Mathematical Physics, Analysis and Geometry最新文献

We aim at understanding for which (complex) values of the potential the pinning partition function vanishes. The pinning model is a Gibbs measure based on discrete renewal processes with power law inter-arrival distributions. We obtain some results for rather general inter-arrival laws, but we achieve a substantially more complete understanding for a specific one parameter family of inter-arrivals. We show, for such a specific family, that the zeros asymptotically lie on (and densely fill) a closed curve that, unsurprisingly, touches the real axis only in one point (the critical point of the model). We also perform a sharper analysis of the zeros close to the critical point and we exploit this analysis to approach the challenging problem of Griffiths singularities for the disordered pinning model. The techniques we exploit are both probabilistic and analytical. Regarding the first, a central role is played by limit theorems for heavy tail random variables. As for the second, potential theory and singularity analysis of generating functions, along with their interplay, will be at the heart of several of our arguments.

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.

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.

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.

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.

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.

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.

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.

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.

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.