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 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
D_{4} triality invariants are modular forms as well as invariant polynomials for a fiber product of the modular group and the Weyl group of type F_{4} . We show that the ring of D_{4} triality invariants satisfying a certain cusp condition is isomorphic to the ring of joint covariants of a binary cubic and a binary quadratic form.
In the present manuscript we address and solve for the first time a nonlocal discrete isoperimetric problem. We consider indeed a generalization of the classical perimeter, what we call a nonlocal bi-axial discrete perimeter, where, not only the external boundary of a polyomino $\mathcal{P}$ contributes to the perimeter, but all internal and external components of $\mathcal{P}$. Furthermore, we find and characterize its minimizers in the class of polyominoes with fixed area $n$. Moreover, we explain how the solution of the nonlocal discrete isoperimetric problem is related to the rigorous study of the metastable behavior of a long-range bi-axial Ising model.
Adinkras are highly structured graphs developed to study 1-dimensional supersymmetry algebras. A cyclic ordering of the edge colors of an Adinkra, or rainbow, determines a Riemann surface and a height function on the vertices of the Adinkra determines a divisor on this surface. We study the induced map from height functions to divisors on the Jacobian of the Riemann surface. In the first nontrivial case, a 5-dimensional hypercube corresponding to a Jacobian given by a product of 5 elliptic curves each with j-invariant 2048, we develop and characterize a purely combinatorial algorithm to compute height function images. We show that when restricted to a single elliptic curve, every height function is a multiple of a specified generating divisor, and raising and lowering vertices corresponds to adding or subtracting this generator. We also give strict bounds on the coefficients of this generator that appear in the collection of all divisors of height functions.
In this article we study domino tilings of a family of finite regions called Aztec diamonds. Every such tiling determines a partition of the Aztec diamond into five sub-regions; in the four outer sub-regions, every tile lines up with nearby tiles, while in the fifth, central sub-region, differently-oriented tiles co-exist side by side. We show that when n is sufficiently large, the shape of the central sub-region becomes arbitrarily close to a perfect circle of radius n/sqrt(2) for all but a negligible proportion of the tilings. Our proof uses techniques from the theory of interacting particle systems. In particular, we prove and make use of a classification of the stationary behaviors of a totally asymmetric one-dimensional exclusion process in discrete time.
We prove that meromorphic differentials omega((0))(n) (z(1), ... , z(n)) which are recursively generated by an involution identity are symmetric in all their arguments z(1), ... , z(n). The proof involves an intriguing combinatorial identity between integer partitions into a given number of parts.
A longstanding conjecture in Phi(4)(4) theory is that primitive graphs dominate the beta function asymptotically at large loop order in the minimal-subtraction scheme. Here, we investigate this issue by exploiting additional combinatorial structure coming from an extension to vectors with O(N) symmetry. For the 0-dimensional case, we calculate the N-dependent generating function of primitive graphs and its asymptotics, including arbitrarily many subleading corrections. We find that the leading asymptotic growth rate becomes visible only above approximate to 25 loops, while data at lower order is suggestive of a wrong asymptotics. Our results also yield the symmetry-factor weighted sum of 3-connected cubic graphs, and the exact asymptotics of Martin invariants. For individual Feynman graphs, we give bounds on their degree in N depending on their coradical degree, and construct the primitive graphs of highest degree explicitly. We calculate the 4D primitive beta function numerically up to 17 loops, and find its behaviour to be qualitatively similar to the 0D case. The locations of zeros quickly approach their large-loop asymptotics at negative integer N, while the growth rate of the beta function differs from the asymptotic prediction even at 17 loops.
The number of standard Young tableaux of a skew shape lambda/mu can be computed as a sum over excited diagrams inside lambda. Excited diagrams are in bijection with certain lozenge tilings, with flagged semistandard tableaux and also non-intersecting lattice paths inside lambda. We give two new proofs of a multivariate generalization of this formula, which allow us to extend the setup beyond standard Young tableaux and the underlying Schur symmetric polynomials. The first proof uses multiple contour integrals. The second one interprets excited diagrams as configurations of a six-vertex model at a free fermion point and derives the formula for the number of standard Young tableaux of a skew shape from the Yang-Baxter equation. The proofs provide frameworks for the derivation of other such formulas.
Let X-1, ... , X-n be probability spaces, let X be their direct product, let Phi(1), ... , Phi(m) : X -> C be random variables, each depending only on a few coordinates of x = (x(1), ... , x(n)), and let f = Phi(1) + ...+ Phi(m). The expectation E e(lambda f,) where lambda is an element of C, appears in statistical physics as the partition function of a system with multi-spin interactions, and also in combinatorics and computer science, where it is known as the partition function of edge-coloring models, tensor network contractions, or a Holant polynomial. Assuming that each Phi(i) is 1-Lipschitz in the Hamming metric of X, that each Phi(i) (x) depends on at most r >= 2 coordinates x(1), ... , x(n) of x is an element of X, and that for each j there are at most c >= 1 functions aii that depend on the coordinate ./ x(j) , we prove that E e(lambda f) not equal 0 provided |lambda| (3c root r-1)(-1) and that the bound is sharp up to a constant factor. Taking a scaling limit, we prove a similar result for functions Phi(1), ... , Phi(m) : R-n-C that are 1-Lipschitz in the 1 pound metric of R-n and where the expectation is taken with respect to the standard Gaussian measure in R-n. As a corollary, the value of the expectation can be efficiently approximated, provided ) lies in a slightly smaller disc.
We study the Factorization Paradox from the bottom up by adapting methods from perturbative renormalization. Just as quantum field theories are plagued with loop divergences that need to be canceled systematically by introducing counterterms, gravitational path integrals are plagued by wormhole contributions that spoil the factorization of the holographic dual. These wormholes must be canceled by some stringy effects in a UV-complete, holographic theory of quantum gravity. In a simple model of two-dimensional topological gravity, we outline a gravitational analog of the recursive BPHZ procedure in order to systematically introduce "counter-wormholes" which parametrize the unknown stringy effects that lead to factorization. Underlying this procedure is a Hopf algebra of symmetries which is analogous to the Connes-Kreimer Hopf algebra underlying perturbative renormalization. The group dual to this Hopf algebra acts to reorganize contributions from spacetimes with distinct topology and can be seen as a gauge group relating various equivalent ways of constructing a factorizing gravitational path integral.
A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and has a trapped endpoint. We show that the generating function enumerating GSAWs on a half-infinite strip of finite height is rational, and we give a procedure to construct a combinatorial finite state machine that allows one to compute this generating function. We then modify this procedure to compute generating functions for GSAWs under two probabilistic models. We derive the mean trapping lengths for GSAWs in strips of height 2 to 5 to gain insight into the empirically known square lattice result of 71 steps. Finally, we prove that the generating functions for Greek key tours (GSAWs on a finite grid that visit every vertex) on a half-infinite strip of fixed height are also rational, allowing us to resolve several conjectures.
We study perfect matchings on the rail yard graphs in which the right boundary condition is given by the empty partition and the left boundary can be divided into finitely many alternating line segments where all the vertices along each line segment are either removed or remained. When the edge weights satisfy certain conditions, we show that the distributions of the locations of certain types of dimers near the right boundary converge to the spectra of independent GUE minor processes. The proof is based on new quantitative analysis of a formula to compute Schur functions at general points discovered by Li (2022).
In this paper, we give the first combinatorial proof of a rationality scheme for the generating series of maps in positive genus enumerated by both vertices and faces, which was first obtained by Bender, Canfield and Richmond in 1993 by purely computational techniques. To do so, we rely on a bijection obtained by the second author in a previous work between those maps and a family of decorated unicellular maps. Our main contribution consists in a fine analysis of this family of maps. As a byproduct, we also obtain a new and simpler combinatorial proof of the rationality scheme for the generating series of maps enumerated by their number of edges, originally obtained computationally by Bender and Canfield in 1991 and combinatorially by the second author in 2019.
We study a combinatorial model of the quantum scalar field with polynomial potential on a graph. In the first quantization formalism, the value of a Feynman graph is given by a sum over maps from the Feynman graph to the spacetime graph (mapping edges to paths). This picture interacts naturally with Atiyah-Segal-like cutting-gluing of spacetime graphs. In particular, one has combinatorial counterparts of the known gluing formulae for Green's functions and (zeta-regularized) determinants of Laplacians.
We consider the dimer model on the Aztec diamond with Fock's weights, which is gauge equivalent to the model with any choice of positive weight function. We prove an explicit, compact formula for the inverse Kasteleyn matrix, thus extending numerous results in the case of periodic graphs. We also show an explicit product formula for the partition function; as a specific instance of the genus 0 case, we recover Stanley's formula. We then use our explicit formula for the inverse Kasteleyn matrix to recover, in a simple way, limit shape results; we also obtain new ones. In doing so, we extend the correspondence between the limit shape and the amoeba of the corresponding spectral curve of arXiv:2306.07482 to the case of non-generic weights.
Matrix models with continuous symmetry are powerful tools for studying quantum gravity and holography. Tensor models have also found applications in holographic quantum gravity. Matrix models with discrete permutation symmetry have been shown to satisfy large N factorisation properties relevant to holography, while also having applications to the statistical analysis of ensembles of real-world matrices. Here we develop 3-index tensor models in dimension D with a discrete symmetry of permutations in the symmetric group S_D. We construct the most general permutation invariant Gaussian tensor model using the representation theory of symmetric groups and associated partition algebras. We define a representation basis for the 3-index tensors, where the two-point function is diagonalised. Inverting the change of basis gives an explicit formula for the two-point function in the tensor basis for general D.
The authors studied in [Ann. Inst. Henri Poincar & eacute; D 9 (2022), 367-433], a complex multi-matrix model with U(N)2 & times;O(D) symmetry, and whose double scaling limit where simultaneously the large-N and large-D limits were taken while keeping the ratio N/D=M finite and fixed. In this double scaling limit, the complete recursive characterization of the Feynman graphs of arbitrary genus for the leading order grade & ell;=0 was achieved. In this current study, we classify the higher order graphs in & ell;. More specifically, & ell;=1 and & ell;=2 with arbitrary genus, in addition to a specific class of two-particle-irreducible (2PI) graphs for higher & ell;>= 3 but with genus zero. Furthermore, we demonstrate that each 2PI graph with a single O(D)-loop with an arbitrary & ell; corresponds to a reduced alternating knot diagram with & ell; crossings as listed in the Rolfsen knot table, or a resulting alternating knot diagram obtained after performing the Tait flyping moves. We generalize to 2PR by considering the connected sum and the Reidemeister move I.
In the context of lattice walk enumeration in cones, we consider the number of walks in the quarter plane with fixed starting and ending points, prescribed step-set and given length. After renormalization, this number may be interpreted as a discrete heat kernel in the quadrant. We propose a new method to compute complete asymptotic expansions of these numbers of walks as their length tends to infinity, based on two main ingredients: explicit expressions for the underlying generating functions in terms of elliptic Jacobi theta functions along with a duality known as Jacobi transformation. This duality allows us to pass from a classical Taylor expansion of the series to an expansion at the critical point of the model. We work through two examples. First, we present our approach on the well-known Kreweras model, which is algebraic, and show how to obtain a complete asymptotic expansion in this case. We then consider a more generic (so-called infinite group) model, and find the associated complete asymptotic expansion. In this second case, we prove the existence of logarithmic terms in the asymptotic expansion, and we relate the coefficients appearing in the expansion to polyharmonic functions. To our knowledge, this is the first time that logarithmic terms have been observed in the asymptotics of a class of lattice walks confined to a quadrant.
In the edge-2star model with hard constraints we prove the existence of an open set of constraint parameters, bisected by a line segment on which there are nonunique entropy-optimal graphons related by a symmetry. At each point in the open set but off the line segment there is a unique entropy-optimizer, bipodal and varying analytically with the constraints. We also show that throughout another open set, containing a different portion of the same line of symmetry, there is instead a unique optimal graphon, varying analytically with the parameters. We explore the extent of these open sets, determining the point at which a symmetric graphon ceases to be a local maximizer of the entropy. Finally, we prove some foundational theorems in a general setting, relating optimal graphons to the Boltzmann entropy and the generic structure of large constrained random graphs.
