Communications in Mathematical Physics最新文献

We study thermodynamic formalism of dynamical systems with non-uniform structure. Precisely, we obtain the uniqueness of equilibrium states for a family of non-uniformly expansive flows by generalizing Climenhaga-Thompson’s orbit decomposition criteria. In particular, such family includes entropy expansive flows. Meanwhile, the essential part of the decomposition is allowed to satisfy an even weaker version of specification, namely controlled specification, thus also extends the corresponding results in Pavlov, R. (On controlled specification and uniqueness of the equilibrium state in expansive systems. Nonlinearity 32(7), 2441–2466 (2019)). Two applications of our abstract theorems are explored. Firstly, we introduce a notion of regularity condition called weak Walters condition, and study the uniqueness of measure of maximal entropy for a suspension flow with roof function satisfying such condition. Secondly, we investigate topologically transitive frame flows on rank one manifolds of nonpositive curvature, which is a group extension of nonuniformly hyperbolic flows. Under a bunched curvature condition and running a Gauss-Bonnet type of argument, we show the uniqueness of equilibrium states with respect to certain potentials.

We prove the essential self-adjointness of the d’Alembertian (square _g), allowing a larger class of spacetimes than previously considered, including those that arise from perturbing Minkowski spacetime by gravitational radiation. We emphasize the fact, proven by Taira in related settings, that all tempered distributions u satisfying (square _g u = lambda u +f) for (lambda in mathbb {C}backslash mathbb {R}) and f Schwartz are Schwartz. The proof is fully microlocal and relatively quick given the “de,sc-” machinery recently developed by the third author.

The geometric description of incompressible hydrodynamics, as geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms, enables notions of curvature in the study of fluids in order to study stability. Formulas for Ricci curvature are often simpler than those for sectional curvature, which typically takes both signs, but the drawback is that Ricci curvature is rarely well-defined in infinite-dimensional spaces. Here we suggest a definition of Ricci curvature in the case of two-dimensional hydrodynamics, based on the finite-dimensional Zeitlin models arising in quantization theory, which gives a natural tool for renormalization. We provide formulae for the finite-dimensional approximations and give strong numerical evidence that these converge in the infinite-dimensional limit, based in part on four new conjectured identities for Wigner 6j symbols. The suggested limiting expression for (average) Ricci curvature is surprisingly simple and demonstrates an average instability for high-frequency modes which helps explain long-term numerical observations of spherical hydrodynamics due to mixing.

In this paper, we investigate the Langevin dynamics of various lattice formulations of the Yang–Mills–Higgs model, with an inverse Yang–Mills coupling (beta ) and a Higgs parameter (kappa ). The Higgs component is either a bounded field taking values in a compact target space, or an unbounded field taking values in a vector space in which case the model also has a Higgs mass parameter m. We study the regime where ((beta ,kappa )) are small in the first case or ((beta ,kappa /m)) are small in the second case. We prove the exponential ergodicity of the dynamics on the whole lattice via functional inequalities. We establish exponential decay of correlations for a broad class of observables, namely, the infinite volume measure exhibits a strictly positive mass gap. Moreover, when the target space of the Higgs field is compact, appropriately rescaled observables exhibit factorized correlations in the large N limit . These extend the earlier results (Shen et al. in Comm Math Phys 400(2):805–851, 2023) on pure lattice Yang–Mills to the case with a coupled Higgs field. Unlike pure lattice Yang–Mills where the field is always bounded, in the case where the coupled Higgs component is unbounded, the control of its behavior is much harder and requires new techniques. Our approach involves a disintegration argument and a delicate analysis of correlations to effectively control the unbounded Higgs component.

We revisit the classification, and give explicit realisations, of unitary irreducible representations of the BMS group. As compared to McCarthy’s seminal work, we make use of a unique, Lorentz-invariant, decomposition of supermomenta into a hard and a soft piece, that we introduce and properly define, to investigate the extent to which generic representations depart from usual Poincaré particles and highlight their relations to gravitational infrared physics. We insist on making wavefunctions as explicit as possible. Similarly, we explain how branching to a Poincaré subgroup works in practice: this is physically relevant because this amounts to reading off the field content of a given BMS state in terms of a choice of gravity vacuum. In particular, we emphasise how different gravity vacua differ in their interpretation of the same BMS state, here again providing concrete examples as well as the general procedure. Finally, we demonstrate on an example that generic BMS particles are flexible enough to encode memory, as opposed to usual Poincaré particles.

By the introduction of locally constant prefactorization algebras at a fixed scale, we show a mathematical incarnation of the fact that observables at a given scale of a topological field theory propagate to every scale over euclidean spaces. The key is that these prefactorization algebras over are equivalent to algebras over the little n-disc operad. For topological field theories with defects, we get analogous results by replacing with the spaces modelling corners . As a toy example in 1d, we quantize, once more, constant Poisson structures.

