ترغب بنشر مسار تعليمي؟ اضغط هنا

Fluctuations of the product of random matrices and generalized Lyapunov exponent

103   0   0.0 ( 0 )
 نشر من قبل Christophe Texier
 تاريخ النشر 2019
  مجال البحث فيزياء
والبحث باللغة English
 تأليف Christophe Texier




اسأل ChatGPT حول البحث

I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products $Pi_n=M_nM_{n-1}cdots M_1$, where $M_i$s are i.i.d.. Following Tutubalin [Theor. Probab. Appl. {bf 10}, 15 (1965)], the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group. The formalism is illustrated by considering products of random matrices from the group $mathrm{SL}(2,mathbb{R})$ where explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the one-dimensional Schrodinger equation where the random potential is a Levy noise (derivative of a Levy process). In this case, I obtain a general formula for the variance of $ln||Pi_n||$ and for the variance of $ln|psi(x)|$, where $psi(x)$ is the wavefunction, in terms of a single integral involving the Fourier transform of the invariant density of the matrix product. Finally I discuss the continuum limit of random matrix products (matrices close to the identity ). In particular, I investigate a simple case where the spectral problem providing the generalized Lyapunov exponent can be solved exactly.



قيم البحث

اقرأ أيضاً

154 - Jacek Grela , Thomas Guhr 2016
We use supersymmetry to calculate exact spectral densities for a class of complex random matrix models having the form $M=S+LXR$, where $X$ is a random noise part $X$ and $S,L,R$ are fixed structure parts. This is a certain version of the external fi eld random matrix models. We found two-fold integral formulas for arbitrary structural matrices. We investigate some special cases in detail and carry out numerical simulations. The presence or absence of a normality condition on $S$ leads to a qualitatively different behavior of the eigenvalue densities.
140 - Christophe Texier 2019
Products of random matrix products of $mathrm{SL}(2,mathbb{R})$, corresponding to transfer matrices for the one-dimensional Schrodinger equation with a random potential $V$, are studied. I consider both the case where the potential has a finite secon d moment $langle V^2rangle<infty$ and the case where its distribution presents a power law tail $p(V)sim|V|^{-1-alpha}$ for $0<alpha<2$. I study the generalized Lyapunov exponent of the random matrix product (i.e. the cumulant generating function of the logarithm of the wave function). In the high energy/weak disorder limit, it is shown to be given by a universal formula controlled by a unique scale (single parameter scaling). For $langle V^2rangle<infty$, one recovers Gaussian fluctuations with the variance equal to the mean value: $gamma_2simeqgamma_1$. For $langle V^2rangle=infty$, one finds $gamma_2simeq(2/alpha),gamma_1$ and non Gaussian large deviations, related to the universal limiting behaviour of the conductance distribution $W(g)sim g^{-1+alpha/2}$ for $gto0$.
In this paper, we study the product of two complex Ginibre matrices and the loop equations satisfied by their resolvents (i.e. the Stieltjes transform of the correlation functions). We obtain using Schwinger-Dyson equation (SDE) techniques the genera l loop equations satisfied by the resolvents. In order to deal with the product structure of the random matrix of interest, we consider SDEs involving the integral of higher derivatives. One of the advantage of this technique is that it bypasses the reformulation of the problem in terms of singular values. As a byproduct of this study we obtain the large $N$ limit of the Stieltjes transform of the $2$-point correlation function, as well as the first correction to the Stieltjes transform of the density, giving us access to corrections to the smoothed density. In order to pave the way for the establishment of a topological recursion formula we also study the geometry of the corresponding spectral curve. This paper also contains explicit results for different resolvents and their corrections.
We investigate eigenvalue moments of matrices from Circular Orthogonal Ensemble multiplicatively perturbed by a permutation matrix. More precisely we investigate variance of the sum of the eigenvalues raised to power $k$, for arbitrary but fixed $k$ and in the limit of large matrix size. We find that when the permutation defining the perturbed ensemble has only long cycles, the answer is universal and approaches the corresponding moment of the Circular Unitary Ensemble with a particularly fast rate: the error is of order $1/N^3$ and the terms of orders $1/N$ and $1/N^2$ disappear due to cancellations. We prove this rate of convergence using Weingarten calculus and classifying the contributing Weingarten functions first in terms of a graph model and then algebraically.
Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these results.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا