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CENTRAL LIMIT THEOREM FOR THE HEAT KERNEL MEASURE ON THE UNITARY GROUP
, 2009
"... We prove that for a finite collection of realvalued functions f1,..., fn on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of (trf1,..., trfn) under the properly scaled heat kernel measure at a given time on the unitary group U( ..."
Abstract

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We prove that for a finite collection of realvalued functions f1,..., fn on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of (trf1,..., trfn) under the properly scaled heat kernel measure at a given time on the unitary group U(N) has Gaussian fluctuations as N tends to infinity, with a covariance for which we give a formula and which is of order N −1. In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results.
Random pure quantum states via unitary Brownian motion
 ELECTRONIC COMMUNICATIONS IN PROBABILITY
, 2013
"... We introduce a new family of probability distributions on the set of pure states of a finite dimensional quantum system. Without any a priori assumptions, the most natural measure on the set of pure state is the uniform (or Haar) measure. Our family of measures is indexed by a time parameter t and i ..."
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We introduce a new family of probability distributions on the set of pure states of a finite dimensional quantum system. Without any a priori assumptions, the most natural measure on the set of pure state is the uniform (or Haar) measure. Our family of measures is indexed by a time parameter t and interpolates between a deterministic measure (t = 0) and the uniform measure (t = ∞). The measures are constructed using a Brownian motion on the unitary group UN. Remarkably, these measures have a UN−1 invariance, whereas the usual uniform measure has a UN invariance. We compute several averages with respect to these measures using as a tool the Laplace transform of the coordinates.
FREE ENERGIES AND FLUCTUATIONS FOR THE UNITARY BROWNIAN MOTION
"... Abstract. We show that the Laplace transforms of traces of words in independant unitary Brownian motions converge towards an analytic function on a non trivial disc. This results allow to study asymptotics of Wilson loops under the unitary YangMills measure on the plane. The limiting objects obtai ..."
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Abstract. We show that the Laplace transforms of traces of words in independant unitary Brownian motions converge towards an analytic function on a non trivial disc. This results allow to study asymptotics of Wilson loops under the unitary YangMills measure on the plane. The limiting objects obtained are shown to be characterized by equations analog to SchwingerDyson’s ones, named here after Makeenko and Migdal. 1.