\section{Docile Perturbations and the Rozansky-Overbay Invariants}

\draftcut \subsection{The Baby \dpg\ Category}

In this section we introduce a ``baby'' version of \dpg, in which the most interesting features of the
``mature'' versions are present, yet some inconveniences regarding weights are censored.

\begin{definition} \label{def:DoPeGDO}
Let $\Omega$ be some ring of ``scalars'' and let
$\eps$ be a formal parameter. Like \gdo, let $\dpg_b$ be the category with
objects finite sets and, if $A$ and $B$ are finite, with $\mor(A\to B)$
the set of ``docile perturbed Gaussians in $\zeta_A\cup z_B$'':
\[ \mor(A\to B) = \left\{\omega\bbe^{Q+P}\right\}, \]
where $\omega$ and $Q$ are $\eps$-independent and otherwise as in Definition~\ref{def:GDO}, and where $P$ is
a power series in $\eps$ of the form $P=\sum_{k\geq 1}P^{(k)}\eps^k$ and where each $P^{(k)}$ is a polynomial
in $\zeta_A\cup z_B$ satisfying the ``docility condition'':
\[ \deg P^{(k)}\leq 2k+2. \]
The composition law of $\dpg_b$ is ``be compatible
with~\eqref{eq:LMcomposition}'' (so this definition becomes complete
only following the discussion of Feynman diagrams below, or in
Section~\ref{ssec:PDE}). \endpar{\ref{def:DoPeGDO}}
\end{definition}

\begin{comment} If we mod out by $\eps^{k_0+1}$ for some $k_0\geq 0$, or
in other words, restrict our attention to $\dpg_b$ ``up to $\eps^{k_0}$'',
then the rank of the space of docile polynomials is polynomial in the
number of variables (cf.\ Comment~\ref{com:QGood}). Hence storing and
manipulating docile polynomials has a chance of being computationally
cheap; later we will see that this is indeed the case.
\end{comment}

We now seek to understand compositions. With the same diagrammatic language as before, we seek to determine
$\omega$, $Q=(E,F,G)$ and $P$, so that the following would hold, where composition is ``all possible
contractions'':
\begin{equation} \label{eq:DoPeGDOCompositions}
  \begin{array}{c} \input{figs/DoPeGDOCompositions.pdf_t} \end{array}
\end{equation}
Looking only at the $\eps$-independent part, it is clear that the composition law for $\omega$ and for $Q$ is
the same as for \gdo~\eqref{eq:gdocompositions} (so \dpg\ is an ``extension'' of \gdo). We just have to find
$P=\sum_{k\geq 1}P^{(k)}\eps^k$ as a function of $Q_{1,2}$ and $P_{1,2}$.

Well, $P^{(k)}$ must get $k$ factors of $\eps$ and it can only get them
from $P_1$ and $P_2$. So $P^{(k)}$ is a sum of diagrams that have at most
$k$ vertices\footnotemark. These vertices can be connected to each other
(including self-connections), or to the outside, either directly, or by
travelling along $E_{1,2}$ lines, or by travelling along $F_2G_1$ or $G_1F_2$ cycles
as before. The latter cycles produce geometric series that sum to either $(I-F_2G_1)^{-1}$ or
$(I-G_1F_2)^{-1}$. We arrive at the following theorem, which we state in a slightly informal manner as a more
rigorous treatment follows in Section~\ref{ssec:PDE}:

\footnotetext{Less than $k$ if a single vertex brings along more
than one factor of $\eps$. Namely, if it comes from $P_{1,2}^{(l)}$, where $l\geq 2$.}

