Abstract
We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index \(\beta = 1,2,4\) respectively) where time corresponds to the number of terms in the product. More generally, we consider the \(\beta \)-Jacobi product process obtained by extrapolating to arbitrary \(\beta > 0\). For fixed time (i.e. number of factors is constant), we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When \(\beta = 2\), our results imply that the right edge converges to a process which interpolates between the Airy point process and a deterministic configuration. This process connects a time-parametrized family of point processes appearing in the works of Akemann–Burda–Kieburg and Liu–Wang–Wang across time. In the arbitrary \(\beta > 0\) case, our results show tightness of the particles near the right edge. The limiting moment formulas correspond to expressions for the Laplace transform of a conjectural \(\beta \)-generalization of the interpolating process.
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References
Ahn, A.: Airy point process via supersymmetric lifts. arXiv preprint arXiv:2009.06839 (2020)
Ahn, A.: Global universality of Macdonald plane partitions, Ann. Inst. Henri Poincaré Probab. Stat. (to appear)
Ahn, A., Strahov, E.: Product matrix processes with symplectic and orthogonal invariance via sym- metric functions. Int. Math. Res. Not. (2021). available at https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnab045/36900978/rnab045.pdf. rnab045
Ahn, A., Van Peski, R.: Lyapunov exponents for truncated unitary and Ginibre matrices. arXiv preprint arXiv:2109.07375 (2021)
Akemann, G., Baik, J., Di Francesco, P. (eds.): The Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2011)
Akemann, G., Burda, Z., Kieburg, M.: From integrable to chaotic systems: Universal local statistics of Lyapunov exponents. EPL (Europhys. Lett.) 126(4), 40001 (2019)
Akemann, G., Burda, Z., Kieburg, M.: Universality of local spectral statistics of products of random matrices. Phys. Rev. E 102(5), 052134 (2020)
Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)
Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Related Fields 158(1–2), 225–400 (2014)
Borodin, A., Corwin, I., Gorin, V., Shakirov, S.: Observables of Macdonald processes. Trans. Am. Math. Soc. 368(3), 1517–1558 (2016)
Borodin, A., Gorin, V.: General \(\beta \) -Jacobi corners process and the Gaussian free field. Commun. Pure Appl. Math. 68(10), 1774–1844 (2015)
Borodin, A., Gorin, V.: Moments match between the KPZ equation and the Airy point process. SIGMA Symmetry Integrability Geom. Methods Appl. 12, Paper No. 102, 7 (2016)
Borodin, A., Gorin, V., Guionnet, A.: Gaussian asymptotics of discrete \(\beta \) -ensembles. Publ. Math. Inst. Hautes Etudes Sci. 125, 1–78 (2017)
Borodin, A., Gorin, V., Strahov, E.: Product matrix processes as limits of random plane partitions (2018)
Borot, G., Guionnet, A.: Asymptotic expansion of \(\beta \) matrix models in the one-cut regime. Commun. Math. Phys. 317(2), 447–483 (2013)
Borot, G., Guionnet, A.: Asymptotic expansion of beta matrix models in the multi-cut regime (2013)
Collins, B., Mingo, J.A., Śniady, P., Speicher, R.: Second order freeness and fluctuations of random matrices. III. Higher order freeness and free cumulants. Doc. Math. 12, 1–70 (2007)
Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264(3), 773–795 (2006)
Coston, N., O’Rourke, S.: Gaussian fluctuations for linear eigenvalue statistics of products of independent IID random matrices. J. Theor. Probab. (2019)
Dimitrov, E., Knizel, A.: Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions. J. Funct. Anal. 276(10), 3067–3169 (2019)
Ding, X., Ji, H.C.: Local laws for multiplication of random matrices and spiked invariant model. arXiv preprint arXiv:2010.16083 (2020)
Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43(11), 5830–5847 (2002)
Dumitriu, I., Paquette, E.: Spectra of overlapping Wishart matrices and the Gaussian free field. Random Matrices Theory Appl. 7(2), 1850003, 21 (2018)
Forrester, P.J.: Log-gases and random matrices, London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton (2010)
Gorin, V., Marcus, A.: Crystallization of random matrix orbits. Int. Math. Res. Not. rny052, no. 1 (2018)
Gorin, V., Shkolnikov, M.: Stochastic Airy semigroup through tridiagonal matrices. Ann. Probab. 46(4), 2287–2344 (2018)
