Closed-form characterization of the equilibrium point in the infinite model

Derive a closed-form expression (or exact characterization) of the equilibrium point vector π for the infinite-population limit of the symmetric-agent model with d = 2, where π satisfies the recursion π_{i+1} = π_i^2 for all i ∈ Z and the balance equation ∑_{i ≥ 1} π_i − ∑_{i ≤ 0} (1 − π_i) = 0. In particular, determine π_0 and thereby the full equilibrium vector π explicitly.

Background

In the large-n limit, the authors model the system as a density-dependent Markov chain and use Kurtz’s theorem to obtain a deterministic ODE system. At equilibrium (for d = 2), the solution satisfies π_{i+1} = π_i^2, reducing the problem to determining π_0 via a lacunary series equation derived from the zero-mean constraint on tokens.

They note that lacunary series generally lack analytic continuations and, to the best of their knowledge, no closed-form exists for this specific series, preventing an explicit solution for π. An explicit characterization of π would yield exact tail probabilities and sharpen the asymptotic analysis of the stationary distribution.