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.

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.

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 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.

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.

The precise renormalizable interactions in the bosonic sector of electroweak theory are intrinsically determined in the autonomous approach to perturbation theory. This proceeds directly on the Hilbert–Fock space built on the Wigner unirreps of the physical particles, with their given masses: those of three massive vector bosons, a photon, and a massive scalar (the “higgs”). Neither “gauge choices” nor an unobservable “mechanism of spontaneous symmetry breaking” is invoked. Instead, to proceed on Hilbert space requires using string-localized fields to describe the vector bosons. In such a framework, the condition of string independence of the ({mathbb {S}})-matrix yields consistency constraints on the coupling coefficients, the essentially unique outcome being the experimentally known one. The analysis can be largely carried out for other configurations of massive and massless vector bosons, paving the way towards consideration of consistent mass patterns beyond those of the electroweak theory.

We prove that all nice holomorphic vertex operator superalgebras (VOSAs) with central charge at most 24 and with non-trivial odd part are unitary, apart from the hypothetical ones arising as fake copies of the shorter moonshine VOSA or of the latter tensorized with a real free fermion VOSA. Furthermore, excluding the ones with central charge 24 of glueing type III and with no real free fermion, we show that they are all strongly graded-local. In particular, they naturally give rise to holomorphic graded-local conformal nets. In total, we are able to prove that 910 of the 969 nice holomorphic VOSAs with central charge 24 and with non-trivial odd part are strongly graded-local, without counting hypothetical fake copies of the shorter moonshine VOSA tensorized with a real free fermion VOSA.

We propose a geometric characterisation of the topological string partition functions associated with the local Calabi–Yau (CY) manifolds used in the geometric engineering of (d=4), ({mathcal {N}}=2) supersymmetric field theories of class ({mathcal {S}}). A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated with the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalization of the tau-functions associated with these jumps define a natural line bundle playing a key role in the geometric characterisation of the topological string partition functions proposed here.

In the perturbative treatment of interacting quantum field theories, if the interaction Lagrangian changes adiabatically in a finite interval of time, secular growths may appear in the truncated perturbative series also when the interaction Lagrangian density is returned to be constant. If this happens, the perturbative approach does not furnish reliable results in the evaluation of scattering amplitudes or expectation values. In this paper we show that these effects can be avoided for adiabatically switched-on interactions, if the spatial support of the interaction is compact and if the background state is suitably chosen. We start considering equilibrium background states and show that, when thermalisation occurs (interaction Lagrangian of spatial compact support), secular effects are avoided. Furthermore, no secular effects pop up if the limit where the Lagrangian is supported everywhere in space is taken after thermalisation (large time limit), in contrast to the reversed order. This result is generalized showing that if the interaction Lagrangian is spatially compact, secular growths are avoided for generic background states which are only invariant under time translation and to states whose explicit dependence of time is not too strong. Finally, as an application, the presented theorems are used to study a complex scalar and a Dirac field, on a background KMS state, in a classical external electromagnetic potential and the contribution to the two point-function given by a generic loop diagram arising from a second order perturbative expansion.

Motivated by the quantization of linearized gravity, we consider gauge-fixed linearized Einstein equations and their Wick rotation near a Cauchy surface. We show that Calderón projectors for the Wick-rotated equations induce Hadamard bi-solutions on the Lorentzian level. On the other hand, we find smoothing obstructions to gauge-invariance and positivity conditions needed in quantization. These obstructions are primarily due to boundary terms arising in the Wick-rotated theory and depend on the boundary conditions.