Annales Henri Poincaré最新文献

We discuss Laplacian spectrum on a finite metric graph with vertex couplings violating the time-reversal invariance. For the class of star graphs, we determine, under the condition of a fixed total edge length, the configurations for which the ground state eigenvalue is maximized. Furthermore, for general finite metric graphs, we provide upper bounds for all eigenvalues.

In the realm of complex systems, dynamics is often modeled in terms of a nonlinear, stochastic, ordinary differential equation (SDE) with either an additive or a multiplicative Gaussian white noise. In addition to a well-established collection of results proving existence and uniqueness of the solutions, it is of particular relevance the explicit computation of expectation values and correlation functions, since they encode the key physical information of the system under investigation. A pragmatically efficient way to dig out these quantities consists of the Martin–Siggia–Rose (MSR) formalism, which establishes a correspondence between a large class of SDEs and suitably constructed field theories formulated by means of a path-integral approach. Despite the effectiveness of this duality, there is no corresponding, mathematically rigorous proof of such correspondence. We address this issue using techniques proper of the algebraic approach to quantum field theories, which is known to provide a valuable framework to discuss rigorously the path-integral formulation of field theories as well as the solution theory both of ordinary and of partial, stochastic differential equations. In particular, working in this framework, we establish rigorously, albeit at the level of perturbation theory, a correspondence between correlation functions and expectation values computed either in the SDE or in the MSR formalism.

Motivated by their appearance as Coulomb branch geometries of Class S theories, we study the image of the local Hitchin map in tame Hitchin systems of type-D with residue in a special nilpotent orbit (mathcal {O}_H). We describe two important features that distinguish it from the type-A case studied in Balasubramanian et al. (Adv Theor Math Phys Ser 26(6):1585–1667, 2022. https://doi.org/10.4310/ATMP.2022.v26.n6.a2, arXiv:2008.01020 [hep-th]). The first feature, which we term even-type constraints, arises iff the partition label ([mathcal {O}_H]) has even parts. Our Hitchin image is non-singular and thus different from the one studied by Baraglia and Kamgarpour. We argue that our Hitchin image always globalizes to being the Hitchin base of an integrable system. The second feature, which we term odd-type constraints, is related to a particular finite group (overline{A}_b(mathcal {O}_H)) being non-trivial. This finite group parametrizes the choices for the local Hitchin base. Additionally, we also show that the finite group (overline{A}_b(mathcal {O}_H)) encodes the size of the dual special piece.

We consider the simplest non-trivial local composite operators in the massless Sine-Gordon model, which are (partial _mu phi , partial _nu phi ) and the stress tensor (T_{mu nu }). We show that even in the finite regime (beta ^2 < 4 pi ) of the theory, these operators need additional renormalisation (beyond the free-field normal-ordering) at each order in perturbation theory. We further prove convergence of the renormalised perturbative series for their expectation values, both in the Euclidean signature and in Minkowski spacetime, and for the latter in an arbitrary Hadamard state. Lastly, we show that one must add a quantum correction (proportional to (hbar )) to the renormalised stress tensor to obtain a conserved quantity.

We obtain precise plateau estimates for the two-point function of the finite-volume weakly coupled hierarchical (|varphi |^4) model in dimensions (d ge 4), for both free and periodic boundary conditions and for any number (n ge 1) of components of the field (varphi ). We prove that, within a critical window around their respective effective critical points, the two-point functions for both free and periodic boundary conditions have a plateau, in the sense that they decay as (|x|^{-(d-2)}) until reaching a constant plateau value of order (V^{-1/2}) (with a logarithmic correction for (d=4)), where V is the size of the finite volume. The two critical windows for free and periodic boundary conditions do not overlap. The dependence of the plateau height on the location within the critical window is governed by an explicit n-dependent universal profile which is independent of the dimension. The proof is based on a rigorous renormalisation group method and extends the method used by Michta et al. (arXiv:2306.00896) to study the finite-volume susceptibility and related quantities. Our results lead to precise conjectures concerning Euclidean (non-hierarchical) models of spin systems and self-avoiding walk in dimensions (d ge 4).

We point out that the geometry of connected totally geodesic compact null hypersurfaces in Lorentzian manifolds is only slightly more specialized than that of Riemannian flows over compact manifolds, the latter mathematical theory having been much studied in the context of foliation theory since the work by Reinhart (Am J Math 81:529–536, 1959). We are then able to import results on Riemannian flows to the horizon case, so obtaining theorems on the dynamical structure of compact horizons that do not rely on (non-)degeneracy assumptions. Furthermore, we clarify the relation between isometric/geodesible Riemannian flows and non-degeneracy conditions. This work also contains some positive results on the possibility of finding, in the degenerate case, lightlike fields tangent to the horizon that have zero surface gravity.

We compute the spectrum of the finite open XX quantum spin chain with transverse magnetic boundary fields in terms of the roots of a two-parameter orthogonal family of Bernstein–Szegö polynomials. Estimates are presented for the locations of these roots. This enables the construction of an explicit eigenbasis diagonalizing the spin chain in terms of Slater determinants built from a one-parameter subfamily of the Bernstein–Szegö polynomials evaluated on the spectrum.

In Cassia et al. (Commun Math Phys 406(1):15, 2025), we established a connection between symplectic cuts of Calabi–Yau threefolds and open topological strings, and used this to introduce an equivariant deformation of the disk potential of toric branes. In this paper, we establish a connection to higher-dimensional Calabi–Yau geometries by showing that the equivariant disk potential arises as an equivariant period of certain Calabi–Yau fourfolds and fivefolds, which encode moduli spaces of one and two symplectic cuts (the maximal case) by a construction of Braverman (J für die Reine und Angew Math 1999 (508):85–98, 1999). Extended Picard–Fuchs equations for toric branes, capturing dependence on both open and closed string moduli, are derived from a suitable limit of the equivariant quantum cohomology rings of the higher Calabi–Yau geometries.

For an additive perturbation of the complex Ginibre ensemble under a deterministic matrix (X_0), under certain assumption on (X_0), we observe that there are only two kinds of local statistical patterns at the spectral edge: GinUE statistics and critical statistics, which, respectively, correspond to regular and quadratic vanishing spectral points. As a continuation of our previous study on critical statistics (Liu and Zhang in Critical edge statistics for deformed GinUEs. Preprint arXiv: 2311.13227), in this paper we establish the local statistics of GinUE type at the regular spectral edge, which is characterized by a repeated erfc integral found in Liu and Zhang (Probab Theory Relat Fields. https://doi.org/10.1007/s00440-024-01318-9).

In this paper, we consider the Hölder continuity of the integrated density of states (IDS). Applying Avila’s almost reducible result and KAM technique, we proved that there exists a dense subset of Liouvillean frequencies (alpha ), for which the IDS of the analytic quasi-periodic Schrödinger operator is ((chi )-(log ))-Hölder continuous for any (chi >1), provided that the subcritical strip of the operator satisfies (h_0 > 2beta (alpha ) ). We also proved the (chi )-Hölder continuity of the IDS for a dense subset of Liouvillean frequencies for operators with (0< chi < frac{1}{2}), if the subcritical strip satisfies (h_0 > frac{8beta }{ 1-2chi }).