A Deceptively Simple Quadratic Recurrence

TL;DR

This paper explores quadratic recurrences, highlighting the challenges in applying linear recurrence techniques and analyzing asymptotic behavior, especially at a critical parameter value where theory and experiments diverge.

Contribution

It provides asymptotic analysis for a quadratic recurrence model across various parameters and discusses the difficulty in understanding the special case at p=1/2.

Findings

Asymptotics are determined for all p ≠ 1/2.

The p=1/2 case remains theoretically challenging.

Discrepancy between theoretical predictions and experimental observations at p=1/2.

Abstract

Standard techniques for treating linear recurrences no longer apply for quadratic recurrences. It is not hard to determine asymptotics for a specific parametrized model over a wide domain of values (all $p \neq 1/2$ here). The gap between theory and experimentation seems insurmountable, however, at a single outlier ($p = 1/2$).