Andrew Price

Off-critical parafermions and the winding angle distribution of the O(n) model

Usinganoff-criticaldeformationoftheidentityofDuminil-CopinandSmirnov, we prove a relationship between half-plane surface critical exponents γ1 and γ11 as well as wedge critical exponents γ2(α) and γ21(α) and the exponent characterizing the winding angle distribution of the O(n) model in the halfplane, or more generally in a wedge of wedge-angle α. We assume only the existence of these exponents and, for some values of n, the conjectured value of the critical point. If we assume their values as predicted by conformal field theory, one gets complete agreement with the conjectured winding angle distribution, as obtained by CFT and Coulomb gas arguments. We also prove the exponent inequality γ1 − γ11 1, and its extension γ2(α) − γ21(α) 1f or the edge exponents. We provide conjectured values for all exponents for n ∈ [−2,2).

Permutations sortable by two stacks in series

We address the problem of the number of permutations that can be sorted by two stacks in series. We do this by first counting all such permutations of length less than 20 exactly, then using a numerical technique to obtain nineteen further coefficients approximately. Analysing these coefficients by a variety of methods we conclude that the OGF behaves as

Numerical studies of Thompson's group F and related groups

S(z) \sim A (1 - \mu \cdot z)^\gamma,

Permutations sortable by deques and by two stacks in parallel

where

The generating function of planar Eulerian orientations.

\mu =12.45 \pm 0.15,

Stakeholder engagement with major infrastructure projects

\gamma= 1.5 \pm 0.3,

Goal setting and performance management

and