The 
-state Potts model on a finite
graph 
assigns to each graph vertex 
a state 
. For a positive integer 
and real interaction parameter 
, set 
. The zero-field partition function can then
be written
 |
(1)
|
where 
is the Kronecker
delta. Equal states on adjacent graph vertices
are favored when 
and disfavored when 
. Potts (1952) introduced
the model as a 
-state
generalization of the Ising model, which is recovered
at 
up to a constant
normalization and a rescaling of 
.
Expanding the product and then summing over the states gives the Fortuin-Kasteleyn or random-cluster model representation
 |
(2)
|
where 
is the number of connected components of the
spanning subgraph 
(Fortuin and Kasteleyn 1972). Indeed, after the graph
edges in 
have imposed equality of incident states, each connected
component may be assigned any one of the 
states. The right side is a polynomial
in 
and 
,
so it extends the partition
function from positive integers 
to arbitrary complex numbers 
and 
.
For the homogeneous random-cluster model in which every graph edge has parameter 
, the corresponding normalizing sum is
 |
(3)
|
Thus 
gives independent bond percolation, while other positive
values of 
weight a configuration of graph edges according to
its number of connected components.
Setting 
and 
, the same expansion identifies the Potts partition
function with an evaluation of the Tutte polynomial:
 |
(4)
|
At 
, adjacent graph vertices
must have different states, and hence
 |
(5)
|
where 
is the chromatic polynomial. Consequently,
the Potts model simultaneously provides state-sum, random-cluster, and graph-polynomial interpretations of the same invariant
of a finite graph (Wu 1982).
Averous et al. (2026) reported conformal invariance of the critical four-state Potts model on the square integer lattice 
. For a tree broadcast model,
the root spin is uniform among 
states, and each child is transmitted independently conditional
on its parent through 
,
where 
and 
. Reconstruction means that
observations of increasingly distant levels retain nonzero
information about the root.
For fixed integers 
and 
, Blanca et al. (2026)
prove that Glauber dynamics on a uniformly random
-regular graph
on 
vertices
mixes in 
individual attempted updates, with high probability,
for every fixed 
and some constant 
.
The quantity 
is the uniqueness threshold of the ferromagnetic model on the infinite 
-regular tree. On 
-vertex cubes or tori in 
,
for fixed 
and 
, the inverse spectral
gap is 
in the same single-update time scale. This gives a mixing
time bound 
. The Swendsen-Wang
dynamics bound on random regular graphs is 
cluster updates. Blanca et al. (2026) credit ChatGPT
with assistance in discovering the mathematical dichotomy underlying these bounds.
