We derive a determinantal expression for the inhomogeneous asymmetric five vertex model in a rectangular geometry with arbitrary boundary conditions at the bottom and top. Standard non-intersecting lattice path, or free fermion, approaches are not applicable and the determinantal form, thus, is not immediate.
The double tensor Hopf algebra has been introduced by Ebrahimi-Fard and Patras to provide an algebraic framework for cumulants in non-commutative probability theory. In this paper, we obtain a cancellation-free formula, represented in terms of Schr & ouml;der trees, for the antipode in the double tensor Hopf algebra. We apply the antipode formula to recover cumulantmoment formulas as well as a new expression for Anshelevich's free Wick polynomials in terms of Schr & ouml;der trees.
We consider the 3-state Potts generating function T(ν,w) of planar triangulations; that is, the bivariate series that counts planar triangulations with vertices coloured in 3 colours, weighted by their size (number of vertices, recorded by the variable w) and by the number of monochromatic edges (variable ν). This series was proved to be algebraic 15 years ago by Bernardi and the first author: this follows from its link with the solution of a discrete differential equation (DDE), and from general algebraicity results on such equations. However, despite recent progresses on the effective solution of DDEs, the exact value of T(ν,w) has remained unknown so far – except in the case ν=0, corresponding to proper colourings and solved by Tutte in the sixties. We determine here this exact value, proving that T(ν,w) satisfies a polynomial equation of degree 11 in T and genus 1 in w and T. We prove that the critical value of ν is ν_c=1+3/√(47), with a critical exponent 6/5 in the series T(ν_c, ·), while the other values of ν yield the usual map exponent 3/2. By duality of the planar Potts model, our results also characterize the 3-state Potts generating function of planar cubic maps, in which all vertices have degree 3. In particular, the annihilating polynomial, still of degree 11, that we obtain for properly 3-coloured cubic maps proves a conjecture by Bruno Salvy from 2009.
We study q-deformed random unitary ensembles associated with the weight function of the Al-Salam-Carlitz orthogonal polynomials, indexed by a parameter a < 0. In the special case a=- 1, the model reduces to the q-deformed Gaussian unitary ensemble. Employing the Flajolet-Viennot theory together with the combinatorics of matchings, we derive an explicit positive-sum expression for the spectral moments. In the double-scaling regime q= e (-lambda/N) , where N denotes the ensemble size and lambda > 0 is fixed, we derive the first two terms in the largeN expansion of the spectral moments. As a consequence, we obtain a closed-form expression for the limiting spectral density. Notably, this density exhibits two successive phase transitions as lambda increases, characterised by a reduction in the number of soft edges from two, to one, and eventually to none. Furthermore, we show that the limiting density coincides with the limiting zero distribution of the Al-Salam-Carlitz orthogonal polynomials under the same scaling
We consider the perturbative renormalisation of the $\Phi^4_d$ model from Euclidean Quantum Field Theory for any, possibly non-integer dimension $d<4$. The so-called BPHZ renormalisation, named after Bogoliubov, Parasiuk, Hepp and Zimmermann, is usually encoded into extraction-contraction operations on Feynman diagrams, which have a complicated combinatorics. We show that the same procedure can be encoded in the much simpler algebra of polynomials in two unknowns $X$ and $Y$, which represent the fourth and second Wick power of the field. In this setting, renormalisation takes the form of a \lq\lq Wick map\rq\rq\ which maps monomials into Bell polynomials. The construction makes use of recent results by Bruned and Hou on multiindices, which are algebraic objects of intermediate complexity between Feynman diagrams and polynomials.
