Abstract:
This thesis is devoted to the study of phase transitions and the associated behaviour in
the Ashkin–Teller (AT) model on the square lattice Z2 and its related models, including
the six- and eight-vertex models, the Potts model and FK percolation. The AT modelwas
introduced in 1943 as a generalisation of the Ising model and may be represented by a
pair of Ising spin configurations with coupling constants J, J′ for each and U for their
pointwise product, the latter describing the interaction within the pair. The thesis can
be divided into three parts.
The first part is a study of the isotropic (J = J′) ferromagneticAT model on the square
lattice Z2.We confirm the presence of a single phase transition when J ≥ U > 0 and two
distinct ones when U > J > 0. In the uniqueness case, we identify the transition at the
self-dual curve. In the non-uniqueness case, we show that the two distinct transition
curves are dual to each other. A significant part is the detailed study of the self-dual
model via its relations to the six-vertex model and self-dual FK percolation. The main
result in this regard is exponential decay of correlations of the single spins and uniform
positivity of correlations of the product, each in finite volume and under the respective
least favourable boundary conditions, when U > J.
In the second part, we investigate the AT model in the presence of negative coupling
constants on the hypercubic lattices Zd in any dimension d ≥ 2. Observing that it suffices
to study the case J, J′ ≥ 0 > U, we confirm both ferromagnetic and antiferromagnetic
behaviour in this regime.We first establish the existence of a partial antiferromagnetic
phase in the isotropic model in a perturbative sub-regime. In d = 2, we show that the
associated staggered six-vertex height function is localised, although it simultaneously
exhibits antiferromagnetic behaviour. We then demonstrate ferromagnetic behaviour
by partitioning another sub-regime into a collection of smooth curves along which the
model undergoes a subcritically sharp order-disorder phase transition, circumventing the
difficulty of the lack of general monotonicity properties in the parameters.
The third part builds on the first one and deals with the associated self-dual Potts model
with q > 4 states on the square lattice Z2. Under order-disorder Dobrushin boundary
conditions on a square box of size n,we verify that the interface between the ordered and
disordered phases is thin, has fluctuations of order √
n and converges when rescaled to
a Brownian bridge. This is achieved by a coupling with a graphical representation of the
AT model, which allows to relate the interface in the Potts model to a subcritical cluster
conditioned to be long, and then developing the celebrated Ornstein–Zernike theory to
deduce convergence of the latter.We also show the analogous statements for self-dual
FK percolation with q > 4. In a subsequent work, which is not part of the thesis, we
further build on this project and establish the so-called wetting phenomenon in the Potts
model.We present the relevant coupling of the Potts model under order-order Dobrushin
boundary conditions and a graphical representation of the AT model, conditioned
to admit a pair of long clusters, which turn out to be repulsive to each other.