Occupation counts of rarely switching Markov chains: stratified local laws, support selection and the boundary layer
Year: 2026
Type: preprint
Abstract: A Markov chain with exit rates $q_i$ runs for $N$ steps with transition matrix $I+(T/N)Q$. On each irreducible face that carries start mass, we prove a local limit theorem for the occupation counts with an explicit constant and relative error $O(1/T)$, uniformly for $1\le T\le\varepsilon N$ with $\varepsilon\max_iq_i<1$ and on compact subsets of the open face. Up to $O(1)$, minus the log-probability of counts $n$ with support $F$ is $|F|-1$ times $\log(N/\sqrt T)$ plus $N$ times the Donsker and Varadhan functional of $I+(T/N)Q$ at $n/N$. Every admissible face has an explicit power of $T$ in its largest probability for large $T\le N^{1/3}$, and a constant given by a finite-dimensional variational formula. For $T=τ\log N$ the face of the mode minimizes $τλ_F+|F|-1$ at first order, with $λ_F$ the exit rate. If an admissible face has $Δ\ge1$ more states than another and an exit rate smaller by $δ>0$, we locate every crossing of their largest probabilities for large $T\le N^{1/3}$ up to $O(1/\log N)$. For two irreducible faces the crossing is unique for large $N$ in $1\le T\le\varepsilon N$, with coefficient $1/2$ of $\log\log N$. Near a facet, the continuous-time occupation density is the facet density times an explicit kernel, with relative error $O(1/T)$ at times $x\le cT$ off the facet, $c$ small. The counts obey a lattice form with extra error $T^2/N$ for $T\le\varepsilon_1\sqrt N$. The kernel is Bessel for $x=o(T)$ at facet profiles where the removed state has equal entry and exit laws, and at a profile where the rate has a first correction, of order $m\ge3$, if and only if $x=o(T^{1-2/m})$. On the complete graph with the uniform start, each face of at least 2 states crosses a vertex exactly once.
Source: arXiv (Cornell University)
Author: Pemmasani, Arjun
Institution: Harvey Mudd College
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220989379