Random walks on Coxeter interchange graphs

TL;DR

This work extends the rapid-mixing paradigm for random walks on graph-interchange structures from classical tournaments (type $A$) to Coxeter tournaments defined on signed graphs of types $B_n$, $C_n$, and $D_n$. By developing the Coxeter interchange graphs $\mathrm{IntGr}(\Phi,{\bf s})$ and analyzing their detailed network structure via Z-frames, trail decompositions, and extended networks, the authors establish contractive couplings under path coupling with type-appropriate metric re-weighting, yielding rapid mixing bounds. They prove connectivity and polynomial diameter, and provide refined mixing-time bounds that depend on degree $d$ and, in type $C_n$, the maximal crystal degree $\gamma$, highlighting the role of crystals in the analysis. The results connect probabilistic sampling on fibers of Coxeter permutahedra $\Pi_\Phi$ with the richer combinatorial and geometric structure of signed graphs, offering potential avenues for approximate counting and further Coxeter-generalizations. Overall, the paper demonstrates that simple random walks on these sophisticated interchange graphs mix rapidly, enabling efficient sampling of Coxeter tournaments with prescribed score sequences and linking discrete probability with Coxeter combinatorics and geometry.

Abstract

A tournament is an orientation of a graph. Vertices are players and edges are games, directed away from the winner. Kannan, Tetali and Vempala and McShine showed that tournaments with given score sequence can be rapidly sampled, via simple random walks on the interchange graphs of Brualdi and Li. These graphs are generated by the cyclically directed triangle, in the sense that traversing an edge corresponds to the reversal of such a triangle in a tournament. We study Coxeter tournaments on Zaslavsky's signed graphs. These tournaments involve collaborative and solitaire games, as well as the usual competitive games. The interchange graphs are richer in complexity, as a variety of other generators are involved. We prove rapid mixing by an intricate application of Bubley and Dyer's method of path coupling, using a delicate re-weighting of the graph metric. Geometric connections with the Coxeter permutahedra introduced by Ardila, Castillo, Eur and Postnikov are discussed.

Figures (30)

• Figure 1: The permutahedron $\Pi_3\subset{\mathbb R}^4$, projected into ${\mathbb R}^3$. Its 24 vertices correspond to the permutations of the standard win sequence ${\bf w}_4=(0,1,2,3)$.
• Figure 2: The Coxeter permutahedron of type $C_3$.
• Figure 3: The snare drum interchange graph ${\rm IntGr}(C_3,{\bf s})$, when ${\bf s}=(-1,0,1)$, is the Cartesian product of a double edge and the crystal (see Figure \ref{['F_crystal']}).
• Figure 4: The tambourine interchange graph ${\rm IntGr}(C_3,{\bf s})$, when ${\bf s}=(0,0,0)$ is the center of the type $C_3$ permutahedron. This graph is the Cartesian product of a single edge and the cube of double edges.
• Figure 5: The cyclic and balanced triangles $\Delta_c$ and $\Delta_b$ are generators in all types $B_n$, $C_n$ and $D_n$.

Theorems & Definitions (40)

• Theorem 1: KMP23
• Definition 2
• Theorem 3: Path coupling, BD97
• Theorem 4
• Theorem 5
• Definition 6
• Definition 7
• Definition 8
• Lemma 9
• proof