Random walks on Coxeter interchange graphs

A tournament is an orientation of the complete graph $K_{n}$, encoded as some $T=(w_{ij}:i>j)$ with all $w_{ij}\in\{0,1\}$. Each edge $\{i,j\}$ in $K_{n}$ is oriented as $i\to j$ if $w_{ij}=1$ or $i\leftarrow j$ if $w_{ij}=0$. We think of each edge as a game, directed away from the winner. The win sequence

lists the total number of wins by each player, where ${\bf e}_{i}\in{\mathbb{Z}}^{n}$ are the standard basis vectors. We let ${\bf w}_{n}=(0,1,\ldots,n-1)$ denote the standard win sequence, corresponding to the transitive (acyclic) tournament in which $w_{ij}=1$ for all $i>j$. In a sense, ${\bf w}_{n}$ and its permutations are as “spread out” as possible.

Results by Rado [29] and Landau [21] imply that the set ${\rm Win}(n)$ of all win sequences is precisely the set of lattice points in the permutahedron $\Pi_{n-1}$, that is, ${\rm Win}(n)={\mathbb{Z}}^{n}\cap\Pi_{n-1}$. We recall that $\Pi_{n-1}$ is a classical polytope in discrete geometry (see, e.g., Ziegler [38]), obtained as the convex hull of ${\bf w}_{n}$ and its permutations, see Figure 1. By Stanley [33], win sequences are in bijection with spanning forests $F\subset K_{n}$. (The volume of $\Pi_{n-1}$ is the number of spanning trees $T\subset K_{n}$.) See Postnikov [28] for generalizations.

It is convenient to make a linear shift

where ${\bf 1}_{n}=(1,\ldots,1)\in{\mathbb{Z}}^{n}$. Note that this re-centers the polytope at the origin ${\bf 0}_{n}=(0,\ldots,0)\in{\mathbb{Z}}^{n}$. The score sequence

associated with the win sequence ${\bf w}(T)$ of a tournament $T$, is given by

This shift corresponds to awarding a $\pm 1/2$ point for each win/loss. We let ${\rm Score}(n)$ denote the set of all possible score sequences.

Although the set ${\rm Score}(n)$ has a simple, geometric description, the set ${\rm Tour}(n,{\bf s})$, of tournaments with given score sequence ${\bf s}$, appears to be quite combinatorially complex.

Kannan, Tetali and Vempala [16] investigated simple random walk as a way of sampling from ${\rm Tour}(n,{\bf s})$. However, rapid mixing was proved only for ${\bf s}$ sufficiently close to ${\bf 0}_{n}$. McShine [25] established rapid mixing in time $O(n^{3}\log n)$, for all ${\bf s}\in{\rm Score}(n)$, by an elegant application of Bubley and Dyer’s [7] method of path coupling, which was relatively new at the time. See Section 2.3 below for an overview. We note that Sarkar [30] has shown that mixing takes $\Omega(n^{3})$ for some sequences.

More specifically, in [16, 25], the random walks are on the interchange graphs ${\rm IntGr}(n,{\bf s})$ introduced by Brualdi and Li [6]. Any two tournaments with the same score sequence have the same number of copies of the cyclic triangle $\Delta_{c}$ (see Figure 5). Moreover, if some $T$ contains a copy $\Delta$ of $\Delta_{c}$, then the tournament $T*\Delta$, obtained by reversing the orientation of all edges in $\Delta$, has the same score sequence as $T$. These observations are the key to exploring ${\rm Tour}(n,{\bf s})$. The graph ${\rm IntGr}(n,{\bf s})$ has a vertex $v(T)$ for each $T\in{\rm Tour}(n,{\bf s})$ and two $v(T_{1}),v(T_{2})$ are neighbors if $T_{2}=T_{1}*\Delta$ for some copy $\Delta\subset T_{1}$ of $\Delta_{c}$. It can be shown that ${\rm IntGr}(n,{\bf s})$ is connected. In this sense, $\Delta_{c}$ generates the set ${\rm Tour}(n,{\bf s})$.

