Universal edge scaling limit of discrete 1d random Schrödinger operator with vanishing potentials

2023/06/29 by Yi Han, Han, Yi · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory in Mathematical Physics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2306.17001

openalex publication_date 2023/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider random Schrödinger operators H_n defined on [0,n]∩ℤ with zero boundary conditions: (H_nψ)_ℓ=ψ_ℓ-1+ψ_ℓ+1+σ\frac\mathfraka(ℓ)n^αψ_ℓ, ℓ=1,⋯,n, ψ₀=ψ_n+1=0, where σ>0 is a fixed constant, \mathfraka(ℓ), ℓ=1,⋯,n, are i.i.d. random variables with mean 0, variance 1 and fast decay. The bulk scaling limit has been investigated in \citekritchevski2011scaling: at the critical exponent α= (1)/(2), the spectrum of H_n, centered at E∈(-2,2)∖\0\ and rescaled by n, converges to the Sch_τ process and does not depend on the distribution of \mathfraka(ℓ). We study the scaling limit at the edge. We show that at the critical value α=(3)/(2), if we center the spectrum at 2 and rescale by n², then the spectrum converges to a new random process depending on σ but not the distribution of \mathfraka(ℓ). We use two methods to describe this edge scaling limit. The first uses the method of moments, where we compute the Laplace transform of the point process, and represent it in terms of integrated local times of Brownian bridges. Then we show that the rescaled largest eigenvalues correspond to the lowest eigenvalues of the random Schrödinger operator -(d²)/(dx²)+σb_x' defined on [0,1] with zero boundary condition, where b_x is a standard Brownian motion. This allows us to compute precise left and right tails of the rescaled largest eigenvalue and compare them to Tracy-Widom beta laws. We also show if we shift the potential \mathfraka(ℓ) by a state-dependent constant and take α=(1)/(2), then for a particularly chosen state-dependent shift, the rescaled largest eigenvalues converge to the Tracy-Widom beta distribution.

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