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Abstract: Abstract The problem of convergence of the joint moments, which depend on two parameters s and h, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes (On the characteristic polynomial of a random unitary matrix and the Riemann zeta function, PhD Thesis, University of Bristol, 2001). Recently, Forrester (Joint moments of a characteristic polynomial and its derivative for the circular \(\beta \) -ensemble, arXiv:2012.08618, 2020) considered the analogous problem for the Circular \(\beta \) -Ensemble (C \(\beta \) E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer s and complex h. In this paper we consider this problem for a generalisation of the C \(\beta \) E, the Circular Jacobi \(\beta \) -ensemble (CJ \(\beta \text {E}_\delta \) ), depending on an additional complex parameter \(\delta \) and we prove convergence of the joint moments for general positive real exponents s and h. We give a representation for the limit in terms of the moments of a family of real random variables of independent interest. This is done by making use of some general results on consistent probability measures on interlacing arrays. Using these techniques, we also extend Forrester’s explicit formula to the case of real s and \(\delta \) and integer h. Finally, we prove an analogous result for the moments of the logarithmic derivative of the characteristic polynomial of the Laguerre \(\beta \) -ensemble. PubDate: 2022-05-14

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Abstract: Abstract Auffinger and Chen (J Stat Phys 157:40–59, 2014) proved a variational formula for the free energy of the spherical bipartite spin glass in terms of a global minimum over the overlaps. We show that a different optimisation procedure leads to a saddle point, similar to the one achieved for models on the vertices of the hypercube. PubDate: 2022-05-03

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Abstract: Abstract The box and ball system (BBS) models the dynamics of balls moving among an array of boxes. The simplest BBS is derived from the ultradiscretization of the discrete Toda equation, which is one of the most famous discrete integrable systems. The discrete Toda equation can be extended to two types of discrete hungry Toda (dhToda) equations, one of which is the equation of motion of the BBS with numbered balls (nBBS). In this paper, based on the ultradiscretization of the other type of dhToda equation, we present a new nBBS in which not balls, but boxes, are numbered. We also investigate conserved quantities with respect to balls and boxes, the solitonical nature of ball motions, and a scattering rule in collisions of balls to clarify the characteristics of the resulting nBBS. PubDate: 2022-04-24

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Abstract: Abstract We consider a Bose gas consisting of N particles in \({\mathbb {R}}^3\) , trapped by an external field and interacting through a two-body potential with scattering length of order \(N^{-1}\) . We prove that low energy states exhibit complete Bose–Einstein condensation with optimal rate, generalizing previous work in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018; 376:1311–1395, 2020), restricted to translation invariant systems. This extends recent results in Nam et al. (Preprint, 2001. arXiv:2001.04364), removing the smallness assumption on the size of the scattering length. PubDate: 2022-04-12

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Abstract: Abstract We use infinite dimensional self-dual \(\mathrm {CAR}\) \(C^{*}\) -algebras to study a \({\mathbb {Z}}_{2}\) -index, which classifies free-fermion systems embedded on \({\mathbb {Z}}^{d}\) disordered lattices. Combes–Thomas estimates are pivotal to show that the \({\mathbb {Z}}_{2}\) -index is uniform with respect to the size of the system. We additionally deal with the set of ground states to completely describe the mathematical structure of the underlying system. Furthermore, the weak \(^{*}\) -topology of the set of linear functionals is used to analyze paths connecting different sets of ground states. PubDate: 2022-03-14 DOI: 10.1007/s11040-022-09421-w

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Abstract: Abstract Thanks to the Birman-Schwinger principle, Weyl’s laws for Birman-Schwinger operators yields semiclassical Weyl’s laws for the corresponding Schrödinger operators. In a recent preprint Rozenblum established quite general Weyl’s laws for Birman-Schwinger operators associated with pseudodifferential operators of critical order and potentials that are product of \(L\!\log \!L\) -Orlicz functions and Alfhors-regular measures supported on a submanifold. In this paper, we show that, for matrix-valued \(L\!\log \!L\) -Orlicz potentials supported on the whole manifold, Rozenblum’s results are direct consequences of the Cwikel-type estimates on tori recently established by Sukochev–Zanin. As applications we obtain CLR-type inequalities and semiclassical Weyl’s laws for critical Schrödinger operators associated with matrix-valued \(L\!\log \!L\) -Orlicz potentials. Finally, we explain how the Weyl’s laws of this paper imply a strong version of Connes’ integration formula for matrix-valued \(L\!\log \!L\) -Orlicz potentials. PubDate: 2022-03-12 DOI: 10.1007/s11040-022-09422-9

