On Essential Self-Adjointness for Magnetic Schrödinger and Pauli Operators on the Unit Disc in $${\mathbb {R}^2}$$

We study the question of magnetic confinement of quantum particles on the unit disk \({\mathbb {D}}\) in \({\mathbb {R}^2}\) , i.e. we wish to achieve confinement solely by means of the growth of the magnetic field \({B(\vec x)}\) near the boundary of the disk. In the spinless case, we show that \({B(\vec x)\ge \frac{\sqrt 3}{2}\cdot\frac{1}{(1-r)^2}-\frac{1}{\sqrt 3}\frac{1}{(1-r)^2\ln \frac{1}{1-r}}}\) , for \({|\vec x|}\) close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants \({\frac{\sqrt 3}{2}}\) and \({-\frac{1}{\sqrt 3}}\) are optimal. This answers, in this context, an open question from Colin de Verdière and Truc (Ann Inst Fourier 2011, Preprint, arXiv:0903.0803v3). We also derive growth conditions for radially symmetric magnetic fields which lead to confinement of spin 1/2 particles.     

On the Convexity of the KdV Hamiltonian

Motivated by perturbation theory, we prove that the nonlinear part \({H^{*}}\) of the KdV Hamiltonian \({H^{kdv}}\), when expressed in action variables \({I = (I_{n})_{n \geqslant 1}}\), extends to a real analytic function on the positive quadrant \({\ell^{2}_{+}(\mathbb{N})}\) of \({\ell^{2}(\mathbb{N})}\) and is strictly concave near \({0}\). As a consequence, the differential of \({H^{*}}\) defines a local diffeomorphism near 0 of \({\ell_{\mathbb{C}}^{2}(\mathbb{N})}\). Furthermore, we prove that the Fourier-Lebesgue spaces \({\mathcal{F}\mathcal{L}^{s,p}}\) with \({-1/2 \leqslant s \leqslant 0}\) and \({2 \leqslant p < \infty}\), admit global KdV-Birkhoff coordinates. In particular, it means that \({\ell^{2}_+(\mathbb{N})}\) is the space of action variables of the underlying phase space \({\mathcal{F}\mathcal{L}^{-1/2,4}}\) and that the KdV equation is globally in time \({C^{0}}\)-well-posed on \({\mathcal{F}\mathcal{L}^{-1/2,4}}\).     

Disappearance of Mott Oscillations in Sub-barrier Elastic Scattering of Identical Nuclei and Atomic Ions

The scattering of identical nuclei at low energies exhibits conspicuous Mott oscillations which can be used to investigate the presence of components in the predominantly Coulomb interaction arising from several physical effects. It is found that at a certain critical value of the Sommerfeld parameter the Mott oscillations disappear and the cross section becomes quite flat. We call this effect Transverse Isotropy (TI). The critical value of the Sommerfeld parameter at which TI sets in is found to be \({\eta_{c} = \sqrt{3s + 2}}\), where s is the spin of the nuclei participating in the scattering. No TI is found in the Mott scattering of identical Fermionic nuclei. The critical center of mass energy corresponding to \({\eta_c}\) is found to be \({E_c = 0.40}\) MeV for \({\alpha + \alpha}\) (s = 0), 1.2 MeV for \({^{6}}\)Li + \({^{6}}\)LI (s = 1) and 7.1 MeV for \({^{10}}\)B + \({^{10}}\)B (s = 3). We further found that the inclusion of the nuclear interaction induces a significant modification in the TI. We suggest measurements at these sub-barrier energies for the purpose of extracting useful information about the nuclear interaction between light heavy ions. We also suggest extending the study of the TI to the scattering of identical atomic ions.     

Hölder Regularity of the Solutions of the Cohomological Equation for Roth Type Interval Exchange Maps

There is a general method for constructing a soliton hierarchy from a splitting \({{L_{\pm}}}\) of a loop group as positive and negative sub-groups together with a commuting linearly independent sequence in the positive Lie algebra \({\mathcal{L}_{+}}\). Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each f in the negative subgroup \({L_-}\) a solution \({u_{f}}\) of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function \({\tau_{f}}\) for each element \({f \in L_-}\). In this paper, we give integral formulas for variations of \({\ln\tau_{f}}\) and second partials of \({\ln\tau_{f}}\), discuss whether we can recover solutions \({u_{f}}\) from \({\tau_{f}}\), and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the \({GL(n,\mathbb{C})}\)-hierarchy.     