The permutahedron $\Pi_{n-1}$ is related with the standard root system of type $A_{n-1}$, as the symmetric group $S_{n}$ is the Weyl group of type $A_{n-1}$. We refer to, e.g., the standard text by Humphreys [14] for background on root systems. We recall that Killing [17] and Cartan [8] classified all (irreducible, crystallographic) root systems (up to isomorphism) as the infinite families $A_{n-1}$, $B_{n}$, $C_{n}$ and $D_{n}$ and the finite exceptional types $E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$ and $G_{2}$.

Coxeter permutahedra $\Pi_{\Phi}$, recently studied by Ardila, Castillo, Eur and Postnikov [2], are obtained by replacing the role of $S_{n}$ in the definition of $\Pi_{n-1}$ with the Weyl group $W_{\Phi}$ of a root system $\Phi$. See, e.g., Figure 2.

The previous works in this series [19, 18] studied the connection between the polytopes $\Pi_{\Phi}$ and Coxeter tournaments, which are related to orientations of signed graphs, as developed by Zaslavsky [35, 36, 37]. As mentioned above, these tournaments involve collaborative and solitaire games, as well as the usual competitive games in classical graph tournaments.

In this work (see Theorem 4) we show that simple random walks mix rapidly on the Coxeter interchange graphs ${\rm IntGr}(\Phi,{\bf s})$. These graphs encode the combinatorics of the sets ${\rm Tour}(\Phi,{\bf s})$, of Coxeter tournaments with a given score sequence ${\bf s}$, and give structural information about the fibers of the Coxeter permutahedra $\Pi_{\Phi}$. We focus on the non-standard types $\Phi=B_{n}$, $C_{n}$ and $D_{n}$.

We also show (see Theorem 5) that all Coxeter interchange graphs are connected and we bound their diameter. In constructing our random walk couplings, we uncover various other fine, structural properties of the graphs ${\rm IntGr}(\Phi,{\bf s})$, and hence the sets ${\rm Tour}(\Phi,{\bf s})$, which might be of independent (algebraic, geometric, etc.) interest.

Path coupling is a powerful method for establishing rapid mixing (see Section 2.3). As already mentioned, path coupling was used in [25]. We will also use this method, however, the application in the Coxeter setting is significantly more delicate.

As discussed above, the interchange graphs in type $A_{n-1}$ are generated by a single neutral tournament, namely, the cyclic triangle $\Delta_{c}$. On the other hand, in the Coxeter setting, there are a number of other generators which play a role (see Figures 6, 7 and 5). A fascinating interplay arises, as these generators can interact in a variety of interesting ways. As such, the Coxeter interchange graphs are much richer in complexity. Likewise, the analysis of random walks on these structures is more involved.

The types increase in difficulty in order $A_{n-1}$, $D_{n}$, $B_{n}$, $C_{n}$. Type $C_{n}$ is especially challenging, due to the presence of loops in some of the generators, which we call clovers (see Figure 7). These generators correspond to double edges in an interchange graph. In particular, a special type of structure, which we call a crystal (see Figure 25), can appear in type $C_{n}$ interchange graphs. The crystal arises in $C_{3}$ when ${\bf s}=(2,1,1)$. Crystals can also be found as subgraphs in larger type $C_{n}$ interchange graphs, for instance, in the snare drum in Figure 3, when ${\bf s}=(-1,0,1)$. In other examples, such as the tambourine in Figure 4, when ${\bf s}=(0,0,0)$ is the center of the polytope, there are no crystals, but interesting structure nonetheless.

The presence of crystals in type $C_{n}$ interchange graphs leads to two main issues. The first is in extending certain natural couplings on various small subgraphs (the extended networks discussed in Sections 5.1 and 5.2) to a unified coupling on the entire interchange graph. To overcome this difficulty, we will prove a number of detailed combinatorial properties of the Coxeter interchange graphs. We classify the types of subgraphs (see Figures 14, 17 and 25), which together form the full graph, and study the ways in which they can intersect. For example, one crucial property (see Lemma 21) is that any two crystals can share at most one single edge. Without this property, it seems that a path coupling argument would not be possible.

The second issue caused by crystals is in obtaining a “contractive” (see Section 2.3) coupling. In applying path coupling, we will need to re-weight the graph metric in a specific way, which accounts for the occurrence of crystals. Loosely speaking, the choice of weights is related to the fact that, in our random walk couplings, crystals work like “switches,” that convert single edges to double edges, and vice versa.