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Abstract: Abstract J-trajectories are arc length parameterized curves in almost Hermitian manifold which satisfy the equation \(\nabla _{{\dot{\gamma }}}{\dot{\gamma }}=q J {\dot{\gamma }}\) . In this paper J-trajectories in the solvable Lie group \(\mathrm {Sol}_0^4\) are investigated. The first and the second curvature of a non-geodesic J-trajectory in an arbitrary LCK manifold whose anti Lee field has constant length are examined. In particular, the curvatures of non-geodesic J-trajectories in \(\mathrm {Sol}_0^4\) are characterized. PubDate: 2022-03-06 DOI: 10.1007/s11040-022-09418-5

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Abstract: Abstract In a 1979 paper, Okamoto introduced the space of initial values for the six Painlevé equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase space, a rational surface with certain exceptional divisors removed. We show that the construction of the space of initial values remains meaningful for certain classes of second-order complex differential equations, and more generally, Hamiltonian systems, where all movable singularities of all their solutions are algebraic poles (by some authors denoted the quasi-Painlevé property), which is a generalisation of the Painlevé property. The difference here is that the initial value problems obtained in the extended phase space become regular only after an additional change of dependent and independent variables. Constructing the analogue of space of initial values for these equations in this way also serves as an algorithm to single out, from a given class of equations or system of equations, those equations which are free from movable logarithmic branch points. PubDate: 2022-03-06 DOI: 10.1007/s11040-022-09417-6

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Abstract: Abstract We study some properties concerning Tsallis and Kaniadakis divergences between two probability measures. More exactly, we prove the pseudo-additivity, non-negativity, monotonicity and find some bounds for the divergences mentioned above. PubDate: 2022-02-24 DOI: 10.1007/s11040-022-09420-x

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Abstract: Abstract The study of set-theoretic solutions of the Yang-Baxter equation, also known as Yang-Baxter maps, is historically a meeting ground for various areas of mathematics and mathematical physics. In this work, we study factorization problems on rational loop groups, which give rise to Yang-Baxter maps on various geometrical objects. We also study the symplectic and Poisson geometry of these Yang-Baxter maps, which we show to be integrable maps in the sense of having natural collections of Poisson commuting integrals. In a special case, the factorization problems we consider are associated with the N-soliton collision process in the n-Manakov system, and in this context we show that the polarization scattering map is a symplectomorphism. PubDate: 2022-02-19 DOI: 10.1007/s11040-022-09419-4

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Abstract: Abstract We constructed involutions for a Halphen pencil of index 2, and proved that the birational mapping corresponding to the autonomous reduction of the elliptic Painlevé equation for the same pencil can be obtained as the composition of two such involutions. PubDate: 2022-02-05 DOI: 10.1007/s11040-022-09416-7

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Abstract: Abstract We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process \(\mathcal {P}_{\lambda }\) in \(\mathbb {R}^{2}\) of intensity λ. In the homogeneous RCM, the vertices at x,y are connected with probability g( x − y ), independent of everything else, where \(g:[0,\infty ) \to [0,1]\) and ⋅ is the Euclidean norm. In the inhomogeneous version of the model, points of \(\mathcal {P}_{\lambda }\) are endowed with weights that are non-negative independent random variables with distribution \(P(W>w)= w^{-\beta }1_{[1,\infty )}(w)\) , β > 0. Vertices located at x,y with weights Wx,Wy are connected with probability \(1 - \exp \left (- \frac {\eta W_{x}W_{y}}{ x-y ^{\alpha }} \right )\) , η,α > 0, independent of all else. The graphs are enhanced by considering the edges of the graph as straight line segments starting and ending at points of \(\mathcal {P}_{\lambda }\) . A path in the graph is a continuous curve that is a subset of the union of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the mid point of each segment located at a distinct point of \(\mathcal {P}_{\lambda }\) . Intersecting lines form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. We derive conditions for the existence of a phase transition and show that there is no percolation at criticality. PubDate: 2022-01-13 DOI: 10.1007/s11040-021-09409-y

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Abstract: Abstract We investigate the BCS critical temperature \(T_c\) in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of \(T_c\) at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory. PubDate: 2022-01-11 DOI: 10.1007/s11040-021-09415-0

