An exactly solvable Markov chain with a fully characterized lumping lattice: equilateral geometry, three-eigenvalue spectrum, and exact coarse-grained relaxation

Abstract

Coarse-graining a Markov chain by merging microstates (lumping) almost always destroys Markovianity: the reduced dynamics acquires memory, and exact reduced descriptions require semi-Markov constructions. We introduce and solve a class of reversible finite Markov chains for which the opposite extreme holds. The class LD(u,a,Delta,s) is a Metropolis walk, at inverse temperature x, on a "double-pencil" family of subsets: the u one-point deletions and a one-point additions of a reference configuration U, with two energy levels separated by Delta, and pair proposals drawn from an ambient alphabet of s sites. We prove that (i) the transition graph is the complete graph K_{u+a} with multiplicity one, a consequence of the equilaterality of the unit Hamming sphere; (ii) the spectrum consists of exactly three distinct eigenvalues, 1, b(x) = 1 - (u + a e^{-x Delta}) / C(s,2) and c = 1 - (u + a) / C(s,2), with multiplicities 1, u and a-1, so that the relaxation gap saturates at u / C(s,2) instead of closing or opening exponentially (a non-Arrhenius relaxation in the presence of Arrhenius populations); and (iii) for x > 0 the lumping lattice is completely characterized: a partition is lumpable if and only if every class that contains excited states either is pure or contains all excited states (the absorbing family), which includes, as boundary cases, all refinements of the two-sector partition and the trivial partition. To our knowledge this is the first nontrivial model whose full lumping lattice is explicitly characterized, answering constructively the recently posed question of whether exact Markovian coarse-graining by lumping is possible beyond special symmetric cases. Every statement is certified by exhaustive or large-sample exact computation with integer arithmetic certificates; a worked instance LD(28,54,1,136), arising from a constant-energy variety of a classical [[137,124,2]] error-correcting code ensemble, is analyzed with all quantities audited to machine precision. We also quantify exactly the share of relaxation entropy production (Kullback-Leibler decay) visible at each coarse-graining level. Limitations (finite size, equilibrium, extreme symmetry) and a driven non-equilibrium extension are discussed.

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