Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions

We study integrals on the unitary group with respect to the Haar measure. We give a combinatorial interpretation in terms of maps of the asymptotic topological expansion, established previously by Guionnet and Novak. The maps we introduce—the maps of unitary type—satisfy Tutte-like equations. It allows us to show that in the perturbative regime, they describe the different orders of the asymptotic topological expansion. Furthermore, they generalize the monotone Hurwitz numbers.

Thomas Buc-d’Alché

Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions

Topological expansion of unitary integrals and maps

幺正积分与映射的拓扑展开

Consider a real point configuration \bm{A} of size n and an integer r \leq n . The vertices of the r -lineup polytope of \bm{A} correspond to the possible orderings of the top r points of the configuration obtained by maximizing a linear functional. The motivation behind the study of lineup polytopes comes from the representability problem in quantum chemistry. In that context, the relevant point configurations are the vertices of hypersimplices and the integer points contained in an inflated regular simplex. The central problem consists in providing an inequality representation of lineup polytopes as efficiently as possible. In this article, we adapt the developed techniques to the quantum information theory setup. The appropriate point configurations become the vertices of products of simplices. A particular case is that of lineup polytopes of cubes, which form a type B analog of hypersimplices, where the symmetric group of type A naturally acts. To obtain the inequalities, we center our attention on the combinatorics and the symmetry of products of simplices to obtain an algorithmic solution. Along the way, we establish relationships between lineup polytopes of products of simplices with the Gale order, standard Young tableaux, and the resonance arrangement.

Federico Castillo

Jean-Philippe Labbé

Lineup polytopes of products of simplices

单纯形乘积的列阵多面体

We study a rational matroid invariant, obtained as the tropicalization of the Feynman period integral. It equals the volume of the polar of the matroid polytope and we give efficient formulas for its computation. This invariant is proven to respect all known identities of Feynman integrals for graphs. We observe a strong correlation between the tropical and transcendental integrals, which yields a method to approximate unknown Feynman periods.

Erik Panzer

Hepp’s bound for Feynman graphs and matroids

Hepp对费曼图和拟阵的界

We consider regions obtained from 120 degree rotationally invariant hexagons by removing a core and three equal satellites (all equilateral triangles) so that the resulting region is both vertically symmetric and 120 degree rotationally invariant, and give simple product formulas for the number of their lozenge tilings. We describe a new method of approach for proving these formulas, and give the full details for an illustrative special case. As a byproduct, we are also able to generalize this special case in a different direction, by finding a natural counterpart of a twenty year old formula due to Ciucu, Eisenkölbl, Krattenthaler, and Zare, which went unnoticed until now. The general case of the original problem will be treated in a subsequent paper. We then work out consequences for the correlation of holes, which were the original motivation for this study.

Mihai Ciucu

Ilse Fischer

Lozenge tilings of hexagons with removed core and satellites

六边形中移除核心和卫星后的菱形拼贴

The purpose of the present paper is to study the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillator (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, and deepen the understanding of the spectrum of the NcHO. We study first the general expression of special values of the spectral zeta function $\zeta_Q(s)$ of the NcHO at $s=n$ $(n=2,3,\dots)$ and then the generating and meta-generating functions for Ap\'ery-like numbers defined through the analysis of special values $\zeta_Q(n)$. Actually, we show that the generating function $w_{2n}$ of such Ap\'ery-like numbers appearing (as the "first anomaly") in $\zeta_Q(2n)$ for $n=2$ gives an example of automorphic integral with rational period functions in the sense of Knopp, but still a better explanation remains to be clarified explicitly for $n>2$. This is a generalization of our earlier result on showing that $w_2$ is interpreted as a $\Gamma(2)$-modular form of weight $1$. Moreover, certain congruence relations over primes for "normalized" Ap\'ery-like numbers are also proven. In order to describe $w_{2n}$ in a similar manner as $w_2$, we introduce a differential Eisenstein series by using analytic continuation of a classical generalized Eisenstein series due to Berndt. The differential Eisenstein series is actually a typical example of the automorphic integral of negative weight. We then have an explicit expression of $w_4$ in terms of the differential Eisenstein series. We discuss also shortly the Hecke operators acting on such automorphic integrals and relating Eichler's cohomology group.

Abbreviation
Journal Impact	1.40
Quartiles(Global)	PHYSICS, MATHEMATICAL(Q2)

ISSN	2308-5827, 2308-5835
h-index	13

Publication Information	Publisher: European Mathematical Society Publishing House，Publishing cycle: ，Journal Type: journal，Open Access Journals: Yes
Basic data	Year of publication: 2014，Proportion of original research papers: 100.00%，Self Citation Rate:6.70%， Gold OA Rate: 69.23%
Average review cycle	100 weeks

Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions

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Topological expansion of unitary integrals and maps

Lineup polytopes of products of simplices

Hepp’s bound for Feynman graphs and matroids

Lozenge tilings of hexagons with removed core and satellites

Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals

Lineup polytopes of products of simplices

Topological expansion of unitary integrals and maps

Local dimer dynamics in higher dimensions

Logarithmic terms in discrete heat kernel expansions in the quadrant

Permutation invariant tensor models and partition algebras