected, editing done and references adde

Cluster pinch-point densities in polygons

In a statistical cluster or loop model such as percolation, or more generally the Potts models or O(n) models, a pinch point is a single bulk point where several distinct clusters or loops touch. In a polygon P harboring such a model in its interior and with 2N sides exhibiting free/fixed side-alternating boundary conditions, "boundary" clusters anchor to the fixed sides of P. At the critical point and in the continuum limit, the density (i.e., frequency of occurrence) of pinch-point events between s distinct boundary clusters at a bulk point w in P is proportional to _P. The w_i are the vertices of P, psi_1^c is a conformal field theory (CFT) corner one-leg operator, and Psi_s is a CFT bulk 2s-leg operator. In this article, we use the Coulomb gas formalism to construct explicit contour integral formulas for these correlation functions and thereby calculate the density of various pinch-point configurations at arbitrary points in the rectangle, in the hexagon, and for the case s=N, in the 2N-sided polygon at the system's critical point. Explicit formulas for these results are given in terms of algebraic functions or integrals of algebraic functions, particularly Lauricella functions. In critical percolation, the result for s=N=2 gives the density of red bonds between boundary clusters (in the continuum limit) inside a rectangle. We compare our results with high-precision simulations of critical percolation and Ising FK clusters in a rectangle of aspect ratio two and in a regular hexagon and find very good agreement.Comment: 31 pages, 1 appendix, 21 figures. In the second version of this article, we have improved the organization, figures, and references that appeared in the first versio

Geometrical properties of parafermionic spin models

We present measurements of the fractal dimensions associated to the geometrical clusters for Z_4 and Z_5 spin models. We also attempted to measure similar fractal dimensions for the generalised Fortuyin Kastelyn (FK) clusters in these models but we discovered that these clusters do not percolate at the critical point of the model under consideration. These results clearly mark a difference in the behaviour of these non local objects compared to the Ising model or the 3-state Potts model which corresponds to the simplest cases of Z_N spin models with N=2 and N=3 respectively. We compare these fractal dimensions with the ones obtained for SLE interfaces.Comment: 18 pages, 10 figures. v2: published versio