Morse functions constructed by random walks

TL;DR

This paper introduces a method to construct random Morse functions on surfaces using random walks, analyzes their properties, and explores the space of such functions with constraints on critical points and boundary conditions.

Contribution

It presents a novel approach to generating and studying Morse functions on surfaces via random walks, including distribution computations and characterization of function spaces.

Findings

Distribution of Morse functions derived from random walks

Identification of a small set of Morse functions close to any given function

Analysis of Morse functions with bounded critical points and boundary conditions

Abstract

We construct random Morse functions on surfaces by random walk and compute related distributions. We study the space of Morse functions through these random variables. We consider subspaces characterized by the surfaces with boundary obtained by cutting the closed domain surface of the Morse function at the levels of regular values. We consider Morse functions having a bounded number of critical points and one single local minimum. We find a small set of Morse functions which are close enough to any other Morse function in the sense that they share the same characterizing surfaces with boundary.