Gorin, V., Sun, Y.: Gaussian fluctuations for products of random matrices. Am. J. Math. 144(2), 287–393 (2022)
Gorin, V., Zhang, L.: Interlacing adjacent levels of \(\beta \) -Jacobi corners processes. Probab. Theory Related Fields 172(3–4), 915–981 (2018)
Guionnet, A., Novak, J.: Asymptotics of unitary multimatrix models: the Schwinger-Dyson lattice and topo- logical recursion. J. Funct. Anal. 268(10), 2851–2905 (2015)
Heckman, G.J.: Root systems and hypergeometric functions. II. Compositio Math. 64(3), 353–373 (1987)
Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions. I. Compositio Math. 64(3), 329–352 (1987)
Heckman, G., Schlichtkrull, H.: Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics, vol. 16. Academic Press, Inc., San Diego (1994)
Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of gaussian analytic functions and deter- minantal point processes, vol. 51. American Mathematical Society (2009)
Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)
Johansson, K.: Determinantal processes with number variance saturation. Commun. Math. Phys. 252(1), 111–148 (2004)
Kieburg, M., Forrester, P.J., Ipsen, J.R.: Multiplicative convolution of real asymmetric and real anti-symmetric matrices. Adv. Pure Appl. Math. 10(4), 467–492 (2019)
Kieburg, M., Kösters, H.: Products of random matrices from polynomial ensembles. Annales de l’institut henri poincaré, probabilités et statistiques 98–126 (2019)
Kieburg, M., Kuijlaars, A.B.J., Stivigny, D.: Singular value statistics of matrix products with truncated unitary matrices. Int. Math. Res. Not. IMRN 11, 3392–3424 (2016)
Kieburg, M., Zhang, J.: Derivative principles for invariant ensembles. arXiv preprint arXiv:2007.15259 (2020)
Kriecherbauer, T., Shcherbina, M.: Fluctuations of eigenvalues of matrix models and their applications (2010)
Liu, D.-Z., Wang, D., Wang, Y.: Lyapunov exponent, universality and phase transition for products of random matrices (2018)
Macdonald, I.G.: Symmetric functions and Hall polynomials, Second, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
Mingo, J.A., Popa, M.: Real second order freeness and Haar orthogonal matrices. J. Math. Phys. 54(5), 051701, 35 (2013)
Negut, A.: Operators on symmetric polynomials (2013)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX)
Opdam, E.M.: Root systems and hypergeometric functions III. Compositio Math. 67(1), 21–49 (1988)
Opdam, E.M.: Root systems and hypergeometric functions. IV. Compositio Math. 67(2), 191–209 (1988)
Peccati, G., Taqqu, M.S.: Wiener chaos: moments, cumulants and diagrams, Bocconi & Springer Series, vol. 1, Springer, Milan; Bocconi University Press, Milan, 2011. A survey with computer implementation, Supplementary material available online
Petz, D., Réy, J.: On asymptotics of large Haar distributed unitary matrices. Period. Math. Hungar. 49(1), 103–117 (2004)
Redelmeier, C.E.I.: Quaternionic second-order freeness and the fluctuations of large symplectically invariant random matrices (2015)
Shcherbina, M.: Fluctuations of linear eigenvalue statistics of \(\beta \) matrix models in the multi-cut regime. J. Stat. Phys. 151(6), 1004–1034 (2013)
Sinai, Y.G., Soshnikov, A.B.: A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Funktsional. Anal. i Prilozhen. 32(2), 56–79, 96 (1998)
Sodin, S.: Several applications of the moment method in random matrix theory, Proceedings of the International Congress of Mathematicians—Seoul (2014). Vol. III, 2014, pp. 451– 475
Soshnikov, A.: Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207(3), 697–733 (1999)
Vasilchuk, V.: On the fluctuations of eigenvalues of multiplicative deformed unitary invariant ensembles. Random Matrices Theory Appl. 5(2), 1650007, 28 (2016)
Voiculescu, D.: Multiplication of certain noncommuting random variables. J. Oper. Theory 18(2), 223–235 (1987)
Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)
Zhang, J., Kieburg, M., Forrester, P.J.: Harmonic analysis for rank-1 randomised horn problems. Lett. Math. Phys. 111(4), 1–27 (2021)
Acknowledgements
I would like to thank my advisor Vadim Gorin for suggesting the study of matrix products in orthogonal and symplectic symmetry classes, for useful discussions, and giving feedback on several drafts. I also thank the anonymous referees for many helpful comments which improved this text. The author was partially supported by National Science Foundation Grant DMS-1664619.
