Exact recursive calculation of circulant permanents: A band of different diagonals inside a uniform matrix

TL;DR

This paper develops a recursive method to compute the permanent of circulant matrices with a band of diagonals, enabling faster calculations and asymptotic analysis, which is crucial for various complex scientific problems.

Contribution

It introduces a finite-order recurrence relation system for circulant matrix permanents with multiple diagonals, advancing computational and analytical capabilities.

Findings

Derived recurrence relations for k=1, 2, 3 diagonals.

Provides a linear-time computational approach.

Lays groundwork for asymptotic analysis of large matrices.

Abstract

We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such recurrence relations which is based on the permanents of the matrices with defects. The proposed system of linear recurrence equations with variable coefficients provides a powerful tool for the analysis of the circulant permanents, their fast, linear time computing and finding their asymptotics in a large-matrix-size limit. The latter problem is an open fundamental problem. Its solution would be tremendously important for a unified analysis of a wide range of the nature's #P-hard problems, including problems in the physics of many-body systems, critical phenomena, quantum computing, quantum field theory, theory of chaos, fractals, theory of graphs, number theory, combinatorics, cryptography, etc.