Generic Rigidity for Circle Diffeomorphisms with Breaks

We prove that \({C^r}\)-smooth (\({r > 2}\)) circle diffeomorphisms with a break, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, are generically, i.e., for almost all irrational rotation numbers, not \({C^{1+\varepsilon}}\)-rigid, for any \({\varepsilon > 0}\). This result complements our recent proof, joint with Khanin (Geom Funct Anal 24:2002–2028, 2014), that such maps are generically \({C^1}\)-rigid. It stands in remarkable contrast to the result of Yoccoz (Ann Sci Ec Norm Sup 17:333–361, 1984) that \({C^r}\)-smooth circle diffeomorphisms are generically \({C^{r-1-\varkappa}}\)-rigid, for any \({\varkappa > 0}\).     

Fully constrained Majorana neutrino mass matrices using $$\varvec{\varSigma (72\times 3)}$$

In 2002, two neutrino mixing ansatze having trimaximally mixed middle (\(\nu _2\)) columns, namely tri-chi-maximal mixing (\(\text {T}\chi \text {M}\)) and tri-phi-maximal mixing (\(\text {T}\phi \text {M}\)), were proposed. In 2012, it was shown that \(\text {T}\chi \text {M}\) with \(\chi =\pm \,\frac{\pi }{16}\) as well as \(\text {T}\phi \text {M}\) with \(\phi = \pm \,\frac{\pi }{16}\) leads to the solution, \(\sin ^2 \theta _{13} = \frac{2}{3} \sin ^2 \frac{\pi }{16}\), consistent with the latest measurements of the reactor mixing angle, \(\theta _{13}\). To obtain \(\text {T}\chi \text {M}_{(\chi =\pm \,\frac{\pi }{16})}\) and \(\text {T}\phi \text {M}_{(\phi =\pm \,\frac{\pi }{16})}\), the type I see-saw framework with fully constrained Majorana neutrino mass matrices was utilised. These mass matrices also resulted in the neutrino mass ratios, \(m_1:m_2:m_3=\frac{\left( 2+\sqrt{2}\right) }{1+\sqrt{2(2+\sqrt{2})}}:1:\frac{\left( 2+\sqrt{2}\right) }{-1+\sqrt{2(2+\sqrt{2})}}\). In this paper we construct a flavour model based on the discrete group \(\varSigma (72\times 3)\) and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric \(3\times 3\) matrix with six complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex six-dimensional representation of \(\varSigma (72\times 3)\). Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.     

Harmonic Pinnacles in the Discrete Gaussian Model

The 2D Discrete Gaussian model gives each height function \({\eta : {\mathbb{Z}^2\to\mathbb{Z}}}\) a probability proportional to \({\exp(-\beta \mathcal{H}(\eta))}\), where \({\beta}\) is the inverse-temperature and \({\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}\) sums over nearest-neighbor bonds. We consider the model at large fixed \({\beta}\), where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an \({L\times L}\) box with 0 boundary conditions concentrates on two integers M, M + 1 with \({M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}\). The key is a large deviation estimate for the height at the origin in \({\mathbb{Z}^{2}}\), dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on \({\eta\geq 0}\) (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where \({H\sim M/\sqrt{2}}\). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5–6):743–798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order \({\sqrt{\log L}}\). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.     

A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index { < 4}

It is well known that the quantum double \({D(N\subset M)}\) of a finite depth subfactor \({N\subset M}\), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor \({N\subset M}\) with index \({[M:N] < 4}\) the quantum double \({D(N\subset M)}\) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of \({E_6}\) can be realized as a \({\mathbb{Z}_2}\)-simple current extension of \({{{\rm SU}(2)}_{10}\times {{\rm Spin}(11)}_1}\) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor \({N\subset M }\) arises from \({\alpha}\)-induction of completely rational nets \({\mathcal{A}\subset \mathcal{B}}\) and there is a net \({\tilde{\mathcal{A}}}\) with the opposite braiding, then the quantum \({D(N\subset M)}\) is realized by completely rational net. We construct completely rational nets with the opposite braiding of \({{{\rm SU}(2)}_k}\) and use the well-known fact that all subfactors with index \({[M:N] < 4}\) arise by \({\alpha}\)-induction from \({{{\rm SU}(2)}_k}\).     

