Asymptotic Behavior of the Selberg Zeta Functions for Degenerating Families of Hyperbolic Manifolds

Let {M _k } be a degenerating sequence of finite volume, hyperbolic manifolds of dimension d, with d = 2 or d = 3, with finite volume limit M _∞. Let \({Z_{M_{k}} (s)}\) be the associated sequence of Selberg zeta functions, and let \({{\mathcal{Z}}_{k} (s)}\) be the product of local factors in the Euler product expansion of \({Z_{M_{k}} (s)}\) corresponding to the pinching geodesics on M _k . The main result in this article is to prove that \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) converges to \({Z_{M_{\infty}} (s)}\) for all \({s \in \mathbf{C}}\)with Re(s) > (d ? 1)/2. The significant feature of our analysis is that the convergence of \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) to \({Z_{M_{\infty}} (s)}\) is obtained up to the critical line, including the right half of the critical strip, a region where the Euler product definition of the Selberg zeta function does not converge. In the case d = 2, our result reproves by different means the main theorem in Schulze (J Funct Anal 236:120–160, 2006).     

Evaluation of spectral dispersion of optical constants of a-Se films from their normal-incidence transmittance spectra using Swanepoel algebraic envelope approach

Spectral dispersions of index of refraction \({n(\lambda )}\) and extinction coefficient \({\kappa (\lambda )}\) of undoped amorphous selenium (a-Se) films of three thicknesses (d?≈?0.5, 0.75, and 1.0 µm) were evaluated by analyzing experimental room-temperature normal-incidence transmittance-wavelength (\({{T_{{\text{exp}}}}(\lambda )} - \lambda\)) data (λ =?400–1100 µm) of their air-supported {a-Se film/thick glass slide}-stacks using Swanepoel’s transmission envelope theory of uniform films. Above a wavelength \({{\lambda _c}\,\, \approx \,\,640\;{\text{nm}}}\), as-measured \({{T_{{\text{exp}}}}(\lambda )}\,\, - \,\lambda\) spectra display well-resolved maxima and minima, with minor shrinkage in transparent and weak absorption regions (750–1100 nm). Below \({\lambda _{\text{c}}}\), a smeared sharp decline of \({{T_{{\text{exp}}}}(\lambda )}\) with decreasing λ, signifying strong absorption in a-Se films and existence of band-tail localized states. For λ > λ _c, the \({n\,(\lambda )}\, - \,\lambda\) data retrieved from algebraic envelope procedures followed a Sellmeier-like dispersion relation, with the best-fit values of high-frequency dielectric constant \({{\varepsilon _\infty }\, \approx \,\,{\text{4.9}}}\), static index of refraction \({{n_{\text{0}}} = n\left( {E\, \to \,{\text{0}}} \right)\,\, \approx \,\,{\text{2.43}}}\), and resonance wavelength \({{\lambda _0}\, \approx \,490\,\,{\text{nm}}}\), which may be assigned to onset of photogeneration in a-Se. Urbach-like dependency of absorption coefficient \({\alpha (h{{\nu }})}\) of a-Se films on photon energy \({h{{\nu }}}\) was realized with an Urbach-tail breadth of 85 meV. All achieved optical parameters were found to be slightly dependent on film thickness. Findings of present algebraic analysis are consistent with reported literature results obtained on the basis of other optical analytical approaches.     

Noncommutative Lévy Processes for Generalized (Particularly Anyon) Statistics

Let \({T=\mathbb R^d}\) . Let a function \({QT^2\to\mathbb C}\) satisfy \({Q(s,t)=\overline{Q(t,s)}}\) and \({|Q(s,t)|=1}\). A generalized statistics is described by creation operators \({\partial_t^\dagger}\) and annihilation operators ? _t , \({t\in T}\), which satisfy the Q-commutation relations: \({\partial_s\partial^\dagger_t = Q(s, t)\partial^\dagger_t\partial_s+\delta(s, t)}\) , \({\partial_s\partial_t = Q(t, s)\partial_t\partial_s}\), \({\partial^\dagger_s\partial^\dagger_t = Q(t, s)\partial^\dagger_t\partial^\dagger_s}\). From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which Q(s, t) is equal to q if s < t, and to \({\bar q}\) if s > t. Here \({q\in\mathbb C}\) , |q| = 1. We start the paper with a detailed discussion of a Q-Fock space and operators \({(\partial_t^\dagger,\partial_t)_{t\in T}}\) in it, which satisfy the Q-commutation relations. Next, we consider a noncommutative stochastic process (white noise) \({\omega(t)=\partial_t^\dagger+\partial_t+\lambda\partial_t^\dagger\partial_t}\) , \({t\in T}\) . Here \({\lambda\in\mathbb R}\) is a fixed parameter. The case λ = 0 corresponds to a Q-analog of Brownian motion, while λ ≠ 0 corresponds to a (centered) Q-Poisson process. We study Q-Hermite (Q-Charlier respectively) polynomials of infinitely many noncommutatative variables \({(\omega(t))_{t\in T}}\) . The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding Lévy processes. To this end, we recursively define Q-cumulants of a field \({(\xi(t))_{t\in T}}\). This allows us to define a Q-Lévy process as a field \({(\xi(t))_{t\in T}}\) whose values at different points of T are Q-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a Q-Lévy process, and derive a Nualart–Schoutens-type chaotic decomposition for such a process.     