The overall coupling used to establish rapid mixing in the Coxeter setting is quite elaborate. See, e.g., Figures 28, 29 and 30 below. The classical type $A_{n-1}$ result [16, 25] is a special case of the argument depicted in Figure 27.

A signed graph ${\mathcal{S}}$ on $[n]=\{1,2,\ldots,n\}$ has a set of signed edges $E({\mathcal{S}})$. The four possible types of edges are:

• •

  negative edges $e^{-}_{ij}$ between two vertices $i$ and $j$,

• •

  positive edges $e^{+}_{ij}$ between two vertices $i$ and $j$,

• •

  half edges $e^{h}_{i}$ with only one vertex $i$, and

• •

  loops $e^{\ell}_{i}$ at a vertex $i$.

We note that classical graphs $G$ correspond to signed graphs ${\mathcal{S}}$ with only negative edges.

In this work, we focus on the complete signed graphs ${\mathcal{K}}_{\Phi}$ of types $\Phi=B_{n}$, $C_{n}$ and $D_{n}$. These signed graphs contain all possible negative and positive edges $e_{ij}^{\pm}$. In type $B_{n}$ (resp. $C_{n}$), all possible half edges $e_{i}^{h}$ (resp. loops $e_{i}^{\ell}$) are also included. We call a signed graph ${\mathcal{S}}$ a $\Phi$-graph if ${\mathcal{S}}\subset{\mathcal{K}}_{\Phi}$.

Most of the results in the literature on classical (type $A_{n-1}$) graph tournaments restricts to the case that $G$ is the complete graph $K_{n}$. We note that the signed graph ${\mathcal{K}}_{A_{n-1}}$ with all possible negative edges (and no other types of signed edges) corresponds to the classical complete graph $K_{n}$.

A Coxeter tournament ${\mathcal{T}}$ on a signed graph ${\mathcal{S}}$ is an orientation of ${\mathcal{S}}$. When ${\mathcal{S}}$ is unspecified, our default assumption will be that ${\mathcal{S}}={\mathcal{K}}_{\Phi}$. More formally, ${\mathcal{T}}=(w_{e}:e\in E({\mathcal{S}}))$, with all $w_{e}\in\{0,1\}$. We think of each $e\in E({\mathcal{S}})$ as a game, and $w_{e}$ as indicating its outcome. (We think of $E({\mathcal{S}})$ as having a natural ordering, so that $(w_{e}:e\in E({\mathcal{S}}))$ holds all information about the orientation of ${\mathcal{S}}$ under ${\mathcal{T}}$. However, we could, somewhat pedantically, instead write ${\mathcal{T}}=\{(e,w_{e}):e\in E({\mathcal{S}})\}$.)

The score sequence is given by, cf. (2),

where ${\bf e}$ is the vector corresponding to the signed edge $e$, given by ${\bf e}_{ij}^{\pm}={\bf e}_{i}\pm{\bf e}_{j}$, ${\bf e}_{i}^{h}={\bf e}_{i}$ and ${\bf e}_{i}^{\ell}=2{\bf e}_{i}$. In other words:

• •

  negative edges $e_{ij}^{-}$ are competitive games in which one of $i,j$ wins and the other loses a $1/2$ point,

• •

  positive edges $e_{ij}^{+}$ are collaborative games in which $i,j$ both win or lose a $1/2$ point,

• •

  half edges $e_{i}^{h}$ are (half edge) solitaire games in which $i$ wins or loses a $1/2$ point, and

• •

  loops $e_{i}^{\ell}$ are (loop) solitaire games in which $i$ wins or loses $1$ point.

If ${\bf s}({\mathcal{T}})={\bf 0}_{n}$ we say that ${\mathcal{T}}$ is neutral.

We let ${\bf s}_{\Phi}$ denote the standard score sequence corresponding to the Coxeter tournament in which all $w_{e}=1$. We note that, in some contexts, ${\bf s}_{\Phi}$ is called the Weyl vector. It is also the sum of the fundamental weights of the root system $\Phi$. See, e.g., [14, 13] for more details.

