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Karen Yeats

Publications and source records attributed to Karen Yeats.

At least 55 records · Page 3Linked to original sources
arXiv2017

c_2 Invariants of Recursive Families of Graphs

The c_2 invariant, defined by Schnetz in 2011, is an arithmetic graph invariant created towards a better understanding of Feynman integrals. This paper looks at some graph families of interest, with a focus on decompleted toroidal grids. Specifically, the c_2 invariant for p=2 is shown to be zero for all decompleted non-skew toroidal grids. We also calculate the c_2 invariant at p=2 for G a family of graphs called X-ladders. Finally, we show these methods can be applied to any graph with a recursive structure, for any fixed p.

math.CO↗
arXiv2017

Diffeomorphisms of quantum fields

We study field diffeomorphisms $Φ(x)= F(ρ(x))=a_0ρ(x)+a_1ρ^2(x)+\ldots=\sum_{j+0}^\infty a_j ρ^{j+1}$, for free and interacting quantum fields $Φ$. We find that the theory is invariant under such diffeomorphisms if and only if kinematic renormalization schemes are used.

math-ph↗
arXiv2017

Some results on an algebro-geometric condition on graphs

Paolo Aluffi, inspired by an algebro-geometric problem, asked when the Kirchhoff polynomial of a graph is in the Jacobian ideal of the Kirchhoff polynomial of the same graph with one edge deleted. We give some results on which graph-edge pairs have this property. In particular we show that multiple edges can be reduced to double edges, we characterize which edges of wheel graphs satisfy the property, we consider a stronger condition which guarantees the property for any parallel join, and we find a class of series-parallel graphs with the property.

math.CO↗
arXiv2016

Towards a Combinatorial Understanding of Lattice Path Asymptotics

We provide a new strategy to compute the exponential growth constant of enumeration sequences counting walks in lattice path models restricted to the quarter plane. The bounds arise by comparison with half-planes models. In many cases the bounds are provably tight, and provide a combinatorial interpretation of recent formulas of Fayolle and Raschel (2012) and Bostan, Raschel and Salvy (2013). We discuss how to generalize to higher dimensions.

math.CO↗
arXiv2016

Terminal chords in connected chord diagrams

Rooted connected chord diagrams form a nice class of combinatorial objects. Recently they were shown to index solutions to certain Dyson-Schwinger equations in quantum field theory. Key to this indexing role are certain special chords which are called terminal chords. Terminal chords provide a number of combinatorially interesting parameters on rooted connected chord diagrams which have not been studied previously. Understanding these parameters better has implications for quantum field theory. Specifically, we show that the distributions of the number of terminal chords and the number of adjacent terminal chords are asymptotically Gaussian with logarithmic means, and we prove that the average index of the first terminal chord is $2n/3$. Furthermore, we obtain a method to determine any next-to${}^i$ leading log expansion of the solution to these Dyson-Schwinger equations, and have asymptotic information about the coefficients of the log expansions.

math.CO↗
arXiv2016

Generalized chord diagram expansions of Dyson-Schwinger equations

Series solutions for a large family of single equation Dyson-Schwinger equations are given as expansions over decorated rooted connected chord diagrams. The analytic input to the new expansions are the expansions of the regularized integrals for the primitive graphs building the Dyson-Schwinger equation. Each decorated chord diagram contributes a weighted monomial in the coefficients of the expansions of the primitives and so indexes the analytic solution in a tightly controlled way.

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arXiv2015

A few $c_2$ invariants of circulant graphs

The $c_2$ invariant is an arithmetic graph invariant introduced by Schnetz and developed by Brown and Schnetz in order to better understand Feynman integrals. This document looks at the special case where the graph in question is a 4-regular circulant graph with one vertex removed; call such a graph a decompletion of a circulant graph. The $c_2$ invariant for the prime $2$ is computed in the case of the decompletion of circulant graphs $C_n(1,3)$ and $C_{2k+2}(1,k)$. For any prime $p$ and for the previous two families of circulant graphs along with the further families $C_n(1,4)$, $C_n(1,5)$, $C_n(1,6)$, $C_n(2,3)$, $C_n(2,4)$, $C_n(2,5)$, and $C_n(3,4)$, the same technique gives the $c_2$ invariant of the decompletions as the solution to a finite system of recurrence equations.

math.CO↗
arXiv2015

Period Preserving Properties of an Invariant from the Permanent of Signed Incidence Matrices

