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 κ1,κ2>0 be the autocatalytic rate constants and put λ*=κ1κ2/(κ1+κ2). For deterministic initial configurations whose density converges to m∈(0,1), the total variation distance to equilibrium at time tn(s)=(log n)/2λ*+s/λ* converges to 𝒢(r(m)e−s) for every fixed s∈ℝ, where 𝒢(r)=dTV(𝒩(r,1),𝒩(0,1)), r(m)>0 is explicit, and 𝒩(a,b2) denotes the Gaussian distribution with mean a and variance b2. For the worst-case distance 𝒟n(t) over all initial configurations, we prove 𝒟n(Tn+s/λ*)→𝒫(s), where Tn=(1/min{κ1,κ2}+1/2λ*)log n.The profile 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, 𝒫 depends on the mutation parameters. Both cutoffs have windows of constant order, and we identify the asymptotically worst boundary configurations.
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BibTeX
@misc{xiang2026cutoff,
title = {{Cutoff profiles and boundary effects in the Togashi-Kaneko model of autocatalytic reactions with weak mutations}},
author = {Yangrui Xiang and Wasiur R. KhudaBukhsh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17775},
eprint = {2610.17775},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17775},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Cutoff profiles and boundary effects in the Togashi-Kaneko model of autocatalytic reactions with weak mutations
AU - Yangrui Xiang
AU - Wasiur R. KhudaBukhsh
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.17775
UR - https://arxiv.org/abs/2610.17775
AB - 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∈(0,1), the total variation distance to equilibrium at time t_n(s)=(log n)/2λ_*+s/λ_* converges to 𝒢(r(m)e^−s) for every fixed s∈ℝ, where 𝒢(r)=d_TV(𝒩(r,1),𝒩(0,1)), r(m)>0 is explicit, and 𝒩(a,b^2) denotes the Gaussian distribution with mean a and variance b^2. For the worst-case distance 𝒟_n(t) over all initial configurations, we prove 𝒟_n(T_n+s/λ_*)→𝒫(s), where T_n=(1/min{κ_1,κ_2}+1/2λ_*)log n.The profile 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, 𝒫 depends on the mutation parameters. Both cutoffs have windows of constant order, and we identify the asymptotically worst boundary configurations.
ER -