As discussed in [2], the Coxeter $\Phi$-permutahedron $\Pi_{\Phi}$ is the convex hull of the orbit of ${\bf s}_{\Phi}$ under the Weyl group $W_{\Phi}$ of type $\Phi$. Thus ${\bf s}_{\Phi}$ is a distinguished vertex of $\Pi_{\Phi}$. Note that the symmetric group $S_{n}$ is the Weyl group of standard type $\Phi=A_{n-1}$ and the Weyl vector is ${\bf s}_{n}={\bf w}_{n}-\frac{n-1}{2}{\bf 1}_{n}$, so $\Pi_{n-1}^{\prime}$ (see (1) above) is the $\Phi$-permutahedron of standard type $\Phi=A_{n-1}$.

In [19], we showed that $\Pi_{\Phi}$ is precisely the set of all possible mean score sequences of random Coxeter tournaments, thereby establishing a Coxeter analogue of a classical result of Moon [26]. The next work in this series [18] focused on deterministic Coxeter tournaments. The set ${\rm Score}(\Phi)$ of all score sequences of Coxeter tournaments was classified, generalizing the classical result of Landau [21] discussed above.

The precise characterization of ${\rm Score}(\Phi)$ is somewhat technical, involving a certain weak sub-majorization condition and additional parity conditions in types $C_{n}$ and $D_{n}$. The proof is constructive, in that it shows how to build a Coxeter tournament with any given score sequence. See [18, Theorem 4] for more details.

The set ${\rm Tour}(\Phi,{\bf s})$ of all Coxeter tournaments on ${\mathcal{K}}_{\Phi}$ with given score sequence ${\bf s}$ was also investigated in [18]. Coxeter analogues of the interchange graphs ${\rm IntGr}(n,{\bf s})$ discussed above were introduced. Recall that the sets ${\rm Tour}(n,{\bf s})$ are generated by the cyclic triangle $\Delta_{c}$. In the Coxeter setting, there are additional generators.

In all types $B_{n}$, $C_{n}$ and $D_{n}$, in addition to $\Delta_{c}$, we also require a balanced triangle $\Delta_{b}$. In type $B_{n}$, there are also three neutral pairs $\Omega_{1}$, $\Omega_{2}$ and $\Omega_{3}$. On the other hand, in type $C_{n}$, there are also two neutral clovers $\Theta_{1}$ and $\Theta_{2}$. See Figures 5, 6 and 7. In figures depicting Coxeter tournaments, we will draw:

• •

  competitive games as edges directed away from their winner,

• •

  collaborative games as solid/dotted lines if won/lost,

• •

  half edge solitaire games as half edges directed away/toward from their (only) endpoint if won/lost, and

• •

  loop solitaire games as solid/dotted loops if won/lost.

The reversal ${\mathcal{T}}^{*}$ of a Coxeter tournament ${\mathcal{T}}=(w_{e}:e\in E({\mathcal{S}}))$ on ${\mathcal{S}}$ is obtained by reversing the outcome of all games in ${\mathcal{T}}$. That is, ${\mathcal{T}}^{*}=(w_{e}^{*}:e\in E({\mathcal{S}}))$, where $w_{e}^{*}=1-w_{e}$. If ${\mathcal{X}}\subset{\mathcal{T}}$, we let ${\mathcal{T}}*{\mathcal{X}}$ denote the Coxeter tournament obtained from ${\mathcal{T}}$ by reversing the outcome of all games in ${\mathcal{X}}$. In particular, ${\mathcal{T}}^{*}={\mathcal{T}}*{\mathcal{T}}$.

The Coxeter interchange graph ${\rm IntGr}(\Phi,{\bf s})$ has a vertex $v({\mathcal{T}})$ for each ${\mathcal{T}}\in{\rm Tour}(\Phi,{\bf s})$. Vertices $v({\mathcal{T}}_{1}),v({\mathcal{T}}_{2})$ are neighbors if ${\mathcal{T}}_{2}={\mathcal{T}}_{1}*{\mathcal{G}}$, for some copy ${\mathcal{G}}\subset{\mathcal{T}}_{1}$ of a type $\Phi$ generator. If ${\mathcal{G}}$ is a neutral clover, we add a double edge, and otherwise we add a single edge.

For instance, the “snare drum,” in Figure 3 above, is ${\rm IntGr}(C_{3},{\bf s})$ when ${\bf s}=(-1,0,1)$.

