Cutoff profiles and boundary effects in the Togashi-Kaneko model of autocatalytic reactions with weak mutations

Abstract

We determine explicit bulk and worst-case total variation cutoff profiles for the configuration law of a two-species Togashi--Kaneko model with a fixed population of $n$ particles and weak mutations. Boundary initial conditions introduce an additional logarithmic growth phase and change both the leading mixing-time coefficient and the limiting cutoff profile. Let $κ_1,κ_2>0$ be the autocatalytic rate constants and put $λ_*=κ_1κ_2/(κ_1+κ_2)$. For deterministic initial configurations whose density converges to $m\in(0,1)$, the total variation distance to equilibrium at time $t_n(s)=\frac{\log n}{2λ_*}+\frac{s}{λ_*}$ converges to $\mathcal G(r(m)e^{-s})$ for every fixed $s\in\mathbb R$, where $\mathcal G(r)=d_{\mathrm{TV}}(\mathcal N(r,1),\mathcal N(0,1))$, $r(m)>0$ is explicit, and $\mathcal N(a,b^2)$ denotes the Gaussian distribution with mean $a$ and variance $b^2$. For the worst-case distance $\mathscr D_n(t)$ over all initial configurations, we prove $\mathscr D_n(T_n+s/λ_*)\to\mathcal P(s)$, where $T_n=\left(\frac{1}{\min\{κ_1,κ_2\}}+\frac{1}{2λ_*}\right)\log n$.The profile $\mathcal P$ is the total variation distance between the equilibrium Gaussian and a Gaussian mixture whose random displacement is determined by a gamma-distributed amplitude arising from early growth at the boundary. Unlike the bulk profile, $\mathcal P$ depends on the mutation parameters. Both cutoffs have windows of constant order, and we identify the asymptotically worst boundary configurations.