Wilsonian renormalization, differential equations and Hopf algebras

In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Then, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally, we work out an analogous construction for the Schwinger-Dyson equations, which yields a bijection between planar $\phi^{3}$ diagrams and a certain class of decorated rooted trees.Comment: 42 pages, 26 figures in PDF format, extended version of a talk given at the conference "Combinatorics and physics" held at Max Planck Institut fuer Mathematik in Bonn in march 2007, some misprints correcte

Directed polymer near a hard wall and KPZ equation in the half-space

We study the directed polymer with fixed endpoints near an absorbing wall, in the continuum and in presence of disorder, equivalent to the KPZ equation on the half space with droplet initial conditions. From a Bethe Ansatz solution of the equivalent attractive boson model we obtain the exact expression for the free energy distribution at all times. It converges at large time to the Tracy Widom distribution $F_4$ of the Gaussian Symplectic Ensemble (GSE). We compare our results with numerical simulations of the lattice directed polymer, both at zero and high temperature.Comment: 7 pages 4 figures one paragraph and one reference adde

Parental Illness and the Labour Supply of Adult Children

An important demographic trend is the aging of the population. As a result, demand for health care services for the sick and elderly is likely to increase. Since care for the sick and elderly is often provided informally by family members, parental illness may have important implications for the labour supply of adult children. Although previous studies show a negative relationship between hours worked and caregiving, they do not account for the potential endogeneity of the parental living arrangement to the child's labour supply. Using panel data and controlling for such endogeneity, I find that caregiving and cohabiting with a sick, elderly parent appear to have smaller effects on labour supply than the past literature suggests. Nonetheless, since cohabiting with a sick elderly parent does have negative effects on the labour supply of women and given that this form of living arrangement is relatively common, the aggregate costs associated with informal caregiving in an intergenerational living arrangement are considerable.aging; labour supply