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.     

T-Duality Simplifies Bulk-Boundary Correspondence

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice \({{{\mathbb{Z}}}^{4}}\), for the weakly coupled n-component \({|\varphi|^{4}}\) spin model for all \({n \ge 1}\), and for the continuous-time weakly self-avoiding walk. For the \({|\varphi|^{4}}\) model, we prove that the critical two-point function has |x|^?2 (Gaussian) decay asymptotically, for \({n \ge 1}\). We also determine the asymptotic decay of the critical correlations of the squares of components of \({\varphi}\), including the logarithmic corrections to Gaussian scaling, for \({n \ge 1}\). The above extends previously known results for n = 1 to all \({n \ge 1}\), and also observes new phenomena for n > 1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the “watermelon” network consisting of p weakly mutually- and self-avoiding walks, for all \({p \ge 1}\), including the logarithmic corrections. This extends a previously known result for p = 1, for which there is no logarithmic correction, to a much more general setting. In addition, for both models, we study the approach to the critical point and prove the existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the n = 0 case of the \({|\varphi|^{4}}\) model, and provides a unified treatment of both models, and of all the above results.     

Parabolic Presentations of the Super Yangian {Y(\mathfrak{gl}_{M|N})} Associated with Arbitrary 01-Sequences

Let μ be an arbitrary composition of M + N and let \({\mathfrak{s}}\) be an arbitrary \({0^{M}1^{N}}\)- sequence. A new presentation, depending on \({\mu \rm and \mathfrak{s}}\), of the super Yangian Y_M|N associated to the general linear Lie superalgebra \({\mathfrak{gl}_{M|N}}\) is obtained.     

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.     

The Vlasov-Poisson Dynamics as the Mean Field Limit of Extended Charges

We consider the discrete Gaussian Free Field in a square box in \({\mathbb{Z}^2}\) of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as \({N \to \infty}\). Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an r_N-neighborhood thereof, we prove that the associated process tends, whenever \({r_N \to \infty}\) and \({r_N/N \to 0}\), to a Poisson point process with intensity measure \({Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}\), where \({\alpha:= 2/\sqrt{g}}\) with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]². In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field.     

Likelihood analysis of the sub-GUT MSSM in light of LHC 13-TeV data

We describe a likelihood analysis using MasterCode of variants of the MSSM in which the soft supersymmetry-breaking parameters are assumed to have universal values at some scale \(M_\mathrm{in}\) below the supersymmetric grand unification scale \(M_\mathrm{GUT}\), as can occur in mirage mediation and other models. In addition to \(M_\mathrm{in}\), such ‘sub-GUT’ models have the 4 parameters of the CMSSM, namely a common gaugino mass \(m_{1/2}\), a common soft supersymmetry-breaking scalar mass \(m_0\), a common trilinear mixing parameter A and the ratio of MSSM Higgs vevs \(\tan \beta \), assuming that the Higgs mixing parameter \(\mu > 0\). We take into account constraints on strongly- and electroweakly-interacting sparticles from \(\sim 36\)/fb of LHC data at 13 TeV and the LUX and 2017 PICO, XENON1T and PandaX-II searches for dark matter scattering, in addition to the previous LHC and dark matter constraints as well as full sets of flavour and electroweak constraints. We find a preference for \(M_\mathrm{in}\sim 10^5\) to \(10^9 \,\, \mathrm {GeV}\), with \(M_\mathrm{in}\sim M_\mathrm{GUT}\) disfavoured by \(\Delta \chi ^2 \sim 3\) due to the \(\mathrm{BR}(B_{s, d} \rightarrow \mu ^+\mu ^-)\) constraint. The lower limits on strongly-interacting sparticles are largely determined by LHC searches, and similar to those in the CMSSM. We find a preference for the LSP to be a Bino or Higgsino with \(m_{\tilde{\chi }^0_{1}} \sim 1 \,\, \mathrm {TeV}\), with annihilation via heavy Higgs bosons H / A and stop coannihilation, or chargino coannihilation, bringing the cold dark matter density into the cosmological range. We find that spin-independent dark matter scattering is likely to be within reach of the planned LUX-Zeplin and XENONnT experiments. We probe the impact of the \((g-2)_\mu \) constraint, finding similar results whether or not it is included.     

