Annales Henri Poincaré最新文献

We perform a rigorous study of the Gibbs statistics of high-density hard-core random configurations on a unit triangular lattice (mathbb {A}_2) and a unit honeycomb graph (mathbb {H}_2), for any value of the (Euclidean) repulsion diameter (D>0). Only attainable values of D are relevant, for which (D^2=a^2+b^2+ab), (a, b in mathbb {Z}) (Löschian numbers). Depending on arithmetic properties of (D^2), we identify, for large fugacities, the pure phases (extreme Gibbs measures) and specify their symmetries. The answers depend on the way(s) an equilateral triangle of side-length D can be inscribed in (mathbb {A}_2) or (mathbb {H}_2). On (mathbb {A}_2), our approach works for all attainable (D^2); on (mathbb {H}_2) we have to exclude (D^2 = 4, 7, 31, 133), where a sliding phenomenon occurs, similar to that on a unit square lattice (mathbb {Z}^2). For all values (D^2) apart from the excluded ones, we prove the coexistence of multiple high-density pure phases. Their number grows at least as (O(D^2)); this establishes the existence of a phase transition. The proof is based on the Pirogov–Sinai theory which, in its original form, requires the verification of key assumptions: finiteness of the set of periodic ground states and the Peierls bound. To establish the Peierls bound, we develop a general method based on the concept of a redistributed area for Delaunay triangles. Some of the presented proofs are computer-assisted. As a by-product of the ground state identification, we solve the disk-packing problem on (mathbb {A}_2) and (mathbb {H}_2) for any value of the disk diameter D.

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 }).

We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in (Ntimes N) non-Hermitian random Gaussian matrices. In a well-known work by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as (Nrightarrow infty ) (Chalker and Mehlig in Phys Rev Lett 81(16):3367–3370, 1998). In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles, which are characterised by correlations between off-diagonal entries controlled by a parameter (tau in [0,1]), with (tau =1) corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite N, for any eigenvalue off the real axis. We further investigate several scaling regimes as (Nrightarrow infty ), both in the limit of strong non-Hermiticity keeping a fixed (tau in [0,1)) and in the weak non-Hermiticity limit, with (tau ) approaching unity in such a way that (N(1-tau )) remains finite.

Generalised hydrodynamics (GHD) describes the large-scale inhomogeneous dynamics of integrable (or close to integrable) systems in one dimension of space, based on a central equation for the fluid density or quasi-particle density: the GHD equation. We consider a new, general form of the GHD equation: we allow for spatially extended interaction kernels, generalising previous constructions. We show that the GHD equation, in our general form and hence also in its conventional form, is Hamiltonian. This holds also including force terms representing inhomogeneous external potentials coupled to conserved densities. To this end, we introduce a new Poisson bracket on functionals of the fluid density, which is seen as our dynamical field variable. The total energy is the Hamiltonian whose flow under this Poisson bracket generates the GHD equation. The fluid density depends on two (real and spectral) variables, and the GHD equation can be seen as a (2+1)-dimensional classical field theory. In its (1+1)-dimensional reduction corresponding to the case without external forces, we further show the system admits an infinite set of conserved quantities that are in involution for our Poisson bracket, hinting at integrability of this field theory.

Traditionally, the evolution of interacting particle systems has been based on an assumption that only the individual components undergo state changes. However, this rigid assumption is not the only possibility. This research explored a class of one-dimensional random processes that evolved in discrete time. During each time step, components in the state zero exhibited the following transitions. First, they could change to one with a probability (beta _0). Second, they could be replaced by a sequence of k consecutive zeros with a probability (beta _k) (where (k=1,ldots ,n)). Moreover, these transitions occurred independent of the events occurring elsewhere in the involved system. Notably, this study revealed an unexpected phenomenon—the occurrence of a first-order phase transition between ergodic and non-ergodic behaviors within this system. Furthermore, in the non-ergodic regime, the existence of an invariant measure distinct from the trivial one was demonstrated.

We consider random n-covers (X_n) of an arbitrary compact hyperbolic surface X. We show that in the large n regime and small window limit, the variance of the smooth spectral statistics of the Laplacian (Delta _rho ) twisted by a unitary representation, obey the universal laws of GOE and GUE random matrices, depending on wether the representation (rho ) preserves or breaks the time reversal symmetry. These results are in accordance with the semiclassical heuristics of Berry (Stochastic processes in classical and quantum systems, Springer, Berlin, 1986; Chaotic behavior of deterministic systems, North-Holland, Amsterdam, 1983) and are a discrete analog of a recent work of Rudnick (Geom. Funct. Anal. 33(6), 1581-1607 2023) for the Weil-Petersson model of random surfaces.

We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil–Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian orthogonal ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. In this note, we show that in this same limit, the (smooth) energy variance for a typical surface is close to the GOE result, a feature called “ergodicity” in the random matrix theory literature.

The asymptotic equipartition property (AEP) in information theory shows that independent and identically distributed (i.i.d.) states behave as uniform states on its typical subspace. In particular, such a phenomenon can be expressed as that asymptotically, the min- and the max-relative entropy under appropriate smoothing coincide with the relative entropy. In this paper, we generalize several such equipartition properties to states on general von Neumann algebras. First, we show that the smooth max-relative entropy of i.i.d. states on a von Neumann algebra has an asymptotic rate given by the quantum relative entropy. In fact, our AEP not only applies to states, but also to quantum channels with appropriate restrictions. In addition, going beyond the i.i.d. assumption, we show that for states that are produced by a sequential process of quantum channels, the smooth max-relative entropy can be upper-bounded by the sum of appropriate channel relative entropies. Our main technical contributions are to extend to the context of general von Neumann algebras a chain rule for quantum channels, as well as an additivity result for the channel relative entropy with a replacer channel.

We prove a new microlocal estimate with normally hyperbolic trapping, which can be applied to Kerr and Kerr-de Sitter spacetimes. We use a new type of symbol class, and corresponding operator class, which is constructed by blowing up the intersection of the unstable manifold and fiber infinity. For scalar wave equations on Kerr and Kerr-de Sitter spacetimes, the extra loss of the microlocal estimates compared with the standard propagation of singularities without trapping is arbitrarily small.

We are concerned in this paper with the real eigenfunctions of Schrödinger operators. We prove an asymptotic upper bound for the number of their nodal domains, which implies in particular that the inequality stated in Courant’s theorem is strict, except for finitely many eigenvalues. Results of this type originated in 1956 with Pleijel’s theorem on the Dirichlet Laplacian and were obtained for some classes of Schrödinger operators by the first author, alone and in collaboration with B. Helffer and T. Hoffmann-Ostenhof. Using methods in part inspired by work of the second author on Neumann and Robin Laplacians, we greatly extend the scope of these previous results.