Localization and delocalization of random interfaces

非均匀随机矩阵的局域化-离域化转变

A localization-delocalization transition for nonhomogeneous random matrices

本文研究了非均匀自伴高斯随机矩阵在谱边缘的局域化-离域化转变，发现该转变由每行非零项数 $d$ 与 $\log N$ 的关系决定，且此现象对稀疏模式具有普适性。

Estimates on path delocalization for copolymers at selective interfaces

We consider a directed random walk model of a random heterogeneous polymer in the proximity of an interface separating two selective solvents. This model exhibits a localization/delocalization transition. A positive value of the free energy corresponds to the localized regime and strong results on the polymer path behavior are known in this case. We focus on the interior of the delocalized phase, which is characterized by the free energy equal to zero, and we show in particular that in this regime there are O(log N) monomers in the unfavorable solvent (N is the length of the polymer). The prev…

Localization and delocalization of eigenvectors for heavy-tailed random matrices

Consider an n x n Hermitian random matrix with, above the diagonal, independent entries with alpha-stable symmetric distribution and 0 < alpha < 2. We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as n goes to infinity. When 1 < alpha < 2 we give vanishing bounds on the Lp-norm of the eigenvectors normalized to have unit L2-norm goes to 0. On the contrary, when 0 < alpha < 2/3, we prove that these eigenvectors are localized.

Local semicircle law under moment conditions. Part II: Localization and delocalization

We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with upper triangular entries being independent identically distributed random variables with mean zero and unit variance. We additionally suppose that $\mathbb E |X_{11}|^{4 + δ} =: μ_{4+δ} < C$ for some $δ> 0$ and some absolute constant $C$. Under these conditions we show that the typical Kolmogorov distance between the empirical spectral distribution function of eigenvalues of $n^{-1/2} {\bf X}$ and Wigner's semicircle law is of order $1/n$ up to some logarithmic correction factor. As a direct consequence of this result we…

Localization-delocalization phenomena for random interfaces

We consider d-dimensional random surface models which for d=1 are the standard (tied-down) random walks (considered as a random ``string''). In higher dimensions, the one-dimensional (discrete) time parameter of the random walk is replaced by the d-dimensional lattice \Z^d, or a finite subset of it. The random surface is represented by real-valued random variables ϕ_i, where i is in \Z^d. A class of natural generalizations of the standard random walk are gradient models whose laws are (formally) expressed as P(dϕ) = 1/Z \exp[-\sum_{|i-j|=1}V(ϕ_i-ϕ_j)] \prod_i dϕ_i, V:\R -> R^+ convex, and with…

Directed polymer in a random medium of dimension 1+3 : multifractal properties at the localization/delocalization transition

We consider the model of the directed polymer in a random medium of dimension 1+3, and investigate its multifractal properties at the localization/delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyse the statistics of the position weights $w_L(\vec r)$ of the end-point of the polymer of length $L$ via the moments $Y_q(L) = \sum_{\vec r} [w_L(\vec r)]^q$. We measure the generalized exponents $τ(q)$ and $\tilde τ(q)$ governing the decay of the typical values $Y^…