A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree

For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥T _c,k =J/arctan?(1/k) the limiting Gibbs measure is unique, and for T<T _c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if \(T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})\), with \(\sqrt{k} then there is a new uncountable set \({\mathcal{G}}_{k,k_{0}}\) of Gibbs measures. Moreover \({\mathcal{G}}_{k,k_{0}}\ne {\mathcal{G}}_{k,k'_{0}}\), for k ₀≠k′₀. Therefore if \(T\in (T_{c,\sqrt{k}}, T_{c,\sqrt{k}+1})\), \(T_{c,\sqrt{k}+1} then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures}\(\cup(\bigcup_{k_{0}:\sqrt{k}.     

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.     

Method of uniform effective field in structure-dynamic approach of nanoionics

In the structure-dynamic approach of nanoionics, the method of a uniform effective field \( {F}_{\mathrm{eff}}^{j,k} \) of a crystallographic planeX ^j has been substantiated for solid electrolyte nanostructures. The \( {F}_{\mathrm{eff}}^{j,k} \)is defined as an approximation of a non-uniform field \( {F}_{\mathrm{dis}}^j \)of X ^j with a discrete- random distribution of excess point charges. The parameters of \( {F}_{\mathrm{eff}}^{j,k} \)are calculated by correction of the uniform Gauss field \( {F}_{\mathrm{G}}^j \) of X ^j . The change in an average frequency of ionic jumps X ^k ?→?X ^k?+?1 between adjacent planes of nanostructure is determined by the sum of field additives to the barrier heights η _k , _k?+?1, and for \( {F}_{\mathrm{G}}^j \) and \( {F}_{\mathrm{dis}}^j \), these sums are the same decimal order of magnitude. For nanostructures with length ~4 nm, the application of \( {F}_{\mathrm{G}}^j \) (as \( {F}_{\mathrm{eff}}^{j,k} \)) gives the accuracy ~20 % in calculations of ion transport characteristics. The computer explorations of the “universal” dynamic response (Reσ ^??∝?ω ⁿ ) show an approximately the same power n < ≈1 for\( {F}_{\mathrm{G}}^j \) and \( {F}_{\mathrm{eff}}^{j,k} \).     

New feature of low $$p_{T}$$ charm quark hadronization in pp collisions at $$\sqrt{s}=7$$s=7 TeV

Treating the light-flavor constituent quarks and antiquarks whose momentum information is extracted from the data of soft light-flavor hadrons in pp collisions at \(\sqrt{s}=7\) TeV as the underlying source of chromatically neutralizing the charm quarks of low transverse momenta (\(p_{T}\)), we show that the experimental data of \(p_{T}\) spectra of single-charm hadrons \(D^{0,+}\), \(D^{*+}\) \(D_{s}^{+}\), \(\varLambda _{c}^{+}\) and \(\varXi _{c}^{0}\) at mid-rapidity in the low \(p_{T}\) range (\(2\lesssim p_{T}\lesssim 7\) GeV/c) in pp collisions at \(\sqrt{s}=7\) TeV can be well understood by the equal-velocity combination of perturbatively created charm quarks and those light-flavor constituent quarks and antiquarks. This suggests a possible new scenario of low \(p_{T}\) charm quark hadronization, in contrast to the traditional fragmentation mechanism, in pp collisions at LHC energies. This is also another support for the exhibition of the soft constituent quark degrees of freedom for the small parton system created in pp collisions at LHC energies.     

Trajectories in Random Monads

Let us have a non-empty finite set S with n>1 elements which we call points and a map M:S→S. After V.I. Arnold, we call such pairs (S, M) monads, but we consider random monads in which all the values of M(?) are random, independent and uniformly distributed in S. We fix some ⊙∈S and consider the infinite sequence M ^t (⊙), t=0,1,2,…?. A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis, Rec, Tra the numbers of visited, recurrent and transient points respectively and study their distributions. The distributions of Vis, Rec, Tra are unimodal. The modes of Rec and Tra equal their minimal values, that is 1 and 0 respectively. The mode of Vis is approximated by \(\sqrt{n}\), plus-minus a constant. The mathematical expectations: \(\mathbb{E}(\mathit{Vis})\) is approximated by \(2 \sqrt{\pi\, n/8}\) plus-minus a constant; \(\mathbb{E}(\mathit{Rec})\) and \(\mathbb{E}(\mathit{Tra})\) are approximated by \(\sqrt{\pi\, n/8}\) plus-minus a constant. For the standard deviations σ(Vis) and σ(Rec)=σ(Tra) respectively we present the approximations

$\sqrt{\frac{4-\pi}{2} \cdot n} \quad\mbox{and}\quad \sqrt{\frac{16-3\pi}{24} \cdot n},$

from which they also deviate at most by a constant. We prove that when n tends to infinity, the correlations Corr(Rec,Tra) and Corr(Rec,Vis)=Corr(Tra,Vis) converge to

$\frac{8-3\pi}{16-3\pi}\quad \mbox{and}\quad \sqrt{\frac{12-3\pi}{16-3\pi}}.$

     

The Scaling Limit of the Critical One-Dimensional Random Schrödinger Operator

We consider two models of one-dimensional discrete random Schrödinger operators

$(H_n\psi)_\ell =\psi_{\ell -1}+\psi_{\ell +1}+v_\ell \psi_\ell$

, \({\psi_0=\psi_{n+1}=0}\) in the cases \({ v_k=\sigma \omega_k/\sqrt{n}}\) and \({ v_k=\sigma \omega_k/ \sqrt{k}}\) . Here ω _k are independent random variables with mean 0 and variance 1.

