Communications in Mathematical Physics最新文献

In this paper, we study small data solutions for the Vlasov–Poisson system with the simplest external potential, for which unstable trapping holds for the associated Hamiltonian flow. We prove sharp decay estimates in space and time for small data solutions to the Vlasov–Poisson system with the repulsive potential (frac{-|x|^2}{2}) in dimension two or higher. The proofs are obtained through a commuting vector field approach. We exploit the uniform hyperbolicity of the Hamiltonian flow, by making use of the commuting vector fields contained in the stable and unstable invariant distributions of phase space for the linearized system. In dimension two, we make use of modified vector field techniques due to the slow decay estimates in time. Moreover, we show an explicit teleological construction of the trapped set in terms of the non-linear evolution of the force field.

We prove that a large class of (Ntimes N) Gaussian random band matrices with band width W exhibits dynamical Anderson localization at all energies when (W ll N^{1/4}). The proof uses the fractional moment method (Aizenman and Molchanov in Commun Math Phys 157(2):245–278, 1993. https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-157/issue-2/Localizationat-large-disorder-and-at-extreme-energies–an/cmp/1104253939.full) and an adaptive Mermin–Wagner style shift.

We explain a construction of (G_2)-instantons on manifolds obtained by resolving (G_2)-orbifolds. This includes the case of (G_2)-instantons on resolutions of (T^7/Gamma ) as a special case. The ingredients needed are a (G_2)-instanton on the orbifold and a Fueter section over the singular set of the orbifold which are used in a gluing construction. In the general case, we make the very restrictive assumption that the Fueter section is pointwise rigid. In the special case of resolutions of (T^7/Gamma ), improved control over the torsion-free (G_2)-structure allows to remove this assumption. As an application, we construct a large number of (G_2)-instantons on the simplest example of a resolution of (T^7/Gamma ). We also construct one new example of a (G_2)-instanton on the resolution of ((T^3 times text {K3})/mathbb {Z}^2_2).

Given a smooth log Calabi–Yau pair (X, D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau–Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (X, D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of Fan et al. (Sel Math (NS) 25(4): Art. 54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z) to compute two-point relative invariants. When D is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of m Gräfnitz et al. (2022) that the proper Landau–Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.

We are concerned with global finite-energy solutions of the three-dimensional compressible Euler–Poisson equations with gravitational potential and general pressure law, especially including the constitutive equation of white dwarf stars. In this paper, we construct global finite-energy solutions of the Cauchy problem for the Euler–Poisson equations with large initial data of spherical symmetry as the inviscid limit of the solutions of the corresponding Cauchy problem for the compressible Navier–Stokes–Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in (L^p), and a more general compensated compactness framework via several new ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum ((rho =0)) and far-field ((rho =infty )) are carefully analyzed. Owing to the generality of the pressure law, only the (W^{-1,p}_{textrm{loc}})-compactness of weak entropy dissipation measures with (pin [1,2)) can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above, so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the (L^p) compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.

We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution (p_{text {noisy}}) and the corresponding noiseless output distribution (p_{text {ideal}}) shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmark F that measures this correlation behaves as (F=text {exp}(-2sepsilon pm O(sepsilon ^2))), where (epsilon ) is the probability of error per circuit location and s is the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution (p_{text {noisy}}) and the uniform distribution (p_{text {unif}}) decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as (p_{text {noisy}}approx Fp_{text {ideal}}+ (1-F)p_{text {unif}}). In other words, although at least one local error occurs with probability (1-F), the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by (O(Fepsilon sqrt{s})). Thus, the “white-noise approximation” is meaningful when (epsilon sqrt{s} ll 1), a quadratically weaker condition than the (epsilon sll 1) requirement to maintain high fidelity. The bound applies if the circuit size satisfies (s ge Omega (nlog (n))), which corresponds to only logarithmic depth circuits, and if, additionally, the inverse error rate satisfies (epsilon ^{-1} ge {tilde{Omega }}(n)), which is needed to ensure errors are scrambled faster than F decays. The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; for example, it was an underlying assumption in complexity-theoretic arguments that noisy random quantum circuits cannot be efficiently sampled classically, even when the fidelity is low. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.

