Searcharxiv⌕ Search

arXiv subjects

Ilia Binder

Publications and source records attributed to Ilia Binder.

27 records · Page 2Linked to original sources
arXiv2014

On fluctuations and localization length for the Anderson model on a strip

We consider the Anderson model on a strip. Assuming that potentials have bounded density with considerable tails we get a lower bound for the fluctuations of the logarithm of the Green's function in a finite box. This implies an effective estimate by $ \exp(CW^2) $ for the localization length of the Anderson model on the strip of width $ W $. The results are obtained, actually, for a more general model with a non-local operator in the vertical direction.

math-ph↗
arXiv2013

Computable Caratheodory Theory

Conformal Riemann mapping of the unit disk onto a simply-connected domain $W$ is a central object of study in classical Complex Analysis. The first complete proof of the Riemann Mapping Theorem given by P. Koebe in 1912 is constructive, and theoretical aspects of computing the Riemann map have been extensively studied since. Carath{é}odory Theory describes the boundary extension of the Riemann map. In this paper we develop its constructive version with explicit complexity bounds.

math.CV↗
arXiv2012

On Optimal Separation of Eigenvalues for a Quasiperiodic Jacobi Matrix

We consider quasiperiodic Jacobi matrices of size N with analytic coefficients. We show that, in the positive Lyapunov exponent regime, after removing some small sets of energies and frequencies, any eigenvalue is separated from the rest of the spectrum by $N^{-1}(\log N)^{-p}$, with p>15.

math.SP↗
arXiv2012

An Estimate on the Number of Eigenvalues of a Quasiperiodic Jacobi Matrix of Size $n$ Contained in an Interval of Size $n^{-C}$

We consider infinite quasi-periodic Jacobi self-adjoint matrices for which the three main diagonals are given via values of real analytic functions on the trajectory of the shift $x\rightarrow x+ω$. We assume that the Lyapunov exponent $L(E_{0})$ of the corresponding Jacobi cocycle satisfies $L(E_{0})\geγ>0$. In this setting we prove that the number of eigenvalues $E_{j}^{(n)}(x)$ of a submatrix of size $n$ contained in an interval $I$ centered at $E_{0}$ with $|I|=n^{-C_{1}}$ does not exceed $(\log n)^{C_{0}}$ for any $x$. Here $n\ge n_{0}$, and $n_{0}$, $C_{0}$, $C_{1}$ are constants depending on $γ$ (and the other parameters of the problem).

math-ph↗
arXiv2010

Computability of Brolin-Lyubich Measure

Brolin-Lyubich measure $λ_R$ of a rational endomorphism $R:\riem\to\riem$ with $°R\geq 2$ is the unique invariant measure of maximal entropy $h_{λ_R}=h_{\text{top}}(R)=\log d$. Its support is the Julia set $J(R)$. We demonstrate that $λ_R$ is always computable by an algorithm which has access to coefficients of $R$, even when $J(R)$ is not computable. In the case when $R$ is a polynomial, Brolin-Lyubich measure coincides with the harmonic measure of the basin of infinity. We find a sufficient condition for computability of the harmonic measure of a domain, which holds for the basin of infinity of a polynomial mapping, and show that computability may fail for a general domain.

math.DS↗
arXiv2008

The rate of convergence of the Walk on Spheres Algorithm

In this paper we examine the rate of convergence of one of the standard algorithms for emulating exit probabilities of Brownian motion, the Walk on Spheres (WoS) algorithm. We obtain the complete characterization of the rate of convergence of WoS in terms of the local geomnetry of a domain.

math.PR↗
arXiv2008

Harmonic measure and winding of random conformal paths: A Coulomb gas perspective

We consider random conformally invariant paths in the complex plane (SLEs). Using the Coulomb gas method in conformal field theory, we rederive the mixed multifractal exponents associated with both the harmonic measure and winding (rotation or monodromy) near such critical curves, previously obtained by quantum gravity methods. The results also extend to the general cases of harmonic measure moments and winding of multiple paths in a star configuration.

cond-mat.stat-mech↗
arXiv2007

On computational complexity of Riemann mapping

In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of boundary is given either as a list of pixels, or by a Turing Machine.

math.CV↗