Trefethen conjecture on the typical growth of GEPP

Prove that for every positive constant ε and every positive constant C, and for all sufficiently large n, the probability that an n-by-n matrix with independent N(0,1) entries has Gaussian elimination with partial pivoting growth greater than n^{1/2+ε} is at most n^{-C}.

Background

The paper discusses the probability distribution of the growth factor of Gaussian elimination with partial pivoting (GEPP) on matrices whose entries are independent standard Gaussian random variables. Earlier numerical experiments suggested exponential decay in the growth factor, but the paper disproves those stronger exponential-tail conjectures by proving an inverse quasi-polynomial lower bound on the probability of large growth.

The remaining conjecture attributed to Trefethen asks for a high-probability upper bound on the typical growth: beyond n{1/2+ε}, the probability of observing larger growth should decay faster than every fixed inverse polynomial n{-C}. The paper's lower bound does not resolve this conjecture, since it is compatible with such an upper bound at polynomial growth scales.

References

\begin{conjecture}[Trefethen] For all positive constants $$ and $C$, and for all sufficiently large $n$, the probability that an $n$-by-$n$ matrix of independent $N (0,1)$ entries has growth greater than $n{1/2 + }$ is at most $n{-C}.$ \end{conjecture}

— Large Growth Happens: Gaussian Elimination with Partial Pivoting on Random Matrices  (2610.08701 - Spielman et al., 6 Oct 2026) in Conjecture [Trefethen], Section 1, Introduction