The bias interval of the elephant random walk
Arjun Pemmasani
math.PR
Nov 10, 2026 · v1
cond-mat.stat-mech
TL;DR
Lean files in the repository machine-check the finite sign-change arguments and the analytic threshold proof, taking the classical weak limit as a named hypothesis.
Abstract
Guérin, Laulin, Raschel, and Simon conjectured that the initial biases giving the superdiffusive elephant random walk a bimodal limit law form an interval centered at one half, and they described the problem as challenging. We prove this. For every superdiffusive memory, strongly biased starts are unimodal, and the limit density has two local maxima exactly when the bias lies in a proper open interval. Outside it, including its endpoints, the density has a unique local maximum. The interval is nonempty exactly above a single memory threshold, numerically about $0.6688$, whose uniqueness follows from a positive-weight integral for the central curvature of the fair-start density. We control sign changes in the finite walk's first and second differences, including those created at the boundary, and pass them to the limit. The bias boundary is continuous and tends to zero at least quadratically fast as memory tends to one.
Problem
Guérin, Laulin, Raschel, and Simon conjectured that the initial biases giving the superdiffusive elephant random walk a bimodal limit density form an open interval centered at 1/2.
Approach
Second differences of the finite walk's count law satisfy a recurrence with positive interior coefficients, so new sign changes can only arise at the boundary. These sign-change bounds are controlled and passed to the limit density. A positive-weight integral for the central curvature of the fair-start density gives a unique memory threshold. Lean files verify the finite sign arguments and the threshold proof, and Arb interval arithmetic certifies the numerical constants.
Results
For every memory a in (1/2,1) the limit density is bimodal exactly when the bias q lies in an open interval (b_a, 1-b_a), and unimodal otherwise, including at the endpoints. The interval is nonempty exactly for a above a unique threshold a_* ≈ 0.6688. The boundary b_a is continuous and tends to zero at least quadratically as a tends to 1.
