If one conditions in the plus phase of the Ising model at low temperature on an interval of length \(N\) to be in the minus phase, is there entropic repulsion? In other words, is the interval surrounded by a large (size \(O(N)\)) contour which has a distance in the direction perpendicular to the interval which grows to infinity as \(N\) increases?
Conjecture: Yes
In [van Enter--Fernandez--Sokal (1993)], the authors explained why this is true if the interval is all minus.
For the low-T regime of the \(1 \over r^2\) one-dimensional Ising model, are the plus and minus Gibbs measures \(g\)-measures?
(P5.2a) Conjecture: The answer is yes at very low temperatures where the truncated correlations are summable.
(P5.2b) Conjecture: The answer is no at intermediate temperatures where the truncated correlations are non-summable.
See [Bissacot--Endo--van Enter--Le Ny (2018)] for \(g\)-measures. In this paper, it was proved that for \(1\over r^{\alpha}\) interactions with \(\alpha < 2\), the answer to problem P5.2 is no.
Question: For the \( 1 \over r^2 \) Potts model, can there be Thouless effect and/or a first order transition?
First-order transitions are impossible for Ising models but can occur for \(q\)-state Potts models for \(1 \over r^{\alpha} \) interactions when q is large and \(\alpha <2\).
Conjecture: A first-order transition could occur also for \(\alpha=2\).
For the occurence of a Thouless effect in the Ising case, see [Simon--Sokal (1981)]. This was finally solved in [Aizenman--Chayes--Chayes--Newman (1988)].
First order transitions are impossible for Ising models:
[Aizenman--Duminil-Copin--Sidoravicius (2015)].
The fact that they can occur for \(q\)-state Potts models for \(1 \over r^{\alpha} \) interactions when q is large and \(\alpha <2\)
was shown in
[Biskup--Chayes--Crawford (2006)].