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The author was partially supported by NSF Grant DMS-1664619 (see acknowledgments).
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Appendices
Appendix A
We demonstrate that \(\beta \)-Jacobi product processes with certain parameters can be realized as the squared singular values of products of truncated Haar-distributed \({\mathbb {U}}_\beta \) matrices for \(\beta = 1,2,4\).
Fix \(\theta \in \{1/2,1,2\}\), and positive integer parameters \(L_1,L_2,\ldots \), N, \(N_1,N_2,\ldots ,\) such that
where \(N_0 := N\). Let \(U_1,U_2,\ldots \) be independent random matrices such that \(U_T\) is a random Haar \({\mathbb {U}}_{2\theta }(L_T)\) matrix. Let \(X_T\) be an \(N_T \times N_{T-1}\) submatrix (or truncation) of \(U_T\),
and \(\mathbf {y}^{(T)} \in {\mathcal {R}}^N\) be the ordered vector of eigenvalues of \(Y_T^*Y_T\).
Theorem A.1
Suppose \(\mathbf {x}^{(1)},\mathbf {x}^{(2)},\ldots \) are independent random vectors in \({\mathcal {R}}^N\) such that \(\mathbf {x}^{(T)} \sim \P _\theta ^{\alpha _T,M_T,N}\) where \(\alpha _T = N_T - N + 1\) and \(M_T = L_T - N_T\). Then \(\left( \mathbf {y}^{(T)}\right) _{T\in {\mathbb {Z}}^+}\) is equal in distribution to
Lemma A.2
Let \(n,n_1,n_2,m \in {\mathbb {Z}}_+\) such that \(n \le n_i \le m\), \(i = 1,2\). Suppose X and \(\widetilde{X}\) are respectively \(n_2 \times n_1\) and \(n_2\times n\) truncations of a random Haar \({\mathbb {U}}_{2\theta }(m)\) matrix. Let W be a fixed \(n_1 \times n\) matrix and \(\widetilde{W} := (W^*W)^{1/2}\). If \(\sigma (A) \in {\mathcal {R}}^N\) denotes the singular values of a matrix A, then \(\sigma (XW) \overset{d}{=} \sigma (\widetilde{X} \widetilde{W})\).