Universal Components of Random Nodal Sets

We give, as L grows to infinity, an explicit lower bound of order \({L^{\frac{n}{m}}}\) for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of P with eigenvalues below L. Here, P denotes an elliptic self-adjoint pseudo-differential operator of order \({m > 0}\), bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed n-dimensional manifold M equipped with some Lebesgue measure. In fact, for every closed hypersurface \({\Sigma}\) of \({\mathbb{R}^n}\), we prove that there exists a positive constant \({p_\Sigma}\) depending only on \({\Sigma}\), such that for every large enough L and every \({x \in M}\), a component diffeomorphic to \({\Sigma}\) appears with probability at least \({p_\Sigma}\) in the vanishing locus of a random section and in the ball of radius \({L^{-\frac{1}{m}}}\) centered at x. These results apply in particular to Laplace–Beltrami and Dirichlet-to-Neumann operators.     

Electronic structure and optical properties of prominent phases of $$\hbox {TiO}_{2}$$: First-principles study

We present the properties of the C-parameter as an event-shape variable. We calculate the coupling constants in the perturbative and also in the non-perturbative parts of the QCD theory, using the dispersive as well as the shape function models. By fitting the corresponding theoretical predictions to our data, we find \(\alpha _{\mathrm {s}} (M_{Z^{0}})\) = 0.117 ± 0.014 and α ₀(μ _I ) = 0.491 ± 0.043 for dispersive model and \(\alpha _{\mathrm {s}} (M_{Z^{0}})\) = 0.124 ± 0.015 and λ ₁ = 1.234 ± 0.052 for the shape function model. Our results are consistent with the world average value of \(\alpha _{\mathrm {s}} (M_{Z^{0}})\) = 0.118 ± 0.002. All these features are explained in the main text.     

New Method of Calculating a Multiplication by using the Generalized Bernstein-Vazirani Algorithm

We present a new method of more speedily calculating a multiplication by using the generalized Bernstein-Vazirani algorithm and many parallel quantum systems. Given the set of real values \(\{a_{1},a_{2},a_{3},\ldots ,a_{N}\}\) and a function \(g:\textbf {R}\rightarrow \{0,1\}\), we shall determine the following values \(\{g(a_{1}),g(a_{2}),g(a_{3}),\ldots , g(a_{N})\}\) simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of \(N\). Next, we consider it as a number in binary representation; M₁ = (g(a₁),g(a₂),g(a₃),…,g(a _N )). By using \(M\) parallel quantum systems, we have \(M\) numbers in binary representation, simultaneously. The speed of obtaining the \(M\) numbers is shown to outperform the classical case by a factor of \(M\). Finally, we calculate the product; \( M_{1}\times M_{2}\times \cdots \times M_{M}. \) The speed of obtaining the product is shown to outperform the classical case by a factor of N × M.     

Spectra of Semi-Infinite Quantum Graph Tubes

The spectrum of a semi-infinite quantum graph tube with square period cells is analyzed. The structure is obtained by rolling up a doubly periodic quantum graph into a tube along a period vector and then retaining only a semi-infinite half of the tube. The eigenfunctions associated to the spectrum of the half-tube involve all Floquet modes of the full tube. This requires solving the complex dispersion relation \({D(\lambda,k_1,k_2)=0}\) with \({(k_1,k_2)\in(\mathbb{C}/2\pi\mathbb{Z})^2}\) subject to the constraint \({a k_1 + bk_2 \equiv 0}\) (mod \({2\pi}\)), where a and b are integers. The number of Floquet modes for a given \({\lambda\in\mathbb{R}}\)  is  \({2\max\left\{ a, b \right\}}\). Rightward and leftward modes are determined according to an indefinite energy flux form. The spectrum may contain eigenvalues that depend on the boundary conditions, and some eigenvalues may be embedded in the continuous spectrum.     