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment

A quantum system (with Hilbert space \({\mathcal {H}_{1}}\)) entangled with its environment (with Hilbert space \({\mathcal {H}_{2}}\)) is usually not attributed to a wave function but only to a reduced density matrix \({\rho_{1}}\). Nevertheless, there is a precise way of attributing to it a random wave function \({\psi_{1}}\), called its conditional wave function, whose probability distribution \({\mu_{1}}\) depends on the entangled wave function \({\psi \in \mathcal {H}_{1} \otimes \mathcal {H}_{2}}\) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of \({\mathcal {H}_{2}}\) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about \({\mu_{1}}\), e.g., that if the environment is sufficiently large then for every orthonormal basis of \({\mathcal {H}_{2}}\), most entangled states \({\psi}\) with given reduced density matrix \({\rho_{1}}\) are such that \({\mu_{1}}\) is close to one of the so-called GAP (Gaussian adjusted projected) measures, \({GAP(\rho_{1})}\). We also show that, for most entangled states \({\psi}\) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval \({[E, E+ \delta E]}\)) and most orthonormal bases of \({\mathcal {H}_{2}}\), \({\mu_{1}}\) is close to \({GAP(\rm {tr}_{2} \rho_{mc})}\) with \({\rho_{mc}}\) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then \({\mu_{1}}\) is close to \({GAP(\rho_\beta)}\) with \({\rho_\beta}\) the canonical density matrix on \({\mathcal {H}_{1}}\) at inverse temperature \({\beta=\beta(E)}\). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function.     

The Exoticness and Realisability of Twisted Haagerup–Izumi Modular Data

The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data \({\mathcal{D}{\rm Hg}}\) fits into a family \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\) , where n ≥  0 and \({\omega\in \mathbb{Z}_{2n+1}}\) . We show \({\mathcal{D}^0 {\rm Hg}_{2n+1}}\) is related to the subfactors Izumi hypothetically associates to the cyclic groups \({\mathbb{Z}_{2n+1}}\) . Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions, etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of Izumi type \({\mathbb{Z}_7, \mathbb{Z}_9}\) and \({\mathbb{Z}_3^2}\) , and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type \({\mathbb{Z}_{11},\mathbb{Z}_{13},\mathbb{Z}_{15},\mathbb{Z}_{17},\mathbb{Z}_{19}}\) (previously, Izumi had shown uniqueness for \({\mathbb{Z}_3}\) and \({\mathbb{Z}_5}\)), and we identify their modular data. We explain how \({\mathcal{D}{\rm Hg}}\) (more generally \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\)) is a graft of the quantum double \({\mathcal{D} Sym(3)}\) (resp. the twisted double \({\mathcal{D}^\omega D_{2n+1}}\)) by affine so(13) (resp. so\({(4n^2+4n+5)}\)) at level 2. We discuss the vertex operator algebra (or conformal field theory) realisation of the modular data \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\) . For example we show there are exactly 2 possible character vectors (giving graded dimensions of all modules) for the Haagerup VOA at central charge c = 8. It seems unlikely that any of this twisted Haagerup-Izumi modular data can be regarded as exotic, in any reasonable sense.     

The Structure of the Kac–Wang–Yan Algebra

The Lie algebra \({\mathcal{D}}\) of regular differential operators on the circle has a universal central extension \({\hat{\mathcal{D}}}\). The invariant subalgebra \({\hat{\mathcal{D}}^+}\) under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum \({\hat{\mathcal{D}}^+}\)-module with central charge \({c \in \mathbb{C}}\), and its irreducible quotient \({\mathcal{V}_c}\), possess vertex algebra structures, and \({\mathcal{V}_c}\) has a nontrivial structure if and only if \({c \in \frac{1}{2}\mathbb{Z}}\). We show that for each integer \({n > 0}\), \({\mathcal{V}_{n/2}}\) and \({\mathcal{V}_{-n}}\) are \({\mathcal{W}}\)-algebras of types \({\mathcal{W}(2, 4,\dots,2n)}\) and \({\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}\), respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group \({{\rm O}(n)}\) and the symplectic group \({{\rm Sp}(2n)}\), respectively. Based on Sergeev’s theorems on the invariant theory of \({{\rm Osp}(1, 2n)}\) we conjecture that \({\mathcal{V}_{-n+1/2}}\) is of type \({\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}\), and we prove this for \({n = 1}\). As an application, we show that invariant subalgebras of \({\beta\gamma}\)-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.     