Proof
Let \(P_{a\times b}\) denote the \(a\times b\) matrix with the \(\min (a,b)\times \min (a,b)\) identity matrix in the upper left corner and 0 elsewhere. The singular value decomposition of M gives
where \(\Sigma = \mathrm {diag}(\sigma (W))\), \(U \in {\mathbb {U}}_{2\theta }(n_1),V \in {\mathbb {U}}_{2\theta }(n)\). Then
Observe that
using the fact that the distributions of X and \(\widetilde{X}\) are invariant under right translation by \({\mathbb {U}}_{2\theta }\). Thus
which proves the lemma. \(\square \)
Proof of Theorem A.1 Set
Let \(\widetilde{X}_1,\widetilde{X}_2,\ldots \) be independent such that \(\widetilde{X}_T\) is an \(N_T \times N\) truncation of a Haar \({\mathbb {U}}_{2\theta }(M_T)\) matrix, and
By Proposition 3.3 and the definition of \(\boxtimes _{2\theta }\), \(\widetilde{\mathbf {y}}^{(T)}\) is the squared singular values vector of \(\widetilde{Y}_T\). We have \(\mathbf {y}^{(1)} \overset{d}{=} \widetilde{\mathbf {y}}^{(1)}\), and for any fixed \(\mathbf {u} \in {\mathcal {U}}^N\),
by Lemma A.2, thus completing our proof. \(\square \)
Appendix B. Observables of Schur Processes
We derive a contour integral formula for joint moments of a special case of Schur processes—Macdonald processes in the case \(q = t\).
For \(q = t\), the Macdonald symmetric functions become the Schur functions
which are independent of t. Thus properties for Macdonald symmetric functions are inherited by the Schur functions. For example, for any countable set of variables X, Y, (2.2) implies
The Schur functions have an explicit form
for details see [42, Chapter I, Sections 3 & 4]
Definition B.1
Suppose \(\mathbf {a} := (a_1,\ldots ,a_N) \in {\mathbb {R}}_+^N\), \(\mathbf {b} \in (b_1,\ldots ,b_M) \in {\mathbb {R}}_+^M\) such that \(a_ib_j < 1\) for \(1 \le i \le N\), \(1 \le j \le M\). Let \(\mathbb {SP}_{\mathbf {a},\mathbf {b}}\) denote the measure on \({\mathbb {Y}}^M = {\mathbb {Y}}\times \cdots \times {\mathbb {Y}}\) where
for \(\mathbf {\lambda } := (\lambda ^1,\ldots ,\lambda ^M) \in {\mathbb {Y}}^M\).
Remark 9
The measure \(\mathbb {SP}_{\mathbf {a},\mathbf {b}}\) is the \(q = t\) case of the so-called ascending Macdonald process. We have
as a consequence of (B.1) and the \(q = t\) case of (2.1).
We prove the following contour integral formula for joint expectations of \({\varvec{p}}_t\) (recall Definition 4.4).
Theorem B.2
Suppose \(\mathbf {a} := (a_1,\ldots ,a_N) \in {\mathbb {R}}_+^N\), \(\mathbf {b} \in (b_1,\ldots ,b_M) \in {\mathbb {R}}_+^M\) such that \(a_ib_j < 1\) for \(1 \le i \le M\), \(1 \le j \le N\). For real \(t_1,\ldots ,t_m > 0\), integers \(1 \le n_m \le \cdots \le n_1 \le M\), and \(\mathbf {\lambda } \sim \mathbb {SP}_{\mathbf {a},\mathbf {b}}\), we have
where the \(z_i\)-contour \({\mathfrak {Y}}_i\) is positively oriented around \(0, \{t_ia_\ell \}_{\ell =1}^N\) but does not encircle \(\{b_\ell ^{-1}\}_{\ell =n_i}^M\), and is encircled by \(t_j^{-1} {\mathfrak {Y}}_j\), \(t_i {\mathfrak {Y}}_j\) for \(j > i\); given that such contours exist.
Proof
Let \(D_t^{x_1,\ldots ,x_n}\) act on functions in \((x_1,\ldots ,x_n)\) by
where \(T_{t,x_i}\) is the t-shift operator
Using (B.2), we have
On the other hand, if \(n \le M\) and f is analytic in a neighborhood of \(\{0\} \cup \{b_i\}_{i=1}^n\) such that f(tw)/f(w) is also analytic in this neighborhood, then by residue expansion,
where the contour is positively oriented around 0, \(\{b_i\}_{i=1}^n\), but no other poles of the integrand. By iterating, we obtain
where the \(w_j\)-contour is positively oriented around 0, \(\{b_j\}_{j=1}^{n_i}\), but no other poles of the integrand, and enclosed by \(t_i w_i\), \(t_j^{-1} w_i\) for \(i < j\); assuming such contours exist. Taking \(z_i = w_i^{-1}\) completes the proof. \(\square \)
Appendix C
We recall the notion of cumulants and some basic properties.