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Abstract: Abstract The aim of this paper is twofold. Firstly we provide necessary and sufficient criteria for the existence of a strict deformation quantization of algebraic tensor products of Poisson algebras, and secondly we discuss the existence of products of KMS states. As an application, we discuss the correspondence between quantum and classical Hamiltonians in spin systems and we provide a relation between the resolvent of Schödinger operators for non-interacting many particle systems and quantization maps. PubDate: 2021-12-24 DOI: 10.1007/s11040-021-09414-1

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Abstract: Abstract We consider the existence of the topological entropy of shift spaces on a finitely generated semigroup whose Cayley graph is a tree. The considered semigroups include free groups. On the other hand, the notion of stem entropy is introduced. For shift spaces on a strict free semigroup, the stem entropy coincides with the topological entropy. We reveal a sufficient condition for the existence of the stem entropy of shift spaces on a semigroup. Furthermore, we demonstrate that the topological entropy exists in many cases and is identical to the stem entropy. PubDate: 2021-12-22 DOI: 10.1007/s11040-021-09411-4

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Abstract: Abstract We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order \(O(\epsilon ^2)\) in the coefficients of the discretization, where \(\epsilon \) is the stepsize. PubDate: 2021-11-28 DOI: 10.1007/s11040-021-09413-2

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Abstract: Abstract Recently, a birational representation of an extended affine Weyl group of type \(A_{mn-1}^{(1)}\times A_{m-1}^{(1)}\times A_{m-1}^{(1)}\) was proposed with the aid of a cluster mutation. In this article we formulate this representation in a framework of a system of q-difference equations with \(mn\times mn\) matrices. This formulation is called a Lax form and is used to derive a generalization of the q-Garnier system. PubDate: 2021-11-27 DOI: 10.1007/s11040-021-09412-3

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Abstract: Abstract In this last decade, an important stochastic model emerged: the Brownian map. It is the limit of various models of random combinatorial maps after rescaling: it is a random metric space with Hausdorff dimension 4, almost surely homeomorphic to the 2-sphere, and possesses some deep connections with Liouville quantum gravity in 2D. In this paper, we present a sequence of random objects that we call \(D\hbox {th}\) -random feuilletages (denoted by \(\mathbf{r}[{D}]\) ), indexed by a parameter \(D\ge 0\) and which are candidate to play the role of the Brownian map in dimension D. The construction relies on some objects that we name iterated Brownian snakes, which are branching analogues of iterated Brownian motions, and which are moreover limits of iterated discrete snakes. In the planar \(D=2\) case, the family of discrete snakes considered coincides with some family of (random) labeled trees known to encode planar quadrangulations. Iterating snakes provides a sequence of random trees \((\mathbf{t}^{(j)}, j\ge 1)\) . The \(D\hbox {th}\) -random feuilletage \(\mathbf{r}[{D}]\) is built using \((\mathbf{t}^{(1)},\ldots ,\mathbf{t}^{(D)})\) : \(\mathbf{r}[{0}]\) is a deterministic circle, \(\mathbf{r}[{1}]\) is Aldous’ continuum random tree, \(\mathbf{r}[{2}]\) is the Brownian map, and somehow, \(\mathbf{r}[{D}]\) is obtained by quotienting \(\mathbf{t}^{(D)}\) by \(\mathbf{r}[{D-1}]\) . A discrete counterpart to \(\mathbf{r}[{D}]\) is introduced and called the \(D\) th random discrete feuilletage with \(n+D\) nodes ( \(\mathbf{R}_{n}[D]\) ). The proof of the convergence of \(\mathbf{R}_{n}[D]\) to \(\mathbf{r}[{D}]\) after appropriate rescaling in some functional space is provided (however, the convergence obtained is too weak to imply the Gromov-Hausdorff convergence). An upper bound on the diameter of \(\mathbf{R}_{n}[D]\) is \(n^{1/2^{D}}\) . Some elements allowing to conjecture that the Hausdorff dimension of \(\mathbf{r}[{D}]\) is \(2^D\) are given. PubDate: 2021-11-27 DOI: 10.1007/s11040-021-09410-5

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Abstract: Abstract We describe the construction of CMC surfaces with symmetries in \(\mathbb {S}^{3}\) and \(\mathbb {R}^{3}\) using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials. PubDate: 2021-11-06 DOI: 10.1007/s11040-021-09397-z

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Abstract: Abstract The orthant model is a directed percolation model on \(\mathbb {Z}^{d}\) , in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster. PubDate: 2021-10-15 DOI: 10.1007/s11040-021-09408-z