We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem.In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the β-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.     

On a Pre-Lie Algebra Defined by Insertion of Rooted Trees

Let K be a field of characteristic zero. For \({n \in \mathbb{N}^{*}}\) , let \({\mathcal{T}^{\,\prime}_{n}}\) be the vector space of non-planar rooted trees with n vertices (Foissy in Bull Sci Math 126, no. 3, 193–239; no. 4, 249–288, 2002). Let \({\vartriangleright}\) be the left pre-Lie product of insertion of a tree inside another defined on infinitesimal characters of the graded Hopf algebra \({\mathcal{H}}\) introduced by Calaque, Ebrahimi-Fard and Manchon. Let \({\mathcal{T}^{\,\prime}=\oplus_{n\geq 2}\mathcal{T}^{\,\prime}_{n}}\) . In this work, we first prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) a pre-Lie algebra generated by the two ladders E ₁ and E ₂ where E ₁ is the ladder with one edge and E ₂ is the ladder with two edges. Second, we prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) is not a free pre-Lie algebra, and we exhibit a family of relations.     

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.     

Compositions,Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes N _α (t), N _β (t), t>0, we have that \(N_{\alpha}(N_{\beta}(t)) \stackrel{\mathrm{d}}{=} \sum_{j=1}^{N_{\beta}(t)} X_{j}\), where the X _j s are Poisson random variables. We present a series of similar cases, where the outer process is Poisson with different inner processes. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form \(N_{\alpha}(\tau_{k}^{\nu})\), ν∈(0,1], where \(\tau_{k}^{\nu}\) is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form Θ(N(t)), t>0, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers.     

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.     

Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-variate Normal Random Polynomials

Let \(z\in \mathbb {C}\), let \(\sigma ^2>0\) be a variance, and for \(N\in \mathbb {N}\) define the integrals

$$\begin{aligned} E_N^{}(z;\sigma ) := \left\{ \begin{array}{ll} {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}}\! (x^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x^2}}{\sqrt{2\pi }}dx&{}\quad \text{ if }\, N=1,\\ {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}^N}\! \prod \prod \limits _{1\le k<l\le N}\!\! e^{-\frac{1}{2N}(1-\sigma ^{-2}) (x_k-x_l)^2} \prod _{1\le n\le N}\!\!\!\!(x_n^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x_n^2}}{\sqrt{2\pi }}dx_n &{}\quad \text{ if }\, N>1. \end{array}\right. \!\!\! \end{aligned}$$

These are expected values of the polynomials \(P_N^{}(z)=\prod _{1\le n\le N}(X_n^2+z^2)\) whose 2N zeros \(\{\pm i X_k\}^{}_{k=1,\ldots ,N}\) are generated by N identically distributed multi-variate mean-zero normal random variables \(\{X_k\}^{N}_{k=1}\) with co-variance \(\mathrm{{Cov}}_N^{}(X_k,X_l)=(1+\frac{\sigma ^2-1}{N})\delta _{k,l}+\frac{\sigma ^2-1}{N}(1-\delta _{k,l})\). The \(E_N^{}(z;\sigma )\) are polynomials in \(z^2\), explicitly computable for arbitrary N, yet a list of the first three \(E_N^{}(z;\sigma )\) shows that the expressions become unwieldy already for moderate N—unless \(\sigma = 1\), in which case \(E_N^{}(z;1) = (1+z^2)^N\) for all \(z\in \mathbb {C}\) and \(N\in \mathbb {N}\). (Incidentally, commonly available computer algebra evaluates the integrals \(E_N^{}(z;\sigma )\) only for N up to a dozen, due to memory constraints). Asymptotic evaluations are needed for the large-N regime. For general complex z these have traditionally been limited to analytic expansion techniques; several rigorous results are proved for complex z near 0. Yet if \(z\in \mathbb {R}\) one can also compute this “infinite-degree” limit with the help of the familiar relative entropy principle for probability measures; a rigorous proof of this fact is supplied. Computer algebra-generated evidence is presented in support of a conjecture that a generalization of the relative entropy principle to signed or complex measures governs the \(N\rightarrow \infty \) asymptotics of the regime \(iz\in \mathbb {R}\). Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.