Definition C.1
For any finite set S, let \(\Theta _S\) be the collection of all set partitions of S, that is
For a random vector \(\mathbf {u} = (u_1,\ldots ,u_m)\) and any \(v_1,\ldots ,v_\nu \in \{u_1,\ldots ,u_m\}\), define the (order \(\nu \)) cumulant
The definition implies that for any random vector \(\mathbf {u}\), the existence of all cumulants of order up to \(\nu \) is equivalent to the existence of all moments of order up to \(\nu \). Note that the cumulants of order 2 are exactly the covariances:
We can also express the cumulant as
For further details see [48, Sections 3.1 & 3.2] wherein the agreement between (C.1) and (C.2) is shown by taking (C.2) as the definition and proving (C.1). Note that (C.2) implies
Lemma C.2
A random vector is Gaussian if and only if all cumulants of order \(\ge 3\) vanish.
We have a useful inversion lemma
Lemma C.3
If K and E are complex-valued functions on the set of nonempty subsets of \([[1,\nu ]]\) such that
then
Proof
Let \(E_{n_1,\ldots ,n_\nu } \in {\mathbb {C}}\) with \(E_{0,\ldots ,0} = 1\). Define the formal power series
For each nonempty \(S \subset [[1,\nu ]]\), let \(K_{n_1,\ldots ,n_\nu } = K(S)\) if \(n_j = \mathbf {1}_{j\in S}\). The relation \(E(\mathbf {t}) = e^{K(\mathbf {t})}\) implies
for any nonempty \(S \subset [[1,\nu ]]\) where \(n_j = \mathbf {1}_{j\in S}\). The relation \(K(\mathbf {t}) = \log E(\mathbf {t})\) now implies (C.3). \(\square \)
Appendix D. Convergence of Laplace transform
We provide calculations which demonstrate that the Laplace transform of the interpolating process converges to that of the picket fence, a Gaussian random variable, and the Airy point process in the respect regimes discussed in Sect. 1.2. We consider each regime in a separate sub-appendix below. The key input for each of these convergence results is the (fixed-time) formula for the Laplace transform of the interpolating process from Theorem 1.4: For \(c_1,\ldots ,c_m > 0\), we have
where the \(u_i\)-contour is a positively oriented contour around \(-c_i,-c_i+1,-c_i+2,\ldots \) which starts and ends at \(+\infty \), and the \(u_j\) contour contains \(u_i + c_i\) and \(u_i - c_j\) for \(j > i\).
1.1 D.1 \(\widehat{T} \rightarrow \infty \): Picket Fence Statistics
We demonstrate the convergence of \(\{\widehat{T}^{-1} {\mathfrak {x}}_i^{(\widehat{T})}\}_{i=1}^\infty \) to picket fence statistics \(\{-i+\tfrac{1}{2}\}_{i=1}^\infty \) by way of Laplace transforms. More precisely, we show the convergence
weakly in probability, as \(\widehat{T} \rightarrow \infty \). This convergence is a consequence of the following:
Theorem D.1
For \(0< \widehat{c} < 1\)
and
Indeed, Theorem D.1 implies the intermediary weak convergence in probability
of finite measures as \(\widehat{T} \rightarrow \infty \) for some \(\varepsilon > 0\) small and fixed, see e.g. [8, Lemma 2.1.7]. This clearly implies (D.2).
Proof
Recall that
Starting from (D.1) with \(m = 1\) and evaluating residues, we have
where the final equality follows from the reflection formula for the Gamma function
Taking \(c = \widehat{T}^{-1} \widehat{c}\) and sending \(\widehat{T} \rightarrow \infty \), we obtain (D.3).
