arXiv 0810.10062008-10math-phmath.MP

Localization on quantum graphs with random edge lengths

Frédéric Klopp, Konstantin Pankrashkin

DOI：: 10.1007/s11005-009-0293-8

英文摘要：

The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple $\ZZ^d$-lattice with $δ$-type boundary conditions at the vertices, and we assume that the edge lengths are randomly independently identically distributed. Under the assumption that the coupling constant at the vertices does not vanish, we show that the operator exhibits the Anderson localization at the bottom of the spectrum almost surely. We also study the case of other spectral edges.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Scaling diagram for the localization length at a band edge

A weak-coupling scaling diagram for the Lyapunov exponent and the integrated density of states near a band edge of a random Jacobi matrix is obtained. The analysis is based on the use of a Fokker-Planck operator describing the drift-diffusion of the Prüfer phases.

2008-02-20math-phmath.MP

Localization lengths for Schroedinger operators on Z^2 with decaying random potentials

We study a class of Schrödinger operators on $\Z^2$ with a random potential decaying as $|x|^{-\dex}$, $0<\dex\leq\frac12$, in the limit of small disorder strength $λ$. For the critical exponent $\dex=\frac12$, we prove that the localization length of eigenfunctions is bounded below by $2^{λ^{-\frac14+η}}$, while for $0<\dex<\frac12$, the lower bound is $λ^{-\frac{2-η}{1-2\dex}}$, for any $η>0$. These estimates "interpolate" between the lower bound $λ^{-2+η}$ due to recent work of Schlag-Shubin-Wolff for $\dex=0$, and pure a.c. spectrum for $\dex>\frac12$ demonstrated in recent work of Bourgai…

2005-10-26math-phmath.MP

$1+1$连续有向聚合物的局域化长度

Localization length of the $1+1$ continuum directed random polymer

本文研究了$1+1$连续有向聚合物的局域化长度，证明了其在热力学极限下依分布收敛，导出了极限分布的显式密度公式，并证实了密度的$\tfrac32$次幂衰减规律。

2023-02-20math-phmath.MPmath.PR

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.

2014-05-04math-phmath.MPmath.SP

Localization and mobility edge for sparsely random potentials

In this paper we consider sparsely random potentials in 5 or more dimensional cubic lattice and exhibit localized and extended states. We identify also the mobility edge for a class of potentials going to infinity at infinity. Our treatment includes a large class of unperturbed selfadjoint opeartors commuting with the usual Laplacian on the lattice.

1998-11-04cond-mat.dis-nnmath-phmath.MP

Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3

We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are similar to those obtained by Schlag, Shubin and Wolff for dimensions one and two. We prove that with probability one, most eigenfunctions have localization lengths bounded from below by $O(\frac{λ^{-2}}{\log\frac1λ})$, where $λ$ is the disorder strength. This is achieved by time-dependent methods which generalize those developed by Erdös and Yau to the lattice and non-Gaussian case. In addition, we show that the macroscopic limit of the corresponding lat…

2005-03-24math-phmath.MPmath.SP

更多相关论文收起更多论文

Localization on a quantum graph with a random potential on the edges

We prove spectral and dynamical localization on a cubic-lattice quantum graph with a random potential. We use multiscale analysis and show how to obtain the necessary estimates in analogy to the well-studied case of random Schroedinger operators.

2006-12-29cond-mat.mes-hallmath-phmath.MP

Anderson Localization at Band Edges for Random Magnetic Fields

We consider a magnetic Schrödinger operator in two dimensions. The magnetic field is given as the sum of a large and constant magnetic field and a random magnetic field. Moreover, we allow for an additional deterministic potential as well as a magnetic field which are both periodic. We show that the spectrum of this operator is contained in broadened bands around the Landau levels and that the edges of these bands consist of pure point spectrum with exponentially decaying eigenfunctions. The proof is based on a recent Wegner estimate obtained in \cite{EH2} and a multiscale analysis.

2011-03-19math-phmath.FAmath.MP

Localization for Linearly Edge Reinforced Random Walks

We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on non-amenable graphs. While we rely on the description of the LRRW as a mixture of Markov chains, the proof does not use the magic formula. We also derive analogous results for the vertex reinforced jump process.

2012-03-19math-phmath.MPmath.PR

一维圆柱上洛伦兹镜面模型局域化长度的多项式界

Polynomial bound for the localization length of Lorentz mirror model on the 1D cylinder

University of Pennsylvania（宾夕法尼亚大学）

通过条件圆柱局域化定理和双开关手术与双计数论证，证明了偶数圆柱上洛伦兹镜面模型和曼哈顿模型局域化长度的多项式上界。

2026-07-28math-phmath.MPmath.PR

Band edge localization beyond regular Floquet eigenvalues

We prove that localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in $L^2(\mathbb{R}^d)$ is universal. By this we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furt…

2020-05-07math-phmath.APmath.MP

Bethe ansatz description of edge-localization in the open-boundary XXZ spin chain

At large values of the anisotropy Δ, the open-boundary Heisenberg spin-1/2 chain has eigenstates displaying localization at the edges. We present a Bethe ansatz description of this `edge-locking' phenomenon in the entire Δ>1 region. We focus on the simplest spin sectors, namely the highly polarized sectors with only one or two overturned spins, i.e., one-particle and two-particle sectors. Edge-locking is associated with pure imaginary solutions of the Bethe equations, which are not commonly encountered in periodic chains. In the one-particle case, at large anisotropies there are two eigenstate…

2013-12-23cond-mat.stat-mechcond-mat.str-elmath-ph

在莫尔晶格边缘观察到的线性和非线性光局域化

Observation of linear and nonlinear light localization at the edges of moiré lattices

本研究首次在光子系统中实验观察了截断莫尔晶格边缘和角落的线性与非线性光局域化现象，发现角落模式与体激发的局域化性质差异显著，并揭示了非线性下从准局域态到表面孤子的转变。

2023-01-05cond-mat.quant-gasmath-phmath.MP

Scalar Wave Propagation in Random, Amplifying Media: Influence of Localization Effects on Length and Time Scales and Threshold Behavior

We present a detailed discussion of scalar wave propagation and light intensity transport in three dimensional random dielectric media with optical gain. The intrinsic length and time scales of such amplifying systems are studied and comprehensively discussed as well as the threshold characteristics of single- and two-particle propagators. Our semi-analytical theory is based on a self-consistent Cooperon resummation, representing the repeated self-interference, and incorporates as well optical gain and absorption, modeled in a semi-analytical way by a finite imaginary part of the dielectric fu…

2011-06-15cond-mat.dis-nncond-mat.mtrl-scimath-ph

一般维度下晶格Anderson-Bernoulli模型边缘附近的局域化

Localization near the edge for the lattice Anderson-Bernoulli model on general dimension

University of Pennsylvania（宾夕法尼亚大学）；Peking University（北京大学）；California Institute of Technology（加州理工学院）

本文在任意维度下证明了具有Bernoulli势的晶格Anderson模型在谱边缘附近的Anderson局域化，解决了Bourgain-Kenig遗留问题，核心创新是概率离散唯一延拓原理（PDUC）。

2026-10-21math-phmath.APmath.DS