The Type III realisation conjecture of Kirkland and Šmigoc
Abstract
Kirkland and Šmigoc constructed a family of stochastic matrices realising the Type III boundary polynomials in the Karpelevič region and conjectured that, conversely, every stochastic realisation of such a polynomial must come from their construction. We prove this conjecture for the full nonzero parameter range 0<α≤1, for genuine Type III reduced Ito polynomials of order n, fα(x)=xy(xq−(1−α))d−αd, where n=qd+y. For 0<α<1, the proof first reduces every realisation to a two-shift cyclic normal form using the Dmitriev--Dynkin boundary theorem. The remaining argument is finite and combinatorial: Coates' coefficient formula and the equality case of a weighted Turán theorem force the q-cycles associated with the backward edges to split into d complete multipartite classes of equal total weight. A circular-arc telescoping argument then converts this additive equality into the product condition required by Kirkland and Šmigoc. The endpoint α=1 is treated separately. We also explain why the closed endpoint α=0 is degenerate: the literal extension to this endpoint fails, because reducible realisations with closed q-cycles and transient states need not contain the global n-cycle.
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BibTeX
@misc{verbeken2026type,
title = {{The Type III realisation conjecture of Kirkland and Šmigoc}},
author = {Brecht F. Verbeken and Vincent Ginis},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.27219},
eprint = {2607.27219},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.27219},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Type III realisation conjecture of Kirkland and Šmigoc
AU - Brecht F. Verbeken
AU - Vincent Ginis
PY - 2026
DA - 2026/07/01
PB - arXiv
DO - 10.48550/arxiv.2607.27219
UR - https://arxiv.org/abs/2607.27219
AB - Kirkland and Šmigoc constructed a family of stochastic matrices realising the Type III boundary polynomials in the Karpelevič region and conjectured that, conversely, every stochastic realisation of such a polynomial must come from their construction. We prove this conjecture for the full nonzero parameter range 0<α≤1, for genuine Type III reduced Ito polynomials of order n, f_α(x)=x^y(x^q−(1−α))^d−α^d, where n=qd+y. For 0<α<1, the proof first reduces every realisation to a two-shift cyclic normal form using the Dmitriev--Dynkin boundary theorem. The remaining argument is finite and combinatorial: Coates' coefficient formula and the equality case of a weighted Turán theorem force the q-cycles associated with the backward edges to split into d complete multipartite classes of equal total weight. A circular-arc telescoping argument then converts this additive equality into the product condition required by Kirkland and Šmigoc. The endpoint α=1 is treated separately. We also explain why the closed endpoint α=0 is degenerate: the literal extension to this endpoint fails, because reducible realisations with closed q-cycles and transient states need not contain the global n-cycle.
ER -