The decision to represent clovers as double edges might seem arbitrary at first sight, however, there is a good reason. As it turns out, rather miraculously, this adjustment makes the type $C_{n}$ interchange graphs degree regular. Furthermore, the degree of ${\rm IntGr}(\Phi,{\bf s})$ is related to distances in $\Pi_{\Phi}$ in the following way. Let $\|{\bf x}\|^{2}=\sum_{i}x_{i}^{2}$ denote the squared length of ${\bf x}\in{\mathbb{R}}^{n}$.

Recall that ${\rm Score}(\Phi)$ is the set of all possible score sequences of Coxeter tournaments on ${\mathcal{K}}_{\Phi}$. As discussed above, this set is classified in [18].

In particular, $d(\Phi,{\bf s})=O(n^{3})$.

We observe that, as ${\bf s}$ moves closer to the center ${\bf 0}_{n}$ of the polytope $\Pi_{\Phi}$, the degree $d(\Phi,{\bf s})$ of ${\rm IntGr}(\Phi,{\bf s})$ increases. This is in line with the intuition that Coxeter tournaments with ${\bf s}$ closer to ${\bf 0}_{n}$ (i.e., closer to being neutral) should contain more copies of the (neutral) generators.

Let us note that ${\bf s}$ with $\|{\bf s}\|^{2}=\|{\bf s}_{\Phi}\|^{2}$ are precisely the vertices of $\Pi_{\Phi}$. For such ${\bf s}$, we have $d(\Phi,{\bf s})=0$, in line with the fact there is a unique Coxeter tournaments with score sequence ${\bf s}$. Indeed, such a tournament is transitive, in the sense that it contains no copy of a neutral generator, and so its interchange graph is single isolated vertex.

In [18], we observed that such a result also holds for graph tournaments, in relation to the standard permutahedron, yielding a geometric interpretation of the classical result (see, e.g., Moon [27]) that any two tournaments with the same win/score sequence have the same number of cyclic triangles.

In closing, let us emphasize the the neutral generators in Figures 5, 6 and 7 were identified in [18]. Theorem 1, proved therein, identifies the degree of the interchange graphs. However, in the current work, we will show (see Theorem 5 below) that the interchange graphs are connected. It is this result that justifies calling these structures “generators,” in the sense that the entire space ${\rm Tour}(\Phi,{\bf s})$ is obtained by iteratively reversing copies of these specific neutral structures.

We recall that an aperiodic, irreducible discrete-time Markov chain $(X_{n})$ on a finite state space $\Omega$ has a unique equilibrium $\pi$ on $S$ such that, for all $x,y\in\Omega$,

as $n\to\infty$. We note that $\pi(y)$ is the asymptotic proportion of time spent at state $y\in\Omega$. The maximal total variation distance from $\pi$ by time $n$,

is non-increasing. The mixing time is defined as

A Markov chain is said to be rapidly mixing if $t_{\rm mix}$ is bounded by a polynomial in $\log|\Omega|$.

Path coupling was introduced by Bubley and Dyer [7]. See, e.g., Aldous and Fill [1, Sec. 12.1.12] or Levin, Peres and Wilmer [22, Sec. 14.2] for reformulations of the original result that are closer in appearance to that of the following.

Consider a connected graph $G=(V,E)$. The graph distance $\delta(x,y)$ is the minimal number of edges in a path between $x$ and $y$. The diameter is $D=\max_{x,y\in V}\delta(x,y)$ is the maximal length of such a path in $G$.

The usual graph distance $\delta$ corresponds to the $w$ for which $w(u,v)=1$ for all $\{u,v\}\in E$.

Often this result is applied with $w=\delta$, and, indeed, this will suffice for us in types $B_{n}$ and $D_{n}$. In this case, path coupling has the intuitive interpretation that if the chain is “contractive” in expectation, then it is rapidly mixing.

On the other hand, in the more complicated type $C_{n}$, we will select a careful re-weighting $w$ that takes into account some of the more intricate features in the interchange graphs of this type.

Finally, note that the type $C_{n}$ interchange graphs are, in fact, multigraphs. Specifically, some pairs of vertices (corresponding to clover reversals) are joined by double edges, as in Figures 3 and 4 above. This is for technical convenience, as it makes the graph regular, and thereby our coupling procedure easier to explain. We note that Theorem 3 still applies, since a Markov chain $(Y_{n})$ on a multigraph $M$ with some double edges is equivalent to the Markov chain $(X_{n})$ on the graph $G$, obtained by collapsing each double edge in $M$ into a single edge, and combining the two associated edge crossing probabilities.

