Moments and cumulants of polynomial random variables on unitarygroups, the Itzykson-Zuber integral, and free probability

By Benoît Collins. 2002

438 citations recorded by the source corpus.

We consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to\infty$. Using this we rederive and strenghthen a result of asymptotic freeness due to Voiculescu. We then study large $d$ asymptotics of matrix model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular we give an explicit formula for $$\lim_{d\to\infty}\frac{\partial^n}{\partial z^n}d^{-2} \log\int_{U_d} e^{zd Tr (XUYU^*)}dU|_{z=0}$$ assuming that the normalized traces $d^{-1} Tr(X^k)$ and $d^{-1} Tr (Y^k)$ converge in the large $d$ limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.

View Paper: https://www.semanticscholar.org/paper/99a70106a45fecac9319510aa56468062b52d29d

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Citations in the corpus

Random matrix techniques in quantum information theory (2015, background): “ are interested in the values of the Weingarten function in the limit n!1. The following result encloses all the information we need for our computations about the asymptotics of the Wg function; see [Col03] for a proof. Theorem 3.6. For a permutation ˙2S p, let Cycles(˙) denote the set of cycles of”
Free Probability and Random Matrices (2014, methodology): “However, we do have a replacement for the Wick formula in this context, which is known as the Weingarten convolution formula, see [57, 60].”
De Finetti theorems for easy quantum groups (2009, unknown): “π ≤ σ, then n|π|W kn(π,σ) = µD(k)(π,σ) +O(n −1), where µD(k) is the Mo¨bius function on the partially ordered set D(k) under the restriction of the order on P(k). Proof. We use a standard method from [11], [12], further developed in [3], [4] and [14]. First note that Gkn = Θ 1/2 kn (1+Bkn)Θ 1/2 kn ,”
Chaos, complexity, and random matrices (2017, unknown): “The exact formula [45, 46] for evaluating these moments is given by ∫ dU U j1 k1 .”
Chaos and complexity by design (2016, background): “The unitary Weingarten function on Sn is given by [55] Wg = 1 n! ∑”
Product of random projections, Jacobi ensembles and universality problems arising from free probability (2004, background): “For a proof in full generality, see [Co l03a], Proposition 2.3.3 p.52 or [ Col03b ], Theorem 3.1.”
A semiclassical ramp in SYK and in gravity (2018, background): “tangled states were generic vectors in the Hilbert space one would expect problems when k! ˘L2k or, roughly, k˘L2. In fact (108) shows that the maximally entangled states are not generic. It is known [66] that the Weingarten functions have simple poles for L= (k 1); (k 2):::(k 2);(k 1). So the rst p”
Quantum variational algorithms are swamped with traps (2022, unknown): “[17] Anna Choromanska, Mikael Henaff, Michael Mathieu, Gerard Ben Arous, and Yann LeCun.”
Unitary designs from statistical mechanics in random quantum circuits. (2019, background): “The general expression for integrating the k -th moment of U ( d ) is [47,48] where we sum over elements of the permutation group S k and deﬁne a δ -function contraction indexed by a permutation σ as δ σ ( (cid:126)ı | (cid:126) ) = δ i 1 ,j σ (1) . . . δ i k ,j σ ( k ) .”
Asymptotics of unitary and orthogonal matrix integrals (2006, background): “c behaviour of IN(V,AN i ) as N goes to inﬁnity. In the most general framework, nothing was known about the convergence of these matrix integrals, except the formal convergence. Namely, it was proved [5] by one 2000 Mathematics Subject Classiﬁcation. 15A52 (46L54). Key words and phrases. Matrix inte”
Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group (2004, methodology): “3.4. Cumulants. Recall that in the work of the first named author [ Col03 ] the asymptotics of the cumulants of the unitary Weingarten functions have been obtained (Theorem 2.15).”
A Fourier view on the R-transform and related asymptotics of spherical integrals (2005, methodology): “They have been then extensively studied in the framework of so-called matrix models that are related to the problem of enumerating maps (after [16], it has been developed in physics for example in [27], [19] or [21], in mathematics in [6] or [11]; a very nice introduction to these links is provided ”