We study the Manhattan and Lorentz mirror models on an infinite cylinder of finite even width n, with the mirror probability p satisfying $$p<Cn^{-1}$$ p < C n - 1 , C a constant. We show that the maximum height along the cylinder reached by a walker is order $$p^{-2}$$ p - 2 . We observe an algebraic structure, which helps organise our argument. The models on the cylinder can be thought of as Markov chains on the Brauer (in the Mirror case) or Walled Brauer (in the Manhattan case) algebra, with the transfer matrix given by multiplication by an element of the algebra.

We study a two-parameter family of quantum spin systems on the complete graph, which is the most general model invariant under the complex orthogonal group. In spin $S=\frac {1}{2}$ it is equivalent to the XXZ model, and in spin $S=1$ to the bilinear-biquadratic Heisenberg model. The paper is motivated by the work of Björnberg, whose model is invariant under the (larger) complex general linear group. In spin $S=\frac {1}{2}$ and $S=1$ we give an explicit formula for the free energy for all values of the two parameters, and for spin $S&gt;1$ for when one of the parameters is non-negative. This allows us to draw phase diagrams and determine critical temperatures. For spin $S=\frac {1}{2}$ and $S=1$, we give the left and right derivatives as the strength parameter of a certain magnetisation term tends to zero, and we give a formula for a certain total spin observable, and heuristics for the set of extremal Gibbs states in several regions of the phase diagrams, in the style of a recent paper of Björnberg, Fröhlich, and Ueltschi. The key technical tool is expressing the partition function in terms of the irreducible characters of the symmetric group and the Brauer algebra. The parameters considered include, and go beyond, those for which the systems have probabilistic representations as interchange processes.