arXiv subjects

Guorui Ma

Publications and source records attributed to Guorui Ma.

5 recordsLinked to original sources

arXiv2026

The sharp threshold for rainbow stackings of random edge-colourings

A rainbow stacking of $m$ independent, uniformly random $r$-edge-colourings of $K_n$ is a tuple of vertex permutations that superimposes the colourings such that no two edges of the same colour overlap. The study of the critical palette size $r$ required for the existence of such stackings was recently initiated by Alon, Defant, and Kravitz [Bull. Lond. Math. Soc., 57, 2025], who bounded the phase transition within a constant-order window around $\frac{m\binom{n}{2}}{2\log(n!)}$. We determine the constant term in this transition. For every fixed $m\ge2$ and every function $\omega(n)\to\infty$, with high probability there is no rainbow stacking if $$r\le \frac{m\binom{n}{2}}{2\log(n!)}+\frac{2m-1}{6}-\frac{\omega(n)}{(\log n)^2},$$ while with high probability there is one if $$r\ge \frac{m\binom{n}{2}}{2\log(n!)}+\frac{2m-1}{6}+\frac{\omega(n)}{(\log n)^2}.$$ Our proof combines a chromatic-polynomial expansion for an auxiliary conflict graph with a refined estimate of the associated weighted permutation sum. Our result yields the exact threshold $\Big\lceil \frac{m\binom{n}{2}}{2\log(n!)}+\frac{2m-1}{6}\Big\rceil$ for a density-one set of integers $n$, resolving a problem of Alon, Defant and Kravitz.

math.CO↗

arXiv2026

Sharp threshold for Hamilton cycles in randomly perturbed sparse graphs

We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $\alpha=\alpha(n)=o(1)$, let $G_{\alpha}$ be an $n$-vertex graph with minimum degree $\delta(G_{\alpha})\ge\alpha n$. We prove that if $$p\ge(1+\varepsilon)\frac{\log(1/\alpha)}{n},$$ then the union $G_{\alpha}\cup G(n,p)$ is Hamiltonian asymptotically almost surely. This significantly strengthens a recent result of Hahn-Klimroth, Maesaka, Mogge, Mohr, and Parczyk by improving the leading constant from 6 to the optimal value of 1. Crucially, we show that this bound on $p$ is best possible when $\alpha n\rightarrow\infty$, thereby establishing the exact probability threshold for Hamiltonicity in this sparse regime. Our proof relies on a robust random expansion lemma, P\'{o}sa's booster lemma, and a sprinkling argument.

math.CO↗

arXiv2026

Geometric construction of superintegrable Poisson projection chains via Poisson centralizers

We introduce a geometric framework for constructing superintegrable systems from Poisson centralizers (commutants) in the Lie-Poisson algebra $S(\mathfrak{g})$ of a complex semisimple Lie algebra. Starting from a chain of reductive subgroups, we study the corresponding invariant Poisson subalgebras and their Poisson centers, and formulate superintegrability in terms of a \emph{Poisson projection chain} of affine Poisson varieties. For a maximal torus $T\subset G$, we prove that the inclusions $S(\mathfrak{g})^G\subset S(\mathfrak{g})^T\subset S(\mathfrak{g})$ determine a superintegrable chain and identify the associated quotient maps $\mathfrak{g}\xrightarrow{\chi_T}\mathfrak{g}//T\xrightarrow{\rho}\mathfrak{g}//G$. The rank (transcendence degree) computations yield the expected dimension split between commuting Hamiltonians and first integrals, and we describe the corresponding symplectic leaves in the intermediate space. Several examples illustrate how the centralizer generators organize into explicit superintegrable Poisson chains.

math-ph↗

arXiv2026

Poisson Centralisers and Polynomial Superintegrability for Magnetic Geodesic Flows on Reductive Homogeneous Spaces

We provide a method for formulating superintegrable magnetic geodesic flows on reductive homogeneous spaces $M=G/A$, with $G$ a compact semisimple Lie group and $A$ a closed subgroup of $G$. In the twisted cotangent bundle $(T^*M,\omega_\varepsilon)$, with $\omega_\varepsilon=\omega_{\mathrm{can}}+\varepsilon\,\pi^*\omega_{\mathrm{KKS}}$ being the canonical plus Kirillov-Kostant-Souriau (KKS) forms, we build two canonical and commuting families of polynomial first integrals: one pulled back from the Lie algebra $\mathfrak{g}$ of $G$ via the magnetic moment map $P$, and one pulled back from a $\mathrm{Ad}(A)$-invariant affine slice of $\mathfrak{m} \cong T_{eA}M$, where $eA$ is the identity of $G/A$. Their common image generates a reduced Poisson algebra obtained from a fiber tensor product, and the natural multiplication map into a Poisson subalgebra of polynomial functions $\mathcal{O}(T^*M) \subset C^\infty(T^*M)$ is Poisson and injective. The center of this fiber tensor product is contained in the Poisson center of the symmetric algebra of $\mathfrak{g}$. In a dense regular locus, the resulting projection chain realises a superintegrable system. As examples, two $\mathrm{SU}(3)$ cases are studied (regular torus and irregular $\mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(1))$ quotients), which illustrate the construction and produce explicit action-angle coordinates.

math-ph↗

arXiv2025

Low-dimensional tori in Calogero-Moser-Sutherland systems

The main result of this paper is an explicit description of the stratification of the phase space of Calogero--Moser--Sutherland (CMS) integrable systems corresponding to Lie groups $SU(n)$. The phase space decomposes into symplectic strata of dimensions $2s$, where $s = 0, 1, \ldots, n - 1$. On each stratum of the positive dimension, we construct natural action-angle coordinates and compute the symplectic form explicitly, showing that every stratum is symplectomorphic to $\mathbb{R}_{> 0}^s \times \mathbb{T}^s$. The zero-dimensional stratum corresponds to the equilibrium point of the multi-time CMS dynamics.

nlin.SI↗