On boundedness of zeros of the independence polynomial of tori

TL;DR

This paper investigates the boundedness of zeros of the independence polynomial of tori, showing boundedness for balanced tori and unboundedness for highly unbalanced ones, with implications for approximation algorithms.

Contribution

It establishes conditions under which zeros of the independence polynomial are bounded or unbounded for sequences of tori, connecting these results to algorithmic implications.

Findings

Zeros are bounded for balanced tori sequences.

Zeros are unbounded for highly unbalanced tori sequences.

Discusses implications for approximation algorithms and surveys related holomorphic dynamics.

Abstract

We study boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. We prove that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori. Here balanced means that the size of the torus is at most exponential in the shortest side length, while highly unbalanced means that the longest side length of the torus is super exponential in the product over the other side lengths cubed. We discuss implications of our results to the existence of efficient algorithms for approximating the independence polynomial on tori. This project was partially inspired by the relationship between zeros of partition functions and holomorphic dynamics, a relationship that in the last two decades played a prominent role in the field. Besides presenting new results, we survey this relationship and its recent consequences.