Communications in Mathematical Physics最新文献

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.

We prove a nonlinear characteristic (C^k)-gluing theorem for vacuum gravitational fields in Bondi gauge for a class of characteristic hypersurfaces near static vacuum n-dimensional backgrounds, (nge 3), with any finite k, with cosmological constant ( Lambda in mathbb {R}), near Birmingham-Kottler backgrounds. This generalises the (C^2)-gluing of Aretakis, Czimek and Rodnianski, carried-out near light cones in four-dimensional Minkowski spacetime.

Smith homomorphisms are maps between bordism groups that change both the dimension and the tangential structure. We give a completely general account of Smith homomorphisms, unifying the many examples in the literature. We provide three definitions of Smith homomorphisms, including as maps of Thom spectra, and show they are equivalent. Using this, we identify the cofiber of the spectrum-level Smith map and extend the Smith homomorphism to a long exact sequence of bordism groups, which is a powerful computation tool. We discuss several examples of this long exact sequence, relating them to known constructions such as Wood’s and Wall’s sequences. Furthermore, taking Anderson duals yields a long exact sequence of invertible field theories, which has a rich physical interpretation. We developed the theory in this paper with applications in mind to symmetry breaking in quantum field theory, which we study in Debray-Devalapurkar-Krulewski-Liu-Pacheco-Tallaj-Thorngren (J High Energy Phys 2025(7):1–48, 2025)

This paper concerns the large Reynolds number limits and asymptotic behaviors of solutions to the 2D steady Navier–Stokes equations in an infinitely long convergent channel. It is shown that for a general convergent infinitely long nozzle whose boundary curves satisfy curvature-decreasing and any given finite negative mass flux, the Prandtl’s viscous boundary layer theory holds in the sense that there exists a Navier–Stokes flow with no-slip boundary condition for small viscosity, which is approximated uniformly by the superposition of an Euler flow and a Prandtl flow. Moreover, the singular asymptotic behaviors of the solution to the Navier–Stokes equations near the vertex of the nozzle and at infinity are determined by the given mass flux, which is also important for the construction of the Prandtl approximation solution due to the possible singularities at the vertex and non-compactness of the nozzle. One of the key ingredients in our analysis is that the curvature-decreasing condition on boundary curves of the convergent nozzle ensures that the limiting inviscid flow is pressure favorable and plays crucial roles in both the Prandtl expansion and the stability analysis.

We consider the Plancherel–Rotach type asymptotics of the biorthogonal polynomials associated with the biorthogonal ensemble having the joint probability density function

where

In the special case where the potential function V is linear, this biorthogonal ensemble arises in the quantum transport theory of disordered wires. We analyze the asymptotic problem via 2-component vector-valued Riemann–Hilbert problems and solve it under the one-cut regular with a hard edge condition. We use the asymptotics of the biorthogonal polynomials to establish sine universality for the correlation kernel in the bulk and provide a central limit theorem with a specific variance for holomorphic linear statistics. As an application of our theory, we establish Ohm’s law (1.13) and universal conductance fluctuation (1.14) for the disordered wire model, thereby rigorously confirming predictions from experimental physics (Washburn and Webb: Adv Phys 35(4):375–422, 1986).

We study the harmonic locus consisting of the monodromy-free Schrödinger operators with rational potential and quadratic growth at infinity. It is known after Oblomkov that it can be identified with the set of all partitions via the Wronskian map for Hermite polynomials. We show that the harmonic locus can also be identified with the subset of Wilson’s Calogero–Moser space that is fixed by the symplectic action of (mathbb C^times .) As a corollary, for the multiplicity-free part of the locus we effectively solve the inverse problem for the Wronskian map by describing the partition in terms of the spectrum of the corresponding Moser matrix. We also compute the characters of the (mathbb C^times )-action at the fixed points, proving, in particular, a conjecture of Conti and Masoero. In the Appendix written by N. Nekrasov there is an alternative proof of this result, based on the space of instantons and the ADHM construction.