Exponential Decay of Loop Lengths in the Loop O(n) Model with Large n

Abstract

Abstract. The loop O(n) model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin O(n) model. It has been conjectured that both the spin and the loop O(n) models exhibit exponential decay of correlations when n > 2. We verify this for the loop O(n) model with large parameter n, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight x. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when nx 6 is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when nx 6 is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.

Unusual changeover in the transition nature of local-interaction Potts models

“…Our results for q c , in particular that of the honeycomb lattice, stand in contrast to the well known result q c = 4 of the usual model [4,5,7,8].…”

“…To conclude our theory we point out that, when applied to (3), the sub-exponential number of simple clusters implies lim N →∞ (Z low ) 1/N = uq 1−ρ so the low temperature free energy density near criticality βf low = − ln(uq 1−ρ ) is minimal if and only if uq −ρ > 1, resulting in u c = q ρ which is equivalent to (7).…”

“…His findings were believed to be lattice independent [6]. Recently, Duminil-Copin et al [7,8] have rigorously confirmed Baxter's predictions using the random cluster representation [9] of the nearest-neighbor interaction model.…”

Approximate ground states of the random-field Potts model from graph cuts

“…These have primarily investigated the phase diagram in (T, ∆)-space for different q and d, where T denotes temperature. For the pure model, the * martin.weigel@complexity-coventry.org temperature-driven transitions are continuous for small q ≤ q c and first-order for large q > q c , with q c = 4 for the square-lattice model with nearest-neighbor interactions [20,21] and q c ≈ 2.8 for the simple-cubic lattice [22]. It is well known that quenched disorder tends to soften first-order transitions [23], and this has even been rigorously established for systems in two dimensions [24].…”

Shifted critical threshold in the loop $ \boldsymbol{O(n)} $ model at arbitrarily small $n$

“…This way, when n = 0, only the loop containing the origin can be observed and it gets a weight proportional to λ |Po| . It is well known that in this case the length of P o admits uniformly bounded exponential moments when λ ∈ (0, 1/µ), with µ = µ(G) being the so-called connective constant of G (see (5) for a definition). The exact value of this constant is known on the hexagonal lattice [7], µ(H) = 1/ 2 + √ 2.…”