Dorey’s Rule and the q-Characters of Simply-Laced Quantum Affine Algebras

Let \({U_q(\widehat{\mathfrak g})}\) be the quantum affine algebra associated to a simply-laced simple Lie algebra \({\mathfrak{g}}\) . We examine the relationship between Dorey’s rule, which is a geometrical statement about Coxeter orbits of \({\mathfrak{g}}\) -weights, and the structure of q-characters of fundamental representations V _i,a of \({U_q(\widehat{\mathfrak g})}\) . In particular, we prove, without recourse to the ADE classification, that the rule provides a necessary and sufficient condition for the monomial 1 to appear in the q-character of a three-fold tensor product \({V_{i,a}\otimes V_{j,b}\otimes V_{k,c}}\) .     

Positive Casimir and Central Characters of Split Real Quantum Groups

We consider the one parameter family \({\alpha \mapsto T_{\alpha}}\) (\({\alpha \in [0,1)}\)) of Pomeau-Manneville type interval maps \({T_{\alpha}(x) = x(1+2^{\alpha} x^{\alpha})}\) for \({x \in [0,1/2)}\) and \({T_{\alpha}(x)=2x-1}\) for \({x \in [1/2, 1]}\), with the associated absolutely continuous invariant probability measure \({\mu_{\alpha}}\). For \({\alpha \in (0,1)}\), Sarig and Gouëzel proved that the system mixes only polynomially with rate \({n^{1-1/{\alpha}}}\) (in particular, there is no spectral gap). We show that for any \({\psi \in L^{q}}\), the map \({\alpha \to \int_0^{1} \psi\, d \mu_{\alpha}}\) is differentiable on \({[0,1-1/q)}\), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For \({\alpha \ge 1/2}\) we need the \({n^{-1/{\alpha}}}\) decorrelation obtained by Gouëzel under additional conditions.     

Cyclotomic Gaudin Models: Construction and Bethe Ansatz

To any finite-dimensional simple Lie algebra \({\mathfrak{g}}\) and automorphism \({\sigma: \mathfrak{g}\to \mathfrak{g}}\) we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of \({U(\mathfrak{g})^{\otimes N}}\) generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case \({\sigma ={\rm id}}\).     

Long-Time Asymptotics for the Focusing NLS Equation with Time-Periodic Boundary Condition on the Half-Line

We consider the focusing nonlinear Schrödinger equation on the quarter plane. The initial data are vanishing at infinity while the boundary data are time- periodic, of the form \({a{\rm e}^{\i\alpha} {\rm e}^{2\i\omega t}}\) . The goal of this paper is to study the asymptotic behavior of the solution of this initial-boundary-value problem. The main tool is the asymptotic analysis of an associated matrix Riemann–Hilbert problem. We show that for \({\omega < -3a^2}\) the solution of the IBV problem has different asymptotic behaviors in different regions. In the region \({x > 4bt}\) , where \({b\mathop{:=} \sqrt{(a^2-\omega)/2} > 0}\) , the solution takes the form of the Zakharov-Manakov vanishing asymptotics. In a region of type \({4bt-\frac{N+1}{2a} {\rm log} t < x < 4bt}\) , where N is any integer, the solution is asymptotic to a train of asymptotic solitons. In the region \({4(b-a\sqrt2)t < x < 4bt}\) , the solution takes the form of a modulated elliptic wave. In the region \({0 < x < 4(b-a\sqrt2)t}\) , the solution takes the form of a plane wave.     

Non-Negative Integral Level Affine Lie Algebra Tensor Categories and Their Associativity Isomorphisms

For a finite-dimensional simple Lie algebra \({\mathfrak{g}}\), we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra \({{\widehat{\mathfrak{g}}}}\) at a fixed level \({\ell\in\mathbb{N}}\) with a certain tensor category of finite-dimensional \({\mathfrak{g}}\)-modules. More precisely, the category of level ? standard \({{\widehat{\mathfrak{g}}}}\)-modules is the module category for the simple vertex operator algebra \({L_{\widehat{\mathfrak{g}}}(\ell, 0)}\), and as is well known, this category is equivalent as an abelian category to \({\mathbf{D}(\mathfrak{g},\ell)}\), the category of finite-dimensional modules for the Zhu’s algebra \({A{(L_{\widehat{\mathfrak{g}}}(\ell, 0))}}\), which is a quotient of \({U(\mathfrak{g})}\). Our main result is a direct construction using Knizhnik–Zamolodchikov equations of the associativity isomorphisms in \({\mathbf{D}(\mathfrak{g},\ell)}\) induced from the associativity isomorphisms constructed by Huang and Lepowsky in \({{L_{\widehat{\mathfrak{g}}}(\ell, 0) - \mathbf{mod}}}\). This construction shows that \({\mathbf{D}(\mathfrak{g},\ell)}\) is closely related to the Drinfeld category of \({U(\mathfrak{g})}\)[[h]]-modules used by Kazhdan and Lusztig to identify categories of \({{\widehat{\mathfrak{g}}}}\)-modules at irrational and most negative rational levels with categories of quantum group modules.     

