# An analytical Lieb-Sokal lemma

## TL;DR

This paper provides analytical estimates for the zeros of real stable polynomials resulting from differential operators, with applications to probability distributions and problems like Kadison-Singer.

## Contribution

It introduces new analytical bounds on zero locations of multiaffine real stable polynomials and applies these to characteristic polynomials in probabilistic and operator theory contexts.

## Key findings

- Derived bounds on zeros of $q(
abla)p$ for multiaffine polynomials
- Applied estimates to expected characteristic polynomials in Strongly Rayleigh distributions
- Connected results to the Kadison-Singer problem

## Abstract

A polynomial $p \in \mathbb{R}[z_1, \cdots, z_n]$ is called real stable if it is non-vanishing whenever all the variables take values in the upper half plane. A well known result of Elliott Lieb and Alan Sokal states that if $p$ and $q$ are $n$ variate real stable polynomials, then the polynomial $q(\partial)p := q(\partial_1, \cdots, \partial_n)p$, is real stable as well. In this paper, we prove analytical estimates on the locations on the zeroes of the real stable polynomial $q(\partial)p$ in the case when both $p$ and $q$ are multiaffine, an important special case, owing to connections to negative dependance in discrete probability. As an application, we prove a general estimate on the expected characteristic polynomials upon sampling from Strongly Rayleigh distributions. We then use this to deduce results concerning two classes of polynomials, mixed characteristic polynomials and mixed determinantal polynomials, that are related to the Kadison-Singer problem.