arXiv math/06081932006-08math.PR

Asymptotics of unitary and othogonal matrix integrals

Benoit Collins, Alice Guionnet, Edouard Maurel-Segala

备注：: 41 pages, important modifications, new section about orthogonal integrals
发表信息：: Adv. Math. 222 (2009), no. 1, 172-215
DOI：: 10.1016/j.aim.2009.03.019

英文摘要：

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large $N$ limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the othogonal group.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

酉积分的拓扑展开与映射

Topological expansion of unitary integrals and maps

本文通过引入满足Tutte方程的酉型映射，为酉群上Haar测度积分的渐近拓扑展开提供了组合解释，并推广了单调Hurwitz数。

2024-11-07math.PR

A power series identity and Bessel-type integrals over unitary groups

In 2008, Lehner, Wettig, Guhr and Wei conjectured a power series identity and showed that it implied a determinantal formula for a Bessel-type integral over the unitary supergroup. The integral is the supersymmetric extension of Bessel-type integrals over the unitary group appearing as partition functions in quantum chromodynamics. The identity is proved by interpreting both sides as the same unitary integral, which can be computed using the Cartan decomposition. An equivalent identity of Schur functions is also given and interpreted probabilistically.

2018-06-06math-phmath.MPmath.PR

Hua type integrals over unitary groups and over projective limits of unitary groups

We discuss some natural maps from a unitary group U(n) to a smaller group U(n-m) (these maps are versions of the Livshic characteristic function). We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces (values of the integrals are products of Gamma-functions). These integrals generalize Hua Loo Keng integrals. We construct inverse limits of unitary groups equipped with analogues of the Haar measure and evaluate some integrals over these inverse limits.

2001-10-18math-phmath.FAmath.MP

Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability

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 gi…

2002-06-06math-phmath.MPmath.PR

Jucys-Murphy Elements and Unitary Matrix Integrals

In this paper, we study the relationship between polynomial integrals on the unitary group and the conjugacy class expansion of symmetric functions in Jucys-Murphy elements. Our main result is an explicit formula for the top coefficients in the class expansion of monomial symmetric functions in Jucys-Murphy elements, from which we recover the first order asymptotics of polynomial integrals over $\U(N)$ as $N \rightarrow \infty$. Our results on class expansion include an analogue of Macdonald's result for the top connection coefficients of the class algebra, a generalization of Stanley and Olsh…

2012-02-07math-phmath.COmath.MP

Integration with respect to the Haar measure on unitary, orthogonal and symplectic group

We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freene…

2004-02-26math-phmath.MPmath.PR

更多相关论文收起更多论文

Feynman path integrals for magnetic Schrödinger operators on infinite weighted graphs

We prove a Feynman path integral formula for the unitary group $ \exp(-itL_{v,θ})$, $t\geq 0$, associated with a discrete magnetic Schrödinger operator $L_{v,θ}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$ |\exp(-itL_{v,θ})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y), $$ which controls the unitary group uniformly in the potentials in terms of a Schrödinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.

2017-09-06math-phmath.FAmath.MP

The Two-Parameter Free Unitary Segal-Bargmann Transform and its Biane-Gross-Malliavin Identification

Motivated by the two-parameter free unitary Segal-Bargmann transform in the form of conditional expectation, we derive the integral transform representation of the two-parameter free unitary Segal-Bargmann transform which coincides to the large-$N$ limit of the two-parameter Segal-Bargmann transform on the unitary group $\mathbb{U}(N)$ and explore its limiting behavior. We also extend the notion of circular systems in order to define a two-parameter free Segal-Bargmann transform and prove a version of Biane-Gross-Malliavin Theorem of the two-parameter free unitary Segal-Bargmann transform.

2016-09-02math.FAmath.OAmath.PR

Recursion relations for Unitary integrals, Combinatorics and the Toeplitz Lattice

The Toeplitz determinants (of increasing size) associated with the symbols $exp{t(z+z^{-1})}$ or $(1-ξ{z})^α (1-ξ{z^{-1}})^β$ satisfy recursion relations, thus expressing all the Toeplitz determinants as a rational function of the first few determinants. A. Borodin found these relations using Riemann-Hilbert methods. The nature of Borodin's relations pointed towards the Toeplitz lattice and its Virasoro algebra, as developed by the authors. In this paper, we take the Toeplitz and Virasoro approach for a fairly large class of symbols, leading to a systematic and simple way of generating such re…

2002-01-30math-phmath.COmath.MP

Weingarten Calculus

We consider the problem of computing the integral $$ \int_{\mathcal{U}(d)} u_{i_1j_1}\cdots u_{i_nj_n} \bar{u}_{i'_1j'_1} \cdots \bar{u}_{i'_{n'}j'_{n'}} dU, $$ where the integration takes place with respect to the probability Haar measure on the unitary group $\mathcal{U}(d)$, and the $u_{ij}$ denotes the $ij$-th entry of a unitary matrix $U$. We present a unified approach connecting classical results, the explicit formula for the integral given by B. Collins and P. Sniady and subsequent works of various authors providing different points of view. Finally we are able to provide an explicit fo…

2021-02-26math.PRmath.RT

Singular value statistics of matrix products with truncated unitary matrices

We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix with a random matrix. We show that the structure of polynomial ensembles and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint singular value density of a product of truncated unitary matrices and its corresponding correlation kernel which can be written as a double contour integral. This leads to hard edge scaling limits that also…

2015-01-16math-phmath.MPmath.PR

Lyapunov exponents for truncated unitary and Ginibre matrices

In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence' statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on $\operatorname{GL}_n(\mathbb{C})$, analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.

2022-01-31math-phmath.MPmath.PR

Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles

This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues' density, and write the universal limit in function of the Kontsevich matrix model ("matrix Airy function", as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral ("matrix Bessel function") known in the literature as generalized Kontsevich matrix model.

2016-10-19math-phmath.MPmath.PR