Mathematical Physics, Analysis and Geometry最新文献

Using Araki–Yamagami’s characterization of quasi-equivalence for quasi-free representations of the CCRs, we provide an abstract criterion for the existence of isomorphisms of second quantization local von Neumann algebras induced by Bogolubov transformations in terms of the respective one particle modular operators. We discuss possible applications to the problem of local normality of vacua of Klein-Gordon fields with different masses.

We consider the macroscopic limit for the space-time density fluctuations in the open symmetric simple exclusion in the quasi-static scaling limit. We prove that the distribution of these fluctuations converge to a gaussian space-time field that is delta correlated in time but with long-range correlations in space.

We study cover times of subsets of ({mathbb {Z}}^2) by a two-dimensional massive random walk loop soup. We consider a sequence of subsets (A_n subset {mathbb {Z}}^2) such that (|A_n| rightarrow infty ) and determine the distributional limit of their cover times ({mathcal {T}}(A_n)). We allow the killing rate (kappa _n) (or equivalently the “mass”) of the loop soup to depend on the size of the set (A_n) to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to (kappa _n^{-1}=|A_n|^{1-8/(log log |A_n|)},) showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order (kappa _n^{-1/2}=|A_n|^{1/2},) if (kappa _n^{-1}) exceeded (|A_n|,) the cover times of all points in a tightly packed set (A_n) (i.e., a square or close to a ball) would presumably be heavily correlated, complicating the analysis. Our result comes close to this extreme case.

This paper focuses on the characteristics of the Hankel determinant generated by a modified singularly Gaussian weight. The weight function is defined as:

where (alpha >1) and (tin (0,1)) are parameters. Using ladder operator techniques, we derive a series of difference formulas for the auxiliary quantities and recurrence coefficients. We present the difference equations for the recurrence coefficients and the logarithmic derivative of the Hankel determinant. We then use the “t-dependence" to obtain the differential identities satisfied by the auxiliary quantities and the logarithmic derivative of the Hankel determinant. To obtain the large n asymptotic expressions of the recurrence coefficients, we use the Coulomb fluid method together with the related difference equations, which depend on n either being odd or even. We also obtain the reduction forms of the second-order differential equations satisfied by the orthogonal polynomials generated by this weight. Two special cases coincide with the bi-confluent Heun equation and the double confluent Heun equation, respectively. Finally, we calculate the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by this weight. Our result not only covers the classical result of Szegö (Trans Am Math Soc 40:450–461, 1936) but also determines our next research direction.

We consider all totally asymmetric simple exclusion processes (TASEPs) whose transition probabilities are given by the Schütz-type formulas and which jump with homogeneous rates. We show that the multi-point distribution of particle positions and the KPZ scaling are described using the probability generating function of the rightmost particle’s jump. For all TASEPs satisfying certain assumptions, we also prove the pointwise convergence of the kernels appearing in the joint distribution of particle positions to those appearing in the KPZ fixed point formula. Our result generalizes the result of Matetski, Quastel, and Remenik [18] in the sense that we provide the KPZ fixed point formulation for a class of TASEPs, instead of for one specific TASEP.

We study a family of Fredholm determinants associated to deformations of the sine kernel, parametrized by a weight function w. For a specific choice of w, this kernel describes bulk statistics of finite temperature free fermions. We establish a connection between these determinants and a system of integro-differential equations generalizing the fifth Painlevé equation, and we show that they allow us to solve an integrable PDE explicitly for a large class of initial data.

We introduce multicritical Schur measures, which are probability laws on integer partitions which give rise to non-generic fluctuations at their edge. They are in the same universality classes as one-dimensional momentum-space models of free fermions in flat confining potentials, studied by Le Doussal, Majumdar and Schehr. These universality classes involve critical exponents of the form (1/(2m+1)), with m a positive integer, and asymptotic distributions given by Fredholm determinants constructed from higher order Airy kernels, extending the generic Tracy–Widom GUE distribution recovered for (m=1). We also compute limit shapes for the multicritical Schur measures, discuss the finite temperature setting, and exhibit an exact mapping to the multicritical unitary matrix models previously encountered by Periwal and Shevitz.

We consider multi-component Kadomtsev-Petviashvili hierarchy of type C (the multi-component CKP hierarchy) originally defined with the help of matrix pseudo-differential operators via the Lax-Sato formalism. Starting from the bilinear relation for the wave functions, we prove existence of the tau-function for the multi-component CKP hierarchy and provide a formula which expresses the wave functions through the tau-function. We also find how this tau-function is related to the tau-function of the multi-component Kadomtsev-Petviashvili hierarchy. The tau-function of the multi-component CKP hierarchy satisfies an integral relation which, unlike the integral relation for the latter tau-function, is no longer bilinear but has a more complicated form.

We provide a concrete characterization of the poly-analytic planar automorphic functions, a special class of non analytic planar automorphic functions with respect to the Appell–Humbert automorphy factor, arising as images of the holomorphic ones by means of the creation differential operator. This is closely connected to the spectral theory of the magnetic Laplacian on the complex plane.

We introduce (delta ) type vertex conditions for beam operators, the fourth-order differential operator, on finite, compact and connected metric graphs. Our study the effect of certain geometrical alterations (graph surgery) of the graph on their spectra. Results are obtained for a class of vertex conditions which can be seen as an analogue of (delta )-conditions for graphs Laplacian. There are a number of possible candidates of (delta ) type conditions for beam operators. We develop surgery principles and record the monotonicity properties of their spectrum, keeping in view the possibility that vertex conditions may change within the same class after certain graph alterations. We also demonstrate the applications of surgery principles by obtaining several lower and upper estimates on the eigenvalues.