Mixing Time of Markov Chains for the 1–2 Model

Abstract

A 1–2 model configuration is a subset of edges of a hexagonal lattice satisfying the constraint that each vertex is incident to 1 or 2 edges. We introduce Markov chains to sample the 1–2 model configurations on the 2D hexagonal lattice and prove that the mixing time of these chains is polynomial in the sizes of the graphs for a large class of probability measures.

Appendix: How to Check the Condition \(F(\ell ,\nu ^{-1})\)

By Theorem 10, in order to prove the strong mixing condition for Gibbs measures of the 1–2 model on all the sufficiently large boxes, it suffices to show that $F(\ell ,\nu ^{-1})$ holds for some fixed positive integer $\ell$. In this section, we investigate efficient algorithms to check the condition $F(\ell ,\nu ^{-1})$.

From the definition of $F(\ell ,\nu ^{-1})$, in order to check the condition $F(\ell ,\nu ^{-1})$, we need to compute the total variation differences of probability measures of 1–2 model configurations on finite graphs of ${{\mathbb {H}}}$ given different boundary conditions. The idea is to apply the measure-preserving correspondence of 1–2 model configurations on ${{\mathbb {H}}}$ and dimer configurations on ${{\mathbb {H}}}_{{\varDelta }}$, and to compute the total variation differences of probability measures for dimer configurations on finite subgraphs of ${{\mathbb {H}}}_{{\varDelta }}$ instead, which is known to be computable by determinants or Pfaffians; see [5, 6, 9,10,11, 14, 20].

By the measure-preserving correspondence of 1–2 model configurations on ${{\mathbb {H}}}$ and dimer configurations on ${{\mathbb {H}}}_{{\varDelta }}$, as well as the domain Markov property, to consider the boundary condition of an $n\times n$ box ${\varLambda }_n$ of ${{\mathbb {H}}}$ in the 1–2 model configurations, it suffices to consider dimer configurations in all the hexagons crossing $\partial {\varLambda }_n$. (For the $3\times 3$ box ${\varLambda }_3$, $\partial {\varLambda }_3$ is given by the blue lines in Fig. 15).

Fig. 15

The green line represents the central line $M_1({\varLambda }_3)$ parallel to the northeastern boundary. The labeled vertices are vertices with the same configurations for $\tau$ and $\tau '$ when computing $\max ^{*}_{\tau ,\tau '}$ (Color figure online)

Each dimer configuration on ${{\mathbb {H}}}_{{\varDelta }}$ satisfies the following two constraints

• each vertex of ${{\mathbb {H}}}$ has exactly one incident preset bisector edge in ${{\mathbb {H}}}_{{\varDelta }}$; and

• around each hexagon of ${{\mathbb {H}}}$, there are an even number of present bisector edges.

Indeed, for any configuration on bisector edges satisfying the above two constraints, the configuration can be uniquely extended to a dimer configuration on ${{\mathbb {H}}}_{{\varDelta }}$. As a result, the influence of boundary conditions to dimer configurations in ${\varLambda }_{\ell }$ depends only on the parity of the number of present bisector edges outside ${\varLambda }_{\ell }$ in each hexagon crossing $\partial {\varLambda }_{\ell }$. Let $\ell \ge 3$. All the hexagons crossed by $\partial {\varLambda }_{\ell }$ can be classified into 3 different types:

• 1. The hexagon has 5 vertices outside ${\varLambda }_{\ell }$, and 1 vertex inside ${\varLambda }_{\ell }$. The two hexagons on the left and right corners of Fig. 15 are of this type.

• 2. The hexagon has 4 vertices outside ${\varLambda }_{\ell }$, and 2 vertices inside ${\varLambda }_{\ell }$. The two hexagons on the top and bottom corners of Fig. 15 are of this type.

• 3. The hexagon has 3 vertices outside ${\varLambda }_{\ell }$ and 3 vertices inside ${\varLambda }_{\ell }$. All the hexagons crossing $\partial {\varLambda }_{\ell }$ but not on the corners are of this type.

We say a hexagon h crossing $\partial {\varLambda }_{\ell }$ is positive (resp. negative) with respect to a boundary condition $\tau$, if in h, an even (resp. odd) number of incident bisector edges to vertices outside ${\varLambda }_{\ell }$ are present in $\tau$. Note that for an admissible boundary condition, there are an even number of hexagons crossing $\partial {\varLambda }_{\ell }$ that are negative with respect to the boundary condition.

We will treat different types of hexagons differently.

If a Type (1) hexagon $h_1$ is positive (resp. negative) with respect to the boundary condition, let e be the bisector edge in $h_1$ incident to the unique vertex v of $h_1$ in ${\varLambda }_{\ell }$; then e must be absent (resp. present), in which case we remove e (resp. all the incident edges of v), and treat the two adjacent hexagons of $h_1$ crossing $\partial {\varLambda }_{\ell }$ as Type-3 hexagons (Type-2 hexagons).

If a Type (2) hexagon $h_2$ is positive (resp. negative) with respect to the boundary condition, then we replace the graph ${{\mathbb {H}}}_{{\varDelta }}$ in the hexagon by an edge (resp. a vertex and two edges), see the left graph (resp. the right graph) of Fig. 17.

Fig. 16

Rooms and Corridors. The largest rhombus represents the $n\times n$ box of ${\varLambda }_n$ of ${{\mathbb {H}}}$; each small rhombus inside ${\varLambda }_n$ represents a $\ell \times \ell$ room; and corridors are represented by the line segments joining different rooms or joining a room to the boundary $\partial {\varLambda }_n$

Fig. 17

Replace a positive (resp. negative) Type (2) hexagon $h_2$ on the boundary by a gadget represented by green lines on the left (resp. right) graph (Color figure online)

If a Type (3) hexagon $h_3$ is positive (resp. negative) with respect to the boundary condition, then we replace the graph ${{\mathbb {H}}}_{{\varDelta }}$ in the hexagon by a gadget in the left graph (resp. the right graph) Fig. 18.

Fig. 18

Replace a positive (resp. negative) Type (3) hexagon $h_3$ on the boundary by a gadget represented by green lines on the left (resp. right) graph (Color figure online)

Therefore, to consider the Gibbs measures for dimer configurations on ${\varLambda }_{\ell }$ with different boundary conditions, it suffices to consider Gibbs measures on different finite graphs. More precisely, the condition $F(\ell ,\nu ^{-1})$ can be checked through the following steps:

• 1. Construct finite subgraphs of ${{\mathbb {H}}}_{{\varDelta }}$ from the finite subgraphs of ${{\mathbb {H}}}$ with given boundary conditions by the procedure described above;

• 2. Give a clockwise odd orientation to each constructed finite subgraph of ${{\mathbb {H}}}_{{\varDelta }}$ (i.e., give each edge on the finite subgraph of ${{\mathbb {H}}}_{{\varDelta }}$ an orientation such that each face of the finite subgraph of ${{\mathbb {H}}}_{{\varDelta }}$ has an odd number of edges oriented in the same way as the clockwise orientation; such an orientation always exists on a planar graph, see [9]);

• 3. Compute the total variational distances of probability measures for dimer configurations on finite subgraphs of ${{\mathbb {H}}}_{{\varDelta }}$ with different boundary conditions; these can be computed by computing determinants and Pfaffians of the corresponding weighted adjacency matrices; see [14].