     

$${\mathbb{Z}_N}$$ -Curves Possessing No Thomae Formulae of Bershadsky–Radul Type

A \({\mathbb{Z}_N}\) -curve is one of the form \({y^{N}=(x-\lambda_{1})^{m_{1}}\cdots(x-\lambda_{s})^{m_{s}}}\) . When N = 2 these curves are called hyperelliptic and for them Thomae proved his classical formulae linking the theta functions corresponding to their period matrices to the branching values λ₁, . . . , λ _s . In his work on Fermionic fields on \({\mathbb{Z}_N}\) -curves with arbitrary N, Bershadsky and Radul discovered the existence of generalized Thomae’s formulae for these curves which they wrote down explicitly in the case in which all rotation numbers m _i equal 1. This work was continued by several authors and new Thomae’s type formulae for \({\mathbb{Z}_N}\) -curves with other rotation numbers m _i were found. In this article we prove that for some choices of the rotation numbers the corresponding \({\mathbb{Z}_N}\) -curves do not admit such generalized Thomae’s formulae.     

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.     

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

For a Hopf algebra B, we endow the Heisenberg double \({\mathcal{H}(B^*)}\) with the structure of a module algebra over the Drinfeld double \({\mathcal{D}(B)}\). Based on this property, we propose that \({\mathcal{H}(B^*)}\) is to be the counterpart of the algebra of fields on the quantum-group side of the Kazhdan–Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) quantum group that is Kazhdan–Lusztig-dual to (p,1) logarithmic conformal models. The corresponding pair \({(\mathcal{D}(B),\mathcal{H}(B^*))}\) is “truncated” to \({(\overline{\mathcal{U}}_{\mathfrak{q}} s\ell2,\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2))}\), where \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)}\) is a \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) module algebra that turns out to have the form \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)=\mathbb{C}_{\mathfrak{q}}[z,\partial]\otimes\mathbb{C}[\lambda]/(\lambda^{2p}-1)}\), where \({\mathbb{C}_{\mathfrak{q}}[z,\partial]}\) is the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\)-module algebra with the relations z ^p  = 0, ? ^p  = 0, and \({\partial z = \mathfrak{q}-\mathfrak{q}^{-1} + \mathfrak{q}^{-2} z\partial}\).     

The Local Limit of the Uniform Spanning Tree on Dense Graphs

Let G be a connected graph in which almost all vertices have linear degrees and let \(\mathcal {T}\) be a uniform spanning tree of G. For any fixed rooted tree F of height r we compute the asymptotic density of vertices v for which the r-ball around v in \(\mathcal {T}\) is isomorphic to F. We deduce from this that if \(\{G_n\}\) is a sequence of such graphs converging to a graphon W, then the uniform spanning tree of \(G_n\) locally converges to a multi-type branching process defined in terms of W. As an application, we prove that in a graph with linear minimum degree, with high probability, the density of leaves in a uniform spanning tree is at least \(e^{-1}-\mathsf {o}(1)\), the density of vertices of degree 2 is at most \(e^{-1}+\mathsf {o}(1)\) and the density of vertices of degree \(k\geqslant 3\) is at most \({(k-2)^{k-2} \over (k-1)! e^{k-2}} + \mathsf {o}(1)\). These bounds are sharp.     

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 \({\sigma ={\rm id}}\).     

Boundedness of Massless Scalar Waves on Reissner-Nordström Interior Backgrounds

We consider solutions of the scalar wave equation \({\Box_g\phi=0}\), without symmetry, on fixed subextremal Reissner-Nordström backgrounds \({(\mathcal{M}, g)}\) with nonvanishing charge. Previously, it has been shown that for ? arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon \({\mathcal{H}^+}\). Using this, we show here that ? is in fact uniformly bounded, \({|\phi| \leq C}\), in the black hole interior up to and including the bifurcate Cauchy horizon \({\mathcal{C}\mathcal{H}^+}\), to which ? in fact extends continuously. The proof depends on novel weighted energy estimates in the black hole interior which, in combination with commutation by angular momentum operators and application of Sobolev embedding, yield uniform pointwise estimates. In a forthcoming companion paper we will extend the result to subextremal Kerr backgrounds with nonvanishing rotation.     

Global Well-Posedness of the Energy-Critical Defocusing NLS on $${\mathbb{R} \times \mathbb{T}^3}$$

We prove global well-posedness in H ¹ for the energy-critical defocusing initial-value problem \({(i\partial_t+\Delta_x)u=u|u|^2,\quad u(0)=\phi,}\) in the semiperiodic setting \({x\in\mathbb{R} \times \mathbb{T}^3}\) .     

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.     

Excited state mass spectra of doubly heavy $$\Xi $$ baryons

In this paper, the mass spectra are obtained for doubly heavy \(\Xi \) baryons, namely, \(\Xi _{cc}^{+}\), \(\Xi _{cc}^{++}\), \(\Xi _{bb}^{-}\), \(\Xi _{bb}^{0}\), \(\Xi _{bc}^{0}\) and \(\Xi _{bc}^{+}\). These baryons consist of two heavy quarks (cc, bb, and bc) with a light (d or u) quark. The ground, radial, and orbital states are calculated in the framework of the hypercentral constituent quark model with Coulomb plus linear potential. Our results are also compared with other predictions, thus, the average possible range of excited states masses of these \(\Xi \) baryons can be determined. The study of the Regge trajectories is performed in (n, \(M^{2}\)) and (J, \(M^{2}\)) planes and their slopes and intercepts are also determined. Lastly, the ground state magnetic moments of these doubly heavy baryons are also calculated.