A Uniform Random-Lattice Tail Bound for the SVP Kissing-Profile Parameter

TL;DR

This paper proves a uniform tail bound for a lattice-dependent parameter related to the shortest vector problem, showing it is typically subexponential in dimension for random lattices.

Contribution

It establishes a stronger, dimension-uniform tail bound for the parameter \\gamma(L) in the Haar--Siegel random-lattice model, confirming it is typically subexponential in dimension.

Findings

\\gamma(L) is bounded by a subexponential function with high probability.

The tail probability of \gamma(L) exceeding T decays as C/T for some constant C.

In the product model, \gamma(L) is eventually bounded by exp(\\sqrt{n}) almost surely.

Abstract

A recent SICOMP paper on classical and quantum algorithms for the shortest vector problem introduced a lattice-dependent parameter \(\gamma(L)\), bounded universally in the exponential sense by \(2^{0.402n+o(n)}\), and conjectured that this parameter is \(2^{o(n)}\) for most lattices. We prove the Haar--Siegel random-lattice version in a stronger, dimension-uniform form. Let \(X_n=\operatorname{SL}_n(\R)/\operatorname{SL}_n(\Z)\), let \(\mu_n\) be its invariant probability measure, and let \(\gamma(L)=\sup_{r\ge1} N_L(r\lambda_1(L))/r^n\), where \(N_L(R)\) counts nonzero vectors of \(L\) of Euclidean norm at most \(R\). For every \(n\ge3\) and every \(T>0\), \[ \mu_n\{L\in X_n:\gamma(L)>T\}\le C T^{-1} \] with an absolute constant \(C\). Consequently, for every sequence \(a_n\to\infty\), \(\gamma(L_n)\le a_n\) with \(\mu_n\)-probability tending to one; in particular \(\gamma(L_n)=2^{o(n)}\) with high probability. In the product model of independent Haar--Siegel lattices, \(\gamma(L_n)\le \exp(\sqrt n)\) eventually almost surely. The proof uses Rogers's second-moment estimate only through a dyadic self-normalization argument around the random scale \(\lambda_1(L)\).