Approximate formulation of the probability that the Determinant or Permanent of a matrix undergoes the least change under perturbation of a single element

TL;DR

This paper develops approximate formulas for the probability that the determinant or permanent of a matrix changes minimally when a single element is perturbed, assuming independence among terms and applying to various matrix classes.

Contribution

It introduces an approximate method assuming independence of permanent terms to estimate minimal change probabilities under perturbations.

Findings

Derived formulas for determinant and permanent change probabilities

Identified new integer sequences related to matrix properties

Applied formulas to different matrix classes

Abstract

In an earlier paper, we discussed the probability that the determinant of a matrix undergoes the least change upon perturbation of one of its elements, provided that most or all of the elements of the matrix are chosen at random and that the randomly chosen elements have a fixed probability of being non-zero. In this paper, we derive approximate formulas for that probability by assuming that the terms in the permanent of a matrix are independent of one another, and we apply that assumption to several classes of matrices. In the course of deriving those formulas, we identified several integer sequences that are not listed on Sloane's Web site.