Elephant Random Walk

The elephant random walk is a random walk whose next step copies or reverses a uniformly chosen earlier step (Schütz and Trimper 2004). Write for its steps and for its position. Given the first steps, choose an index uniformly from 1, 2, ..., , independently of the past, and set equal to the selected step with copying probability , reversing it otherwise. The initial bias is , with .

In the superdiffusive regime , the normalized position converges almost surely to a limit with a probability density function. If is its density given , the density at initial bias is . Pemmasani (2026) proved that, for each , there is such that this density has exactly two local maxima iff . Outside this open interval, including its endpoints, it has a unique local maximum.

There is a unique threshold , numerically about 0.66881, such that exactly for (Pemmasani 2026). The corresponding copying probability is . For the fair start , if is the positive mode above the threshold, there are constants such that

			(1)
			(2)

as . The function is continuous, is continuously differentiable above the threshold, and tends to zero as . Its monotonicity over the whole interval remains open.

For every initial bias and every finite number of steps, with , the distribution of the number of positive steps has at most two modal plateaus, including possible endpoint modes (Pemmasani 2026). Pemmasani credits Claude with a mathematical strategy for the threshold analysis and GPT-6 Astra with help resolving a proof gap concerning positivity of .