Articles you may be interested inEmbedded Gaussian unitary ensembles with U(Ω)⊗SU(r) embedding generated by random twobody interactions with SU(r) symmetryWe consider the mixed matrix moments for the complex Ginibre ensemble. These are well-known. We consider the relation to the expected overlap functions of Chalker and Mehlig. This leads to new asymptotic problems for the overlap. We obtain some results, but we also state some remaining open problems. C 2015 AIP Publishing LLC. FIG. 1. Some analogous elements in random matrix and spin glass theory. (Proofs may differ considerably.) 013301-3 M. Walters and S. Starr J. Math. Phys. 56, 013301 (2015)Another easy fact is that, due to symmetry, m n (p, q) = 0 unless p(1) + · · · + p(k) = q(1) + · · · + q(k). And, of course, m 0 = 1. Using this and the method of induction it is easy to prove Theorem 2.1.

For a positive number q the Mallows measure on the symmetric group is the probability measure on Sn such that Pn,q(π) is proportional to q-to-the-power-inv(π) where inv(π) equals the number of inversions: inv(π) equals the number of pairs i < j such that πi > πj. One may consider this as a mean-field model from statistical mechanics. The weak large deviation principle may replace the Gibbs variational principle for characterizing equilibrium measures. In this sense, we prove absence of phase transition, i.e., phase uniqueness. The Mallows model is a frustration free, mean-field modelThe Hamiltonian in (2) is a mean-field Hamiltonian. Mean-field Hamiltonians have the property that considering a subsystem, the inverse-temperature β needs to be rescaled because of the explicit dependence of H n on n. The consideration of a sub-system is sometimes known as the cavity method for complicated problems, which are most amenable to inductive analysis, removing one particle at a time. See for instance [22] as an indication of the physics approach to this method or [32] for the mathematical side. Temperature renormalization means that if there is an explicit formula for the optimizer of the Mallows model on [0, 1] 2 in the thermodynamic limit, then restricting attention to the sub-squares Λ ij (θ 1 , θ 2 ), the restriction of the measure to these sets may be an optimizer for different choices of β, due to dilution on the sub-squares. (We will refer to the Λ ij (θ 1 , θ 2 )'s as "sub-squares" even though they are rectangles.)Of course, since the model is a mean-field model, there is an interaction between all particles in [0, 1] 2 , including between different sub-squares. But then there is a special symmetry of the model. In two dimensions, it is common to see conformally invariant models, which is the symmetry one finds from local rotational symmetry as well as dilation covariance. For the Mallows model, instead of SO(2) symmetry the group which leaves the model invariant is the group of hyperbolic rotations SO + (1, 1),

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We explore a method introduced by Chatterjee and Ledoux in a paper on eigenvalues of principle submatrices. The method provides a tool to prove concentration of measure in cases where there is a Markov chain meeting certain conditions, and where the spectral gap of the chain is known. We provide several additional applications of this method. These applications include results on operator compressions using the Kac walk on $SO(n)$ and a Kac walk coupled to a thermostat, and a concentration of measure result for the length of the longest increasing subsequence of a random walk distributed under the invariant measure for the asymmetric exclusion process.Comment: 13 pages. Additional applications added and method clarified. in Electronic Communications in Probability [Online], 20 (2015): 1-13. We