A 4-point Feynman diagram in scalar $ϕ^4$ theory is represented by a graph $G$ which is obtained from a connected 4-regular graph by deleting a vertex. The associated Feynman integral gives a quantity called the period of $G$ which is invariant under a number of meaningful graph operations - namely, planar duality, the Schnetz twist, and it also does not depend on the choice of vertex which was deleted to form $G$. In this article we study a graph invariant we call the graph permanent, which was implicitly introduced in a paper by Alon, Linial and Meshulam. The graph permanent applies to any graph $G = (V,E)$ for which $|E|$ is a multiple of $|V| - 1$ (so in particular to graphs obtained from a 4-regular graph by removing a vertex). We prove that the graph permanent, like the period, is invariant under planar duality and the Schnetz twist when these are valid operations, and we show that when $G$ is obtained from a $2k$-regular graph by deleting a vertex, the graph permanent does not depend on the choice of deleted vertex.

math.CO↗
arXiv2014

Tree hook length formulae, Feynman rules and B-series

We consider weighted generating functions of trees where the weights are products of functions of the sizes of the subtrees. This work begins with the observation that three different communities, largely independently, found substantially the same result concerning these series. We unify these results with a common generalization. Next we use the insights of one community on the problems of another in two different ways. Namely, we use the differential equation perspective to find a number of new interesting hook length formulae for trees, and we use the body of examples developed by the combinatorial community to give quantum field theory toy examples with nice properties.

math.CO↗
arXiv2014

Forbidden minors for graphs with no first obstruction to parametric Feynman integration

We give a characterization of 3-connected graphs which are planar and forbid cube, octahedron, and $H$ minors, where $H$ is the graph which is one $Δ-Y$ away from each of the cube and the octahedron. Next we say a graph is Feynman 5-split if no choice of edge ordering gives an obstruction to parametric Feynman integration at the fifth step. The 3-connected Feynman 5-split graphs turn out to be precisely those characterized above. Finally we derive the full list of forbidden minors for Feynman 5-split graphs of any connectivity.

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arXiv2013

Some combinatorial interpretations in perturbative quantum field theory

This paper will describe how combinatorial interpretations can help us understand the algebraic structure of two aspects of perturbative quantum field theory, namely analytic Dyson-Schwinger equations and periods of scalar Feynman graphs. The particular examples which will be looked at are, a better reduction to geometric series for Dyson-Schwinger equations, a subgraph which yields extra denominator reductions in scalar Feynman integrals, and an explanation of a trick of Brown and Schnetz to get one extra step in the denominator reduction of an important particular graph.

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arXiv2012

A chord diagram expansion coming from some Dyson-Schwinger equations

We give an expression for the solution to propagator-type Dyson-Schwinger equations with one primitive at 1 loop as an expansion over rooted connected chord diagrams. Along the way we give a refinement of a classical recurrence of rooted connected chord diagrams, and a representation in terms of binary trees.

math.CO↗
arXiv2012

Properties of the corolla polynomial of a 3-regular graph

We investigate combinatorial properties of a graph polynomial indexed by half-edges of a graph which was introduced recently to understand the connection between Feynman rules for scalar field theory and Feynman rules for gauge theory. We investigate the new graph polynomial as a stand-alone object.

math.CO↗
arXiv2012

Properties of c_2 invariants of Feynman graphs

The c_2 invariant of a Feynman graph is an arithmetic invariant which detects many properties of the corresponding Feynman integral. In this paper, we define the c_2 invariant in momentum space and prove that it equals the c_2 invariant in parametric space for overall log-divergent graphs. Then we show that the c_2 invariant of a graph vanishes whenever it contains subdivergences. Finally, we investigate how the c_2 invariant relates to identities such as the four-term relation in knot theory.

math.AG↗
arXiv2011

A four-vertex, quadratic, spanning forest polynomial identity

The classical Dodgson identity can be interpreted as a quadratic identity of spanning forest polynomials, where the spanning forests used in each polynomial are defined by how three marked vertices are divided among the component trees. We prove an analogous result with four marked vertices.

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arXiv2011

Characteristic Points of Recursive Systems

Characteristic points have been a primary tool in the study of a generating function defined by a single recursive equation. We investigate the proper way to adapt this tool when working with multi-equation recursive systems.

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arXiv2011

On the set of zero coefficients of a function satisfying a linear differential equation

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$. Given that $P_d(z)$ is a nonzero constant polynomial, we show that the set of $n\in \mathbb{N}$ for which $f(n)=0$ is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem-Mahler-Lech theorem, which assumes that $f(n)$ satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a power series satisfying a homogeneous linear differential equation with rational function coefficients.

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