Oriented signed graphs were studied by Zaslavsky [37]. From this point of view (see [37, Fig. 1]), each oriented signed edge in an oriented signed graph ${\mathcal{S}}$ is the union of at most two directed half edges. We modify this idea, by adding a named endpoint to each half-edge, which we call a match. This will allow us to prove certain structural facts using graph theory techniques. We call such a structure a Z-frame.

This concept is fairly general, and not all Z-frames correspond to a Coxeter tournament ${\mathcal{T}}$ on some ${\mathcal{S}}\subset{\mathcal{K}}_{\Phi}$. However, each such ${\mathcal{T}}$ has a unique representation as a Z-frame ${\bf Z}({\mathcal{T}})$. We think of ${\bf Z}({\mathcal{T}})$ as revealing the “inner directed graph structure” of ${\mathcal{T}}$. Players in ${\bf Z}({\mathcal{T}})$ correspond to vertices in ${\mathcal{T}}$. Recall that each game in ${\mathcal{T}}$ corresponds to an oriented signed edge. Each such game is associated with a match in ${\bf Z}({\mathcal{T}})$.

We will think of edges directed away/toward players $v\in V$ as positively/negatively charged. (That being said, positive/negative edges in a Z-frame should not be confused with positive/negative edges in a Coxeter tournament.)

We say that $\bf Z$ is neutral if all players $v\in V$ have net zero charge, i.e., ${\rm deg}^{+}(v)-{\rm deg}^{-}(v)=0$, where ${\rm deg}^{\pm}(v)$ is the number of positive/negative edges incident to $v$. We put ${\rm deg}(v)={\rm deg}^{+}(v)+{\rm deg}^{-}(v)$. Note that, if $\bf Z$ is neutral then ${\rm deg}(v)$ is even.

See Figure 8 for an example of a Coxeter tournament ${\mathcal{T}}$ and its corresponding Z-frame ${\bf Z}({\mathcal{T}})$.

The reason for the above terminology is that directed edges in ${\bf Z}({\mathcal{T}})$ with a positive/negative charge correspond to positive/negative contributions to the score sequence ${\bf s}({\mathcal{T}})$. Note that, in competitive/collaborative games, the two charges are opposing/aligned (regardless of the outcome of the game). See Figure 1.

Our aim is to decompose neutral Coxeter tournaments into irreducible neutral parts. In this section, we will do this for Z-frames in general.

Recall that a trail is a walk in which no edge is visited twice.

Note that the edges along a trail in a $Z$-frame do not repeat, and are connected to each other in alternation by a player/match. Also note that neutral open trails start and end at distinct final matches (the left and rightmost matches along the trail). See Figure 9.

Let ${\bf Z}$ be a neutral Z-frame. We say that ${\bf Z}$ is reducible if it contains a non-empty neutral ${\bf Z}^{\prime}\subsetneq{\bf Z}$. Otherwise, ${\bf Z}$ is irreducible.

Each vertex $v$ in a neutral trail has even degree, since $\mathrm{deg}^{+}(v)=\mathrm{deg}^{-}(v)$ and $\mathrm{deg}(v)=\mathrm{deg}^{+}(v)+\mathrm{deg}^{-}(v)$. As discussed, vertices $v$ in a neutral ${\bf Z}$ have even $\mathrm{deg}(v)$. Our next result observes that if ${\bf Z}$ is irreducible, then all $\mathrm{deg}(v)\leqslant 4$.

Applying the results of the previous section, we obtain the following result for Z-frames ${\bf Z}({\mathcal{T}})$ of Coxeter tournaments ${\mathcal{T}}$.

Note that a tournament ${\mathcal{T}}$ on a signed graph ${\mathcal{S}}$ is neutral if and only if its Z-frame ${\bf Z}({\mathcal{T}})$ is neutral. Naturally, we call such a ${\mathcal{T}}$ irreducible if ${\bf Z}({\mathcal{T}})$ is irreducible, and reducible otherwise.

Finally, we turn our attention to the main result of this section.

