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.