Communications in Mathematical Physics最新文献

We prove the Eigenstate Thermalization Hypothesis for general Wigner-type matrices in the bulk of the self-consistent spectrum, with optimal control on the fluctuations for obs ervables of arbitrary rank. As the main technical ingredient, we prove rank-uniform optimal local laws for one and two resolvents of a Wigner-type matrix with regular observables. Our results hold under very general conditions on the variance profile, even allowing many vanishing entries, demonstrating that Eigenstate Thermalization occurs robustly across a diverse class of random matrix ensembles, for which the underlying quantum system has a non-trivial spatial structure.

In this paper, we continue our investigation of the triple Hodge integrals satisfying the Calabi–Yau condition. For the tau-functions, which generate these integrals, we derive the complete families of the Heisenberg–Virasoro constraints. We also construct several equivalent versions of the cut-and-join operators. These operators describe the algebraic version of topological recursion. For the specific values of parameters associated with the KdV reduction, we prove that these tau-functions are equal to the generating functions of intersection numbers of (psi ) and (kappa ) classes. We interpret this relation as a symplectic invariance of the Chekhov–Eynard–Orantin topological recursion and prove this recursion for the general (Theta )-case.

The ‘brane quantisation’ is a quantisation procedure developed by Gukov and Witten (Adv Theor Math Phys 13(5):1445–1518, 2009). We implement this idea by combining it with the tilting theory and the minimal resolutions. This way, we can realistically compute the deformation quantisation on the space of observables acting on the Hilbert space. We apply this procedure to certain quantisation problems in the context of generalised Kähler structure on ({mathbb {P}}^2). Our approach differs from and complements that of Bischoff and Gualtieri (Commun Math Phys 391(2):357–400, 2022). We also benefitted from an important technical tool: a combinatorial criterion for the Maurer–Cartan equation, developed by Barmeier and Wang (Deformations of path algebras of quivers with relations, 2020. arXiv:2002.10001).

The investigation of rigorous justification of the hydrodynamic limits in bounded domains has seen significant progress in recent years. While some headway has been made for the diffuse-reflection boundary case (Esposito et al. in Ann PDE 4:1–119, 2018; Ghost effect from Boltzmann theory. arXiv:2301.09427, 2023; Jang and Kim in Ann PDE 7:103, 2021), the more intricate in-flow boundary case, where the leading-order boundary layer effect cannot be neglected, still poses an unresolved challenge. In this study, we tackle the stationary and evolutionary Boltzmann equations, considering the in-flow boundary conditions within both convex and non-convex bounded domains, and demonstrate their diffusive limits in (L^2). Our approach hinges on a groundbreaking insight: a remarkable gain of (varepsilon ^{frac{1}{2}}) in the kernel estimate, which arises from a meticulous selection of test functions and the careful application of conservation laws. Additionally, we introduce a boundary layer with a grazing-set cutoff and investigate its BV regularity estimates to effectively control the source terms in the remainder equation with the help of the Hardy’s inequality.

We show that a simple telescoping sum trick, together with the triangle inequality and a tensorisation property of expected-contractive coefficients of random channels, allow us to achieve general simultaneous decoupling for multiple users via local actions. Employing both old (Dupuis et al. in Commun Math Phys 328:251–284, 2014) and new methods (Dupuis in IEEE Trans Inf Theory 69:7784–7792, 2023), we obtain bounds on the expected deviation from ideal decoupling either in the one-shot setting in terms of smooth min-entropies, or the finite block length setting in terms of Rényi entropies. These bounds are essentially optimal without the need to address the simultaneous smoothing conjecture, which remains unresolved. This leads to one-shot, finite block length, and asymptotic achievability results for several tasks in quantum Shannon theory, including local randomness extraction of multiple parties, multi-party assisted entanglement concentration, multi-party quantum state merging, and quantum coding for the quantum multiple access channel. Because of the one-shot nature of our protocols, we obtain achievability results without the need for time-sharing, which at the same time leads to easy proofs of the asymptotic coding theorems. We show that our one-shot decoupling bounds furthermore yield achievable rates (so far only conjectured) for all four tasks in compound settings, which are additionally optimal for entanglement of assistance and state merging.

Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a (d^{p+q})-dimensional matrix variable that commutes with (U^{otimes p} otimes {bar{U}}^{otimes q}), for all (U in textrm{U}(d)). Solving such problems naively can be prohibitively expensive even if (p+q) is small but the local dimension d is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in d, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand–Tsetlin basis, which we obtain by adapting a general method inspired by the Okounkov–Vershik approach. To illustrate potential applications of our framework, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.

Lattice gauge theories are lattice approximations of the Yang–Mills theory in physics. The abelian lattice Higgs model is one of the simplest examples of a lattice gauge theory interacting with an external field. In a previous paper (Forsström et al. in Math Phys 4(2):257–329, 2023), we calculated the leading order term of the expected value of Wilson loop observables in the low-temperature regime of the abelian lattice Higgs model on ({mathbb {Z}}^4,) with structure group (G = {mathbb {Z}}_n) for some (n ge 2.) In the absence of a Higgs field, these are important observables since they exhibit a phase transition which can be interpreted as distinguishing between regions with and without quark confinement. However, in the presence of a Higgs field, this is no longer the case, and a more relevant family of observables are so-called open Wilson lines. In this paper, we extend and refine the ideas introduced in Forsström et al. (Math Phys 4(2):257–329, 2023) to calculate the leading order term of the expected value of the more general Wilson line observables. Using our main result, we then calculate the leading order term of several natural ratios of expected values and confirm the behavior predicted by physicists.

In the context of 3-manifolds, determining the asymptotic expansion of the Witten–Reshetikhin–Turaev invariants and constructing the topological field theory that provides their categorification remain important unsolved problems. Motivated by solving these problems, Gukov–Pei–Putrov–Vafa refined the Witten–Reshetikhin–Turaev invariants from a physical point of view. From a mathematical point of view, we can describe that they introduced new q-series invariants for negative definite plumbed manifolds and conjectured that their radial limits coincide with the Witten–Reshetikhin–Turaev invariants. In this paper, we prove their conjecture. In our previous work, the author attributed this conjecture to the holomorphy of certain meromorphic functions by developing an asymptotic formula based on the Euler–Maclaurin summation formula. However, it is challenging to prove holomorphy for general plumbed manifolds. In this paper, we address this challenge using induction on a sequence of trees obtained by repeating “pruning trees,” which is a special type of the Kirby moves.

The Josephson–Anderson relation, valid for the incompressible Navier–Stokes solutions which describe flow around a solid body, equates the power dissipated by drag instantaneously to the flux of vorticity across the flow lines of the potential Euler solution considered by d’Alembert. Its derivation involves a decomposition of the velocity field into this background potential-flow field and a solenoidal field corresponding to the rotational wake behind the body, with the flux term describing a transfer from the interaction energy between the two fields and into kinetic energy of the rotational flow. We establish the validity of the Josephson–Anderson relation for the weak solutions of the Euler equations obtained in the zero-viscosity limit, with one transfer term due to inviscid vorticity flux and the other due to a viscous skin-friction anomaly. Furthermore, we establish weak forms of the local balance equations for both interaction and rotational energies. We define nonlinear spatial fluxes of these energies and show that the asymptotic flux of interaction energy to the wall equals the anomalous skin-friction term in the Josephson–Anderson relation. However, when the Euler solution satisfies a condition of vanishing normal velocity at the wall, then the anomalous term vanishes. In this case, we can show also that the asymptotic flux of rotational energy to the wall must vanish and we obtain in the rotational wake the Onsager–Duchon–Robert relation between viscous dissipation anomaly and inertial dissipation due to scale-cascade. In this way we establish a precise connection between the Josephson–Anderson relation and the Onsager theory of turbulence, and we provide a novel resolution of the d’Alembert paradox.

The non-local degenerate Cahn–Hilliard equation is derived from the Vlasov equation with long range attraction. We study the local limit as the delocalization parameter converges to 0. The difficulty arises from the degeneracy which requires compactness estimates, but all necessary a priori estimates can be obtained only on the nonlocal quantities yielding almost no information on the limiting solution itself. We introduce a novel condition on the nonlocal kernel which allows us to exploit the available nonlocal a priori estimates. The condition, satisfied by most of the kernels appearing in the applications, can be of independent interest. Our approach is flexible and systems can be treated as well.