Asymptotic behavior of the non-linear dynamics for (pi,k) across rounds

Characterize the asymptotic behavior as i → ∞ of the distribution (pi,k)k∈N describing the proportion of balls that lie in boxes with exactly k balls after round i in the three-throw mystery-balls process with r = ⌊τ t⌋ boxes per throw and independent uniform throws; specifically, analyze the associated non-linear recurrence to determine the existence and form of any limiting distribution and the conditions on τ under which it converges to zero mass versus a positive residual mass.

Background

To analyze the success probability of the coefficient-recovery game, the paper defines pi,k as the expected fraction of balls in boxes with k occupants after round i and derives a non-linear update rule relating (pi+1,k) to (pi,k). Winning corresponds to the total remaining fraction tending to zero; losing corresponds to convergence to a positive limit.

While the authors provide numerical evidence and reliable approximations for (pi,k), they explicitly state that a full analytical description of its asymptotic behavior remains out of reach in the paper, leaving open the rigorous characterization of the dynamics and its dependence on τ.