Off-Diagonal Decay of Toric Bergman Kernels

We study the off-diagonal decay of Bergman kernels \({\Pi_{h^k}(z,w)}\) and Berezin kernels \({P_{h^k}(z,w)}\) for ample invariant line bundles over compact toric projective kähler manifolds of dimension m. When the metric is real analytic, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) where \({D(z,w)}\) is the diastasis. When the metric is only \({C^{\infty}}\) this asymptotic cannot hold for all \({(z,w)}\) since the diastasis is not even defined for all \({(z,w)}\) close to the diagonal. Our main result is that for general toric \({C^{\infty}}\) metrics, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) as long as w lies on the \({\mathbb{R}_+^m}\)-orbit of z, and for general \({(z,w)}\), \({{\rm lim\,sup}_{k \to \infty} \frac{1}{k} {\rm log} P_{h^k}(z,w) \,\leq\, - D(z^*,w^*)}\) where \({D(z, w^*)}\) is the diastasis between z and the translate of w by \({(S^1)^m}\) to the \({\mathbb{R}_+^m}\) orbit of z. These results are complementary to Mike Christ’s negative results showing that \({P_{h^k}(z,w)}\) does not have off-diagonal exponential decay at “speed” k if \({(z,w)}\) lies on the same \({(S^1)^m}\)-orbit.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

The Feynman Propagator on Perturbations of Minkowski Space

The singular values squared of the random matrix product \({Y = {G_{r} G_{r-1}} \ldots G_{1} (G_{0} + A)}\), where each \({G_{j}}\) is a rectangular standard complex Gaussian matrix while A is non-random, are shown to be a determinantal point process with the correlation kernel given by a double contour integral. When all but finitely many eigenvalues of A*A are equal to bN, the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of \({0 < b < 1}\) is independent of b, and is in fact the same as that known for the case b =  0 due to Kuijlaars and Zhang. The critical regime of b =  1 allows for a double scaling limit by choosing \({{b = (1 - \tau/\sqrt{N})^{-1}}}\), and for this the critical kernel and outlier phenomenon are established. In the simplest case r =  0, which is closely related to non-intersecting squared Bessel paths, a distribution corresponding to the finite shifted mean LUE is proven to be the scaling limit in the supercritical regime of \({b > 1}\) with two distinct scaling rates. Similar results also hold true for the random matrix product \({T_{r} T_{r-1} \ldots T_{1} (G_{0} + A)}\), with each \({T_{j}}\) being a truncated unitary matrix.     

Quantum Symmetries and Strong Haagerup Inequalities

In this paper, we consider families of operators \({\{x_r\}_{r \in \Lambda}}\) in a tracial C*-probability space \({({\mathcal{A}}, \varphi)}\) , whose joint *-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups \({\{H_n^+\}_{n \in \mathbb {N}}}\) . We prove a strong form of Haagerup’s inequality for the non-self-adjoint operator algebra \({{\mathcal{B}}}\) generated by \({\{x_r\}_{r \in \Lambda}}\) , which generalizes the strong Haagerup inequalities for *-free R-diagonal families obtained by Kemp–Speicher (J Funct Anal 251:141–173, 2007). As an application of our result, we show that \({{\mathcal{B}}}\) always has the metric approximation property (MAP). We also apply our techniques to study the reduced C*-algebra of the free unitary quantum group \({U_n^+}\) . We show that the non-self-adjoint subalgebra \({{\mathcal{B}}_n}\) generated by the matrix elements of the fundamental corepresentation of \({U_n^+}\) has the MAP. Additionally, we prove a strong Haagerup inequality for \({{\mathcal{B}}_n}\) , which improves on the estimates given by Vergnioux’s property RD (Vergnioux in J Oper Theory 57:303–324, 2007).