Central limit theorem for the heat kernel measure on the unitary group

Abstract

We prove that for a finite collection of real-valued functions $f_{1},...,f_{n}$ on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of $(\tr f_{1},...,\tr f_{n})$ 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.Comment: 44 page

Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ U N and $${\mathbb {GL}}_N$$ GL N

“…We prove Theorem 1.2 (on page 29) incorporating some estimates from [31] with a Fourier cutoff argument. Note that in [31], the (Gaussian) fluctuations of the empirical integrals U f d ν N t are computed: They are on the scale of the Sobolev space H 1/2 (U) as t → ∞. We conjecture that the O(1/N 2 p−1 ) in (1.4) can be improved to O(1/N 2 ), and that therefore the a.s. convergence holds for f ∈ H p (U) for any p > 1 2 .…”

“…This is the approach taken in [30,31], and instead, the heat kernel is evaluated at time t/N to compensate. In that sense, our limiting concentration results can be interpreted as statements about the heat kernel in a neighborhood of t = 0.…”

“…Theorem 1.2 is treated first, separately, with specialized techniques adapted from [31]. We then proceed with Theorem 1.7, and then derive Theorems 1.3 and 1.5 essentially as special cases.…”

“…The following general lemma was given in [31,Proposition 4.1]; it is adapted from the now well-known techniques in [22] and attributable to earlier work of Talagrand.…”

An Estimation Method of Water Drive Displacement Based on Formation Permeability Distribution and Inverse Problem Modeling

“…The left endpoint is log min or − 3 − on the -axis, and the right endpoint is log max or + 3 + on that axis, and so forth: > 0 ( = log = 0.5 ⋅ (log min + log max )). According to the area superposition principle of a normal distribution curve [19]: selecting = log as the starting point and step size Δ , subdivide the probability curve. If area summation of these form-closed figures can infinitely approach 1, we can obtain the values of the left endpoint log min and right endpoint log max and so forth; log max = 2 ⋅ log − log min .…”