Conformal Mappings and Dispersionless Toda Hierarchy

Let \({\mathfrak{D}}\) be the space consists of pairs (f, g), where f is a univalent function on the unit disc with f(0) = 0, g is a univalent function on the exterior of the unit disc with g(∞) = ∞ and f′(0)g′(∞) = 1. In this article, we define the time variables \({t_n, n\in \mathbb{Z}}\), on \({\mathfrak{D}}\) which are holomorphic with respect to the natural complex structure on \({\mathfrak{D}}\) and can serve as local complex coordinates for \({\mathfrak{D}}\) . We show that the evolutions of the pair (f, g) with respect to these time coordinates are governed by the dispersionless Toda hierarchy flows. An explicit tau function is constructed for the dispersionless Toda hierarchy. By restricting \({\mathfrak{D}}\) to the subspace Σ consists of pairs where \({f(w)=1/\overline{g(1/\bar{w})}}\), we obtain the integrable hierarchy of conformal mappings considered by Wiegmann and Zabrodin [31]. Since every C ¹ homeomorphism γ of the unit circle corresponds uniquely to an element (f, g) of \({\mathfrak{D}}\) under the conformal welding \({\gamma=g^{-1}\circ f}\), the space Homeo _C (S ¹) can be naturally identified as a subspace of \({\mathfrak{D}}\) characterized by f(S ¹) = g(S ¹). We show that we can naturally define complexified vector fields \({\partial_n, n\in \mathbb{Z}}\) on Homeo _C (S ¹) so that the evolutions of (f, g) on Homeo _C (S ¹) with respect to ? _n satisfy the dispersionless Toda hierarchy. Finally, we show that there is a similar integrable structure for the Riemann mappings (f ^?1, g ^?1). Moreover, in the latter case, the time variables are Fourier coefficients of γ and 1/γ ^?1.     

Typical Gibbs Configurations for the 1d Random Field Ising Model with Long Range Interaction

We study one–dimensional Ising spin systems with ferromagnetic, long–range interaction decaying as n ^?2+α , \({\alpha \in [0,\frac 12]}\), in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, which are gaussian or subgaussian with variance θ. We show that when the temperature and the variance of the randomness are sufficiently small, with overwhelming probability with respect to the random fields, the typical configurations, within intervals centered at the origin whose length grow faster than any power of θ ^?1, are intervals of + spins followed by intervals of ? spins whose typical length is \({ \simeq\,\theta^{-\frac{2}{(1-2\alpha)}}}\) for 0 ≤ α < 1/2 and between \({ e^{\frac{1}{\theta}}}\) and \({e^{\frac 1 {\theta^{2}}}}\) for α = 1/2.     

Schur Polynomials and The Yang-Baxter Equation

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map \({g \mapsto R (g)}\) from \({ \rm{GL}(2, \mathbb{C}) \times \rm{GL}(1, \mathbb{C})}\) to End \({(V \otimes V)}\) , where V is a two-dimensional vector space such that if \({g, h \in G}\) then R ₁₂(g)R ₁₃(gh) R ₂₃(h) = R ₂₃(h) R ₁₃(gh)R ₁₂(g). Here R _{i j} denotes R applied to the i, j components of \({V \otimes V \otimes V}\) . The image of this map consists of matrices whose nonzero coefficients a ₁, a ₂, b ₁, b ₂, c ₁, c ₂ are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a ₁ a ₂ + b ₁ b ₂ ? c ₁ c ₂ = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial s _λ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.     

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.     

Braidings of Tensor Spaces

Let V be a braided vector space, i.e., a vector space together with a solution \({\hat{R}\in {{End}}(V\otimes V)}\) of the Yang–Baxter equation. Denote \({T(V):=\bigoplus_k V^{\otimes k}}\) . We associate to \({\hat{R}}\) a one-parameter family of solutions \({T(\hat{R})\in {\rm End}(T(V)\otimes T(V))}\) of the Yang–Baxter equation on the tensor space T (V). Main ingredients of the solution are braid analogues of the binomial coefficients and of the Pochhammer symbols. The association \({\hat{R}\rightsquigarrow T(\hat{R})}\) is functorial with respect to V.     

Fluctuations for the Ginzburg-Landau $${\nabla \phi}$$ Interface Model on a Bounded Domain

We study the massless field on \({D_n = D \cap \tfrac{1}{n} \mathbf{Z}^2}\), where \({D \subseteq \mathbf{R}^2}\) is a bounded domain with smooth boundary, with Hamiltonian \({\mathcal {H}(h) = \sum_{x \sim y} \mathcal {V}(h(x) - h(y))}\). The interaction \({\mathcal {V}}\) is assumed to be symmetric and uniformly convex. This is a general model for a (2 + 1)-dimensional effective interface where h represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt: h(x) = n x · u + f(x) for \({x \in \partial D_n,\,u \in \mathbf{R}^2}\), and f : R ² → R continuous. We prove that the fluctuations of linear functionals of h(x) about the tilt converge in the limit to a Gaussian free field on D, the standard Gaussian with respect to the weighted Dirichlet inner product \({(f,g)_\nabla^\beta = \int_D \sum_i \beta_i \partial_i f_i \partial_i g_i}\) for some explicit β = β(u). In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of h are asymptotically described by SLE(4), a conformally invariant random curve.