Exponential concentration of terminal stochastic attrition survivors ($\mathbb{P}(|X^N-\sqrt{1-{\alpha }^2}|>\delta )\le 2e^{-(1-{\alpha }^2){\delta }^{2}N/256}$)

Start the symmetric stochastic attrition process at $(N,\lfloor \alpha N\rfloor )$, with $0<\alpha <1$, and let $X^N$ be the final first-population size divided by $N$. The bound holds for all sufficiently large $N$, depending on $\alpha ,\delta >0$. At absorption, one coordinate is zero. If the first survives, the terminal squared difference is $N^2(X^N{)}^2$; if it is extinguished, that difference is nonpositive. In either case the indicated deviation forces an error of at least $\sqrt{1-{\alpha }^2}\delta N^2$ from the limiting squared invariant. The initial rounding error and total conditional drift are only $O(N)$. Apply Azuma-Hoeffding inequality to the centered squared-population near-invariant for stochastic attrition, using its $2N$ increments bounded by $4N$, to obtain the displayed exponent.