We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE(({alpha },eta )) of continuum planar aggregation models. The class includes regularized versions of the Hastings–Levitov family HL(({alpha })) and continuum versions of the family of dielectric-breakdown models, where the local attachment intensity for new particles is specified as a negative power (-eta ) of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner–Kufarev equation, where the driving measure is made to depend on the solution and on the parameter ({zeta }={alpha }+eta ). Our results are subject to a subcriticality condition ({zeta }leqslant 1): this includes HL(({alpha })) for ({alpha }leqslant 1) and also the case ({alpha }=2,eta =-1) corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL(({alpha })) at ({alpha }=1), consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein–Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if ({zeta }leqslant 1).

We explore three-dimensional gravity with negative cosmological constant via canonical quantization. We focus on chiral gravity which is related to a single copy of (text {PSL}(2,mathbb {R})) Chern-Simons theory and is simpler to treat in canonical quantization. Its phase space for an initial value surface (Sigma ) is given by the appropriate moduli space of Riemann surfaces. We use geometric quantization to compute partition functions of chiral gravity on three-manifolds of the form (Sigma times {{,textrm{S},}}^1), where (Sigma ) can have asymptotic boundaries. Most of these topologies do not admit a classical solution and are thus not amenable to a direct semiclassical path integral computation. We use an index theorem that expresses the partition function as an integral of characteristic classes over phase space. In the presence of n asymptotic boundaries, we use techniques from equivariant cohomology to localize the integral to a finite-dimensional integral over (overline{mathcal {M}}_{g,n}), which we evaluate in low genus cases. Higher genus partition functions quickly become complicated since they depend in an oscillatory way on Newton’s constant. There is a precise sense in which one can isolate the non-oscillatory part which we call the fake partition function. We establish that there is a topological recursion that computes the fake partition functions for arbitrary Riemann surfaces (Sigma ). There is a scaling limit in which the model reduces to JT gravity and our methods give a novel way to compute JT partition functions via equivariant localization.

We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows us to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation, which allows us to represent the Duhamel expansion for the real-time dynamics in terms of Euclidean correlation functions, for which precise decay estimates are proved using fermionic cluster expansion.

A unitary and strongly rational vertex operator algebra (VOA) ({mathbb {V}}) is called strongly unitary if all irreducible ({mathbb {V}})-modules are unitarizable. A strongly unitary VOA ({mathbb {V}}) is called completely unitary if for each unitary ({mathbb {V}})-modules ({mathbb {W}}_1,{mathbb {W}}_2) the canonical non-degenerate Hermitian form on the fusion product ({mathbb {W}}_1boxtimes {mathbb {W}}_2) is positive. It is known that if ({mathbb {V}}) is completely unitary, then the modular category (textrm{Mod}^textrm{u}({mathbb {V}})) of unitary ({mathbb {V}})-modules is unitary (Gui in Commun Math Phys 372(3):893–950, 2019), and all simple VOA extensions of ({mathbb {V}}) are automatically unitary and moreover completely unitary (Gui in Int Math Res Not 2022(10):7550–7614, 2022; Carpi et al. in Commun Math Phys 1–44, 2023). In this paper, we give a geometric characterization of the positivity of the Hermitian product on ({mathbb {W}}_1boxtimes {mathbb {W}}_2), which helps us prove that the positivity is always true when ({mathbb {W}}_1boxtimes {mathbb {W}}_2) is an irreducible and unitarizable ({mathbb {V}})-module. We give several applications: (1) We show that if ({mathbb {V}}) is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group G, and if ({mathbb {V}}^G) is strongly unitary, then ({mathbb {V}}^G) is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if ({mathbb {V}}) is unitary and strongly rational, and if ({mathbb {U}}) is a simple current extension which is unitarizable as a ({mathbb {V}})-module, then ({mathbb {U}}) is a unitary VOA.