Applying (D.1) again, we have
where the \(u_i\)-contour is positively oriented around \(-c,-c+1,-c+2,\ldots \) starting and ending at \(+\infty \) for \(i = 1,2\), and the \(u_2\)-contour encloses \(u_1 + c\) and \(u_1 - c\). Taking \(c = \widehat{T}^{-1} \widehat{c}\) and sending \(\widehat{T} \rightarrow \infty \), we obtain
where the \(u_i\)-contour is positively oriented around \({\mathbb {R}}_{\ge 0}\), starting and ending at \(+\infty \) for \(i = 1,2\), and the \(u_2\)-contour encloses the \(u_1\)-contour. Since the integrand decays as \(u_1 \rightarrow +\infty \) and is analytic as a function of \(u_1\) in the domain enclosed by the \(u_1\)-contour, the right hand side evaluates to 0. This proves (D.4). \(\square \)
1.2 D.2 \(\widehat{T} \rightarrow \infty \): Gaussian Fluctuations
We show that
as \(\widehat{T} \rightarrow \infty \) using Laplace transforms. More precisely, we prove the following:
Theorem D.2
For \(0< \widehat{c} < 1\), we have
and for \(0< \widehat{c}_1, \widehat{c}_2 < 1/2\),
We first show how Theorem D.2 implies (D.6):
Proof
(Proof of (D.6) using Theorem D.2) Let \(\rho ^{(\widehat{T})}_k\) denote the \(k\hbox {th}\) correlation function of \(\left\{ \widehat{T}^{-1/2}\left( {\mathfrak {x}}_i^{(\widehat{T})} + \tfrac{\widehat{T}}{2} \right) \right\} _{i=1}^\infty \). We first note that (D.7) implies
weakly as \(\widehat{T} \rightarrow \infty \). The problem is that \(\rho ^{(\widehat{T})}_1\) is not quite the density of \(\widehat{T}^{-1/2}\left( {\mathfrak {x}}_1^{(\widehat{T})} + \tfrac{\widehat{T}}{2}\right) \), which we denote by \(p^{(\widehat{T})}\), and what we want is
weakly as \(\widehat{T} \rightarrow \infty \). For this, we need an argument which tells us that \(\rho ^{(\widehat{T})}_1\) is approximately given by \(p^{(\widehat{T})}\).
Observe that
as \(\widehat{T} \rightarrow \infty \), where the last line follows from Theorem D.2. As a consequence, we have
for any \(a \in {\mathbb {R}}\). Since
where
by e.g. [33, (1.2.2) & (1.2.3)], we find that
as \(\widehat{T} \rightarrow \infty \), and in particular
Therefore
where the first equality uses [33, (1.2.2) & (1.2.3)] again. Combining the above with (D.9), we obtain (D.10) as desired. \(\square \)
We now prove Theorem D.2.
Proof of Theorem D.2 We have the expansion (D.5) from the proof of Theorem D.1. Starting from there and taking \(c = \widehat{c} \widehat{T}^{-1/2}\), we have
The \(n = 0\) term dominates the summation as \(\widehat{T} \rightarrow \infty \) and we obtain (D.9).
To prove (D.8), we obtain a similar residue expansion for (D.1) when \(m = 2\):
where we have also applied the reflection formula for the Gamma function as in the proof of Theorem D.1. The second summation is over distinct \(n_1,n_2 \in \{0,1,2,\ldots \}\). Taking \(c_i = \widehat{c}_i \widehat{T}^{-1/2}\), the exponential terms dictate growth of each term as \(\widehat{T} \rightarrow \infty \). Thus we find that only the \(n = 0\) term in the first summation survives in the limit and (D.8) follows. \(\square \)
1.3 D.3 Airy point process
The process \(\zeta _1^{(\widehat{T})},\zeta _2^{(\widehat{T})},\ldots \) given by
converges to the Airy point process \({\mathfrak {a}}_1,{\mathfrak {a}}_2,\ldots \) as \(\widehat{T} \rightarrow 0\). In this sub-appendix we demonstrate, by way of a direct computation on Laplace transforms which is not fully rigorous, that
Theorem D.3
For \(\widehat{c}_1,\ldots ,\widehat{c}_m > 0\),
Proof Idea
From (D.11), it suffices to show
as \(\widehat{T} \rightarrow 0\). The Laplace transform for the correlation functions of the Airy point process can be computed explicitly, see [1, Appendix B], as
where the \(z_i\)-contour is a vertical contour from \(-\mathbf {i}\infty \) to \(\mathbf {i}\infty \) such that
whenever \(1 \le i < j \le m\).
