Jonathan Leake

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math.CO2026

Trickle-down Theorems via C-Lorentzian Polynomials II: Pairwise Spectral Influence and Improved Dobrushin's Condition

Jonathan Leake, Shayan Oveis Gharan

Let $I^¼$ be a probability distribution on a multi-state spin system on a set $V$ of sites; equivalently, a $d$ -partite simplicial complex with distribution $I^¼$ on maximal faces. F…

math.CO2025

Optimal Trickle-Down Theorems for Path Complexes via C-Lorentzian Polynomials with Applications to Sampling and Log-Concave Sequences

Jonathan Leake, Kasper Lindberg, Shayan Oveis Gharan

Let $X$ be a $d$ -partite $d$ -dimensional simplicial complex with parts $T_{1}, \dots, T_{d}$ and let $I^¼$ be a distribution on the facets of $X$ . Informally, we say $(X, I^¼)$ is a path co…

math.CO2024

Compatibility of Real-Rooted Polynomials with Mixed Signs

Jonathan Leake, Nick Ryder

We characterize compatible families of real-rooted polynomials, allowing both positive and negative leading coefficients. Our characterization naturally generalizes the same-sign c…

math.CO2024

From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP

Leonid Gurvits, Nathan Klein, Jonathan Leake

We give simply exponential lower bounds on the probabilities of a given strongly Rayleigh distribution, depending only on its expectation. This resolves a weak version of a problem…