On the Spielman-Teng Conjecture

TL;DR

This paper proves that for an $n\times n$ iid subgaussian matrix $M$ with zero-mean, unit-variance entries, the probability that the least singular value satisfies ${\sigma_n(M) \leq \varepsilon n^{-1/2}}$ is asymptotically $(1+o(1))\varepsilon$ up to an exponentially small error, extending the Spielman--Teng conjecture to the subgaussian setting. The authors develop a multi-stage proof: a geometric reduction that ties the problem to a last-row inner product against a kernel vector, a truncation scheme to focus on the smallest singular directions, a Gaussian replacement (Lindeberg-type exchange) leveraging quasi-randomness, and a rescaling that reinterprets the problem in the Gaussian regime where the distribution is well understood. Central techniques include anti-concentration via LCD controls, bootstrapping arguments for regularity events, and universality results (Tao–Vu) that transfer insights from Gaussian matrices to the broader subgaussian class. The work establishes submicroscopic universality of the least singular value down to exponentially small scales (up to a $1+o(1)$ factor) and provides a robust framework for analyzing fine-scale spectral properties beyond previously known polynomial scales. Consequently, the paper delivers a near-complete resolution of the Spielman--Teng conjecture in the subgaussian setting and strengthens the universality philosophy for random matrices.

Abstract

Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$ and variance $1$ and let $σ_n(M)$ denote the least singular value of $M$. We prove that \[\mathbb{P}\big( σ_{n}(M) \leq \varepsilon n^{-1/2} \big) = (1+o(1)) \varepsilon + e^{-Ω(n)}\] for all $0 \leq \varepsilon \ll 1$. This resolves, up to a $1+o(1)$ factor, a seminal conjecture of Spielman and Teng.

Theorems & Definitions (74)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof : Proof of Lemma \ref{['lem:geometric-reduction']}