This paper studies observability inequalities for heat equations on both bounded domains and the whole space ({mathbb {R}}^d). The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content. Notably, the aforementioned inequality holds not only for all sets with Hausdorff dimension s for any (sin (d-1,d]), but also for certain sets of Hausdorff dimension exactly (d-1). On the whole space ({mathbb {R}}^d), we establish the observability inequality for observation sets that are thick at the scale of the log-type Hausdorff content. Furthermore, we prove that for the 1-dimensional heat equation on an interval, the Hausdorff content we have chosen is an optimal scale for the observability inequality. To obtain these observability inequalities, we use the adapted Lebeau-Robbiano strategy of Duyckaerts and Miller (J. Funct. Anal. 2012). For this purpose, we prove the following results at scale of the log-type Hausdorff content, the former being derived from the latter: we establish a fractal version of spectral inequality/Logvinenko-Sereda uncertainty principle; we develop a quantitative propagation of smallness of analytic functions; we build up a Remez-type inequality; and more fundamentally, we provide an upper bound for the log-type Hausdorff content of a set where a monic polynomial is small, based on an estimate by Lubinsky (J. Inequal. Appl. 1997), which is ultimately traced back to the classical Cartan Lemma. In addition, we set up a capacity-based slicing lemma (related to the log-type gauge functions) and establish a quantitative relationship between Hausdorff contents and capacities. These tools are crucial in the studies of the aforementioned propagation of smallness in high-dimensional situations.

Motivated by various applications, unbounded Hamiltonian simulation has recently garnered great attention. Quantum Magnus algorithms, designed to achieve commutator scaling for time-dependent Hamiltonian simulation, have been found to be particularly efficient for such applications. When applied to unbounded Hamiltonian simulation in the interaction picture, they exhibit an unexpected superconvergence phenomenon. However, existing proofs are limited to the spatially continuous setting and do not extend to discrete spatial discretizations. In this work, we provide the first superconvergence estimate in the fully discrete setting with a finite number of spatial discretization points N, and show that it holds with an error constant uniform in N. The proof is based on the two-parameter symbol class, which, to our knowledge, is applied for the first time in algorithm analysis. The key idea is to establish a semiclassical framework by identifying two parameters through the discretization number and the time step size rescaled by the operator norm, such that the semiclassical uniformity guarantees the uniformity of both. This approach may have broader applications in numerical analysis beyond the specific context of this work.

We show a relation between quantum learning theory and algorithmic hardness. We use the existence of efficient, local learning algorithms for energy estimation—such as the classical shadows algorithm—to prove that finding near-ground states of disordered quantum systems exhibiting a certain topological property is impossible in the average case for Lipschitz quantum algorithms. A corollary of our result is that many standard quantum algorithms fail to find near-ground states of these systems, including time-T Lindbladian dynamics from an arbitrary initial state, time-T quantum annealing, phase estimation to T bits of precision, and depth-T variational quantum algorithms, whenever T is less than some universal constant times the logarithm of the system size. To achieve this, we introduce a generalization of the overlap gap property (OGP) for quantum systems that we call the quantum overlap gap property (QOGP). This property is defined by a specific topological structure over representations of low-energy quantum states as output by an efficient local learning algorithm. We prove that preparing low-energy states of systems which exhibit the QOGP is intractable for quantum algorithms whose outputs are stable under perturbations of their inputs. We then prove that the QOGP is satisfied for a sparsified variant of the quantum p-spin model, giving the first known algorithmic hardness-of-approximation result for quantum algorithms in finding the ground state of a non-stoquastic, noncommuting quantum system. Our resulting lower bound for quantum algorithms optimizing this model using Lindbladian evolution matches (up to constant factors) the best-known time lower bound for classical Langevin dynamics optimizing classical p-spin models. For this reason we suspect that finding ground states of typical instances of these quantum spin models using quantum algorithms is, in practice, as intractable as the classical p-spin model is for classical algorithms. Inversely, we show that the Sachdev–Ye–Kitaev model does not exhibit the QOGP, consistent with previous evidence that the model is rapidly mixing at low temperatures.

In this paper, we construct invariant Gibbs dynamics for the hyperbolic (Phi ^{k+1}_2)-model (namely, defocusing stochastic damped nonlinear wave equation forced by an additive space-time white noise) on the plane. (i) For this purpose, we first revisit the construction of a (Phi ^{k+1}_2)-measure on the plane. More precisely, by establishing coming down from infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane, we first construct a (Phi ^{k+1}_2)-measure on the plane as a limit of the (Phi ^{k+1}_2)-measures on large tori. (ii) We then construct invariant Gibbs dynamics for the hyperbolic (Phi ^{k+1}_2)-model on the plane, by taking a limit of the invariant Gibbs dynamics on large tori constructed by the first two authors with Gubinelli and Koch (Int Math Res Not 21:16954–16999,2022). Here, our main strategy is to develop further the ideas from a recent work on the hyperbolic (Phi ^3_3)-model on the three-dimensional torus by the first two authors and Okamoto (Mem Eur Math Soc 16, 2025), and to study convergence of the so-called enhanced Gibbs measures, for which coming down from infinity for the associated SNLH with positive regularity plays a crucial role. By combining wave and heat analysis together with ideas from optimal transport theory, we then conclude global well-posedness of the hyperbolic (Phi ^{k+1}_2)-model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting (Phi ^{k+1}_2)-measure on the plane under the dynamics of the parabolic (Phi ^{k+1}_2)-model.