The assumption that ${\mathcal{T}}$ is on ${\mathcal{K}}_{\Phi}$ is crucial, since not all of the games in the generators used to reverse ${\mathcal{I}}$ will be in ${\mathcal{I}}$ itself. However, after the series of reversals, only the games in ${\mathcal{I}}$ will have been reoriented, i.e., all games in ${\mathcal{T}}\setminus{\mathcal{I}}$ will be restored to their initial orientations.

Finally, using the reversing lemma, we will prove the main result of this section.

In this section, we will classify the possibilities for $(N,{\mathcal{D}})$. As we will see, this is the key to applying path coupling (Theorem 3 above) in Section 6 below. In the classical case of type $A_{n-1}$ there is only one possibility for $N$ (a “single diamond”), and this is the reason why such a simple contractive coupling (as in Figure 27 below) is possible. As it turns out, this continues to hold in types $B_{n}$ and $D_{n}$, but the underlying reasons are more complicated. Type $C_{n}$, on the other hand, is significantly more complex, as then the structure of $N$ can take various other forms.

It can be seen that any two distinct generators ${\mathcal{G}}_{1}\neq{\mathcal{G}}_{2}$ are either disjoint ${\mathcal{G}}_{1}\cap{\mathcal{G}}_{2}=\emptyset$ or else have exactly one game $g$ in common ${\mathcal{G}}_{1}\cap{\mathcal{G}}_{2}=\{g\}$. In this case, we say that ${\mathcal{G}}_{1},{\mathcal{G}}_{2}$ are adjacent.

Note that if a path of length two from ${\mathcal{T}}_{1}$ to ${\mathcal{T}}_{2}$ passes through some ${\mathcal{T}}_{12}$, then there are two generators ${\mathcal{G}}_{1},{\mathcal{G}}_{2}\subset{\mathcal{T}}_{12}$ such that ${\mathcal{T}}_{i}={\mathcal{T}}_{12}*{\mathcal{G}}_{i}$, for $i\in\{1,2\}$. In this way, every such path of length two from ${\mathcal{T}}_{1}$ to ${\mathcal{T}}_{2}$ is determined by a midpoint ${\mathcal{T}}_{12}$ and a pair of generators ${\mathcal{G}}_{1},{\mathcal{G}}_{2}\subset{\mathcal{T}}_{12}$.

There are three possible networks when ${\mathcal{G}}_{1},{\mathcal{G}}_{2}$ are disjoint. We call these the single, double and quadruple diamonds. See Figure 14. Recall that double edges in ${\rm IntGr}$ correspond to neutral clover reversals. All other types of reversals (neutral triangles and pairs) are represented as single edges.

The following result classifies the types of networks $N({\mathcal{T}}_{1},{\mathcal{T}}_{2})$ when ${\mathcal{G}}_{1},{\mathcal{G}}_{2}$ are disjoint.

The case that ${\mathcal{G}}_{1},{\mathcal{G}}_{2}$ are adjacent is more involved. It is useful to note that, in this case, the difference ${\mathcal{D}}={\mathcal{T}}_{1}\setminus{\mathcal{T}}_{2}$ is a neutral tournament with exactly four games on either three or four vertices. Even so, there are a number of cases to consider, and the key to a concise argument is grouping symmetric cases together. For a Coxeter tournament ${\mathcal{T}}$, we define its projection graph $\pi({\mathcal{T}})$ to be the graph obtained by changing each:

• •

  oriented negative/positive edge (i.e., competitive/collaborative game) into an undirected edge,

• •

  oriented half edge (i.e., half edge solitaire game) into an undirected half edge,

• •

  oriented loop (i.e., loop solitaire game) into an undirected loop.

There are a number of ways that two generators ${\mathcal{G}}_{1},{\mathcal{G}}_{2}$ can be adjacent. However, there are only four possibilities for their projected difference $\pi({\mathcal{D}})$. We call these the square, tent, fork and hanger, as in Figure 16. The following observation is essentially self-evident, and can be verified by an elementary case analysis. We omit the proof.

In addition to the single and double diamond networks in Figure 14, there are two additional networks that can occur when ${\mathcal{G}}_{1},{\mathcal{G}}_{2}$ are adjacent. We call these the split and heavy diamonds, see Figure 17.

The following result classifies the types of networks $N({\mathcal{T}}_{1},{\mathcal{T}}_{2})$ when ${\mathcal{G}}_{1},{\mathcal{G}}_{2}$ are adjacent.