# Mean and variance of the Potts model in the hottest state

I’ve just arXiv’d another revision of my paper on the PFAB algorithm: arXiv:1503.08066v3 [stat.CO]. It includes a rather elegant proof of the exact mean and variance of the sufficient statistic S(**y**) in the *hottest* state, when the inverse temperature . The proof by my co-author Geoff Nicholls holds for any Potts model with first-order neighbours. That is, the nearest 4 neighbours in a 2D lattice (or 6 neighbours in 3D). For posterity, I present my rather clunkier proof below, which involves induction on dimension for a rectangular lattice.

The Potts (1952) model is an example of a Gibbs random field on a regular lattice, where each node can take values in the set . The Ising model can be viewed as a special case, when *q*=2. The size of the configuration space is therefore , where n is the number of nodes. The dual lattice defines undirected edges between neighbouring nodes . If the nodes in a 2D lattice with c columns are indexed row-wise, the nearest (first-order) neighbours are , except at the boundary. Nodes situated on the boundary of the domain have less than four neighbours. The total number of unique edges is thus for a square lattice, or if the lattice is rectangular.

The sufficient statistic of the Potts model is the sum of all like neighbour pairs:

where is the Kronecker delta function, which equals 1 if a = b and equals 0 otherwise. ranges from 0, when all of the nodes form a chequerboard pattern, to when all of the nodes have the same value. The likelihood of the Potts model is thus:

The normalising constant of the Potts model is intractable for any non-trivial lattice, since it requires a sum over the configuration space:

When the inverse temperature , simplifies to , hence the labels are independent and uniformly-distributed.

### Theorem 1

The sum over configuration space of the sufficient statistic of the

q-state Potts model on a rectangular 2D lattice is.

### Proof

For a *q*=2 state Potts model on a lattice with *n*=4 nodes and edges, contains 16 possible configurations:

. This can also be written as .

Now consider a rectangular lattice with *r* > 1 rows and *c* > 1 columns, so that and the dual lattice . The size of the configuration space is . Assume that the sum over configuration space is equal to . This sum can be decomposed into within each row, plus between rows.

If this lattice is extended by adding another row (or equivalently, another column), then (or otherwise, ) and the dual lattice . The nodes in this new row can take possible values, so the size of the configuration space is now . will increase proportional to for the new row, plus for the connections with its adjacent row:

Q.E.D.

### Theorem 2

The expectation of the

q-state Potts model on a rectangular 2D lattice is when the inverse temperature .

### Proof

The proof follows from Theorem 1 by noting that and hence:

Q.E.D.

### Theorem 3

The sum over configuration space of the square of the sufficient statistic of the

q-state Potts model on a rectangular 2D lattice is

### Proof

For a *q*=2 state Potts model on a lattice with *n*=4 nodes and edges, . This can also be written as .

Now assume for a rectangular lattice with *r *> 1 rows and *c* > 1 columns that

This can be decomposed into .

If we extend the lattice by adding another row, then

Q.E.D.

### Theorem 4

The variance of the

q-state Potts model on a rectangular 2D lattice is when the inverse temperature .

### Proof

The proof follows from Theorems 1 and 3:

Q.E.D.