%\Needspace{0mm} % 23mm is not enough
\null

\parpic[r]{\input{figs/FD.pdf_t}}
\begin{theorem}
\picskip{4}
In a composition as in~\eqref{eq:DoPeGDOCompositions} the term $P^{(k)}$ in $P$ is the sum of all
connected Feynman diagrams as on the right, each divided by the order of its automorphism group, and in which the
vertices are determined by $P_1$ and $P_2$ and in which there are five types of propagators (all sampled on
the right):

\begin{enumerate}
\item A $P_1$-to-$P_2$ propagator which equals $(I-F_2G_1)^{-1}$.
\item A $P_1$-to-$P_1$ propagator which equals $(I-F_2G_1)^{-1}F_2$.
\item A $P_2$-to-$P_2$ propagator which equals $G_1(I-G_1F_2)^{-1}$.
\item A Greek-to-$P_2$ propagator which equals $E_1(I-F_2G_1)^{-1}$.
\item A $P_1$-to-Latin propagator which equals $(I-F_2G_1)^{-1}E_2$.
\end{enumerate}

The figure here depicts a contribution to $P^{(4)}$. In general the valencies of vertices may be higher and
self-contractions of two edges coming out of the same vertex are allowed.\qed
\end{theorem}

\parshape 1 0in \textwidth
\begin{proposition}
$\dpg_b$, as defined in Definition~\ref{def:DoPeGDO} and with composition as in the above
theorem, is indeed a category. Namely, with notation as in Equation~\eqref{eq:DoPeGDOCompositions} and with
$P$ as in the theorem, if $P_1$ and $P_2$ are docile then so is $P$.
\end{proposition}

\parshape 1 0in \textwidth
\noindent{\em Proof.} Consider a diagram contributing to $P$ that has $m$ vertices
$v_1,\ldots,v_m$. Each $v_i$ comes from either $P_1$ or $P_2$ and brings
along some power $k_i$ of $\eps$, so the diagram overall contributes a
term $T$ in which the power of $\eps$ is $k=\sum_{i=1}^mk_i$. We need to
show that the degree of $T$ in the Greek and Latin variables satisfies
$\deg T\leq 2k+2$. Indeed, by the docility of $P_1$ and $P_2$ each
$v_i$ contributes at most $2k_i+2$ to that degree. Also, the diagram
is connected\footnote{Da liegt der Hund begraben. Had we used
$\omega\bbe^QP$ instead of $\omega\bbe^{Q+P}$ for the morphisms of \dpg\ 
we'd have had no connectedness here and the docility bound would have
been $\deg P^{(k)}\leq 4k$, leading to slower computations.} so it has
at least $m-1$ edges, and each one contracts to variables, so each one
reduces the overall degree by $2$. So $\deg T\leq\left(\sum_{i=1}^m
2k_i+2\right)-2(m-1)=2k+2$. \qed

\vskip 1mm

{\red MORE. Add a ``formula'' version and a demo.}

\draftcut \subsection{Algebra by means of Partial Differential Equations} \label{ssec:PDE}

Much as we love intuitive graphical reasonings such as in the previous
sections, we also like the more solid grounds of algebra. Hence we repeat
the content of Sections~\ref{ssec:GDO} and~\ref{ssec:baby} in a purely
algebraic language (as it turns out, it is the language of partial
differential equations, though they are only used with power series,
and hence we remain in pure algebra).

Recall from Comment~\ref{com:DoubleContraction} that in order to compute compositions of generating functions we
need to evaluate contractions like
$\left.\bbe^{\sum\partial_{z_b}\partial_{\zeta_b}}(\calL\cdot\calM)\right|_{z_b=\zeta_b=0}$. This inspires the
following slightly more general definition:

\begin{definition} Let $B$ be a finite set, let $F$ be a $B\times B$ matrix, and let $\calE$ be a power
series in variables that include the variable $z_B$. Set the ``partial contraction'' and the ``full
contraction'' of $\calE$ using $F$ to be
\[ [F\colon\calE]_B\coloneqq \bbe^{\frac12\sum_{i,j\in B} F_{ij}\partial_{z_i}\partial_{z_j}}\calE
  \quad\text{and}\quad
  \langle F\colon\calE\rangle_B\coloneqq \left.[F\colon\calE]_B\right|_{z_B\to 0}.
\]
\end{definition}