Therefore, we use the formula (D.1) with \(c_i = 2^{1/3} \widehat{T}^{-2/3} \widehat{c}_i\) and send \(\widehat{T} \rightarrow 0\) to see the desired asymptotics of (D.1) with the help of the explicit formula (D.13). The analysis of (D.1) relies on steepest descent which we outline. Set
Then after changing variables \(v_i = -u_i\) in (D.1), we have
where the contour conditions for \(u_i\) translates to the condition that \(v_i - c_i\) and \(v_i + c_j\) are contained in the \(v_j\) contour for \(1 \le i < j \le m\). Observe that
where \(\psi \) is the digamma function. The asymptotics below hold for \(|\arg v| < \pi - \delta \) below for some fixed \(\delta > 0\) (which we note does not correspond to the entire range of our contour integration). We have the asymptotic expansion
as \(|v| \rightarrow \infty \), which can be differentiated arbitrarily many times for asymptotic expansions of the derivatives of \(f_{\widehat{T}}\). In particular, we find
Thus, if \(c \asymp T^{-2/3}\) and \(v \asymp \widehat{T}^{-1}\) as \(\widehat{T} \rightarrow 0\), we have
as \(|v| \rightarrow \infty \), where we see that the first summand on the right hand side is positive for \(\widehat{T}\) large and is the leading order term from (D.15). Therefore, we want to localize the integral at a saddle point of the first term. Then we have a solution \(v_{\widehat{T}}\) to
which satisfies
from which we have
using the expansion (D.15). If \(v - v_{\widehat{T}} = O(\widehat{T}^{-2/3 + \varepsilon })\) and \(c \asymp \widehat{T}^{-2/3}\) for \(\varepsilon > 0\) very small but fixed, then (D.16) implies
Writing \(c = 2^{1/3}\widehat{T}^{-2/3} \widehat{c}\) for \(\widehat{c} > 0\) fixed and \(z = 2^{-1/3} \widehat{T}^{2/3}(v - v_{\widehat{T}})\), we obtain
By choosing appropriate steepest descent contours, which we omit the details of here, we may localize the integral in (D.14) near \(v = v_{\widehat{T}}\). Using the asymptotics above, localizing the integrals near \(v_{\widehat{T}}\) (ignoring the details of justifying this localization), and changing variables \(z_i = 2^{-1/3} \widehat{T}^{2/3} (v_i - v_{\widehat{T}})\) in (D.14) and recalling \(c_i = 2^{1/3} \widehat{T}^{-2/3} \widehat{c}_i\), we obtain
where the \(z_i\)-contours are vertical contours from \(-\mathbf {i}\infty \) to \(\mathbf {i}\infty \) such that
whenever \(1 \le i < j \le m\). After changing variables so that \(z_i\) is replaced by \(z_i + \frac{\widehat{c}}{2}\), we recognize that the right hand side becomes
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Ahn, A. Fluctuations of \(\beta \)-Jacobi product processes. Probab. Theory Relat. Fields 183, 57–123 (2022). https://doi.org/10.1007/s00440-022-01109-0
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DOI: https://doi.org/10.1007/s00440-022-01109-0
Keywords
- Products of random matrices
- Singular values
- Beta ensemble
- Macdonald processes
Mathematics Subject Classification
- 15B52
- 60B20
- 33D52