\begin{note} To ensure convergence one must assume some ``smallness'' condition on either $F$ or
$\calE$. We defer this to a later point.
\end{note}

\begin{note}
In the above definition, $\calE$ replaces the product $\calL\cdot\calM$ of~\eqref{eq:DoubleContraction}, we
restrict to a single ``type'' of variables $z_B$ instead of the $z_B\cup\zeta_B$
of~\eqref{eq:DoubleContraction} (so $B$ here is ``twice'' the $B$ of~\eqref{eq:DoubleContraction}), and
instead of a pairing matrix of the form $\begin{pmatrix}0&I\\I&0\end{pmatrix}$ as
in~\eqref{eq:DoubleContraction}, we allow a general matrix $F$. This will become beneficial soon.
\end{note}

\begin{note} The computations of $[F\colon\cdot]_B$ and of $\langle F\colon\cdot\rangle_B$ are equivalent by
``soft'' means: $[F\colon\cdot]_B$ clearly determines $\langle F\colon\cdot\rangle_B$, and we also have
$[F\colon\calE]_B=\left.\left\langle F\colon\left.\calE\right|_{z_b\to
z_b+z'_b}\right\rangle\right|_{z'_b\to z_b}$, where $z'_B$ is a new set of variables indexed by $B$. The
full contraction $\langle F\colon\cdot\rangle_B$ is used in~\eqref{eq:DoubleContraction}, yet the partial
contraction $[F\colon\cdot]_B$ is easier to manipulate as below.
\end{note}

Let $\lambda$ be a formal variable and let $\calZ_\lambda\coloneqq[\lambda F\colon\calE]_B$. Then
$\calZ_\lambda$ (and hence all we care about in this section) is
determined by the following initial value problem, a heat equation:
\begin{equation} \label{eq:heat}
  \calZ_0=\calE \qquad\text{and}\qquad
  \partial_\lambda\calZ_\lambda = \frac12\sum_{i,j\in B}F_{ij}\partial_{z_iz_j}\calZ_\lambda.
\end{equation}
Yet we like to write generating functions as exponentials\footnote{The equations become non-linear, but
as we will see later, their solutions lie in smaller spaces, allowing for more efficient
manipulations.}, and hence the following proposition:

\begin{proposition} With $E=\log\calE$ and $Z_\lambda=\log\calZ_\lambda$ Equation~\eqref{eq:heat}
becomes
\begin{equation} \label{eq:sythesis}
  Z_0=E \qquad\text{and}\qquad
  \partial_\lambda Z_\lambda = \frac12\sum_{i,j\in B}F_{ij}\left(
    \partial_{z_iz_j}Z_\lambda + (\partial_{z_i}Z_\lambda)(\partial_{z_j}Z_\lambda)
  \right).
\end{equation}
\end{proposition}

\begin{proof} Simply substitute $\calZ_\lambda=\bbe^{Z_\lambda}$ into~\eqref{eq:heat} and carry out the
differentiations.
\end{proof}

A sometimes-useful alternative to~\eqref{eq:sythesis} is to allow $F$
to be implicitly dependent on $\lambda$ in an arbitrary (differentiable)
manner with the condition $\left.F\right|_{\lambda=0}=0$ and to suppress
the $\lambda$ subscript in $Z_\lambda$. The resulting equation is
\begin{equation} \tag{\ref{eq:sythesis}'}
  \left.Z\right|_{\lambda=0}=E \qquad\text{and}\qquad
  \partial_\lambda Z = \frac12\sum_{i,j\in B}(\partial_\lambda F_{ij})\left(
    \partial_{z_iz_j}Z + (\partial_{z_i}Z)(\partial_{z_j}Z)
  \right).
\end{equation}

We call Equation~\eqref{eq:sythesis} (and its variant
Equation~(\ref{eq:sythesis}')) ``the synthesis equation'',
as it governs how the ``vertices'' in $E$ merge and contract
to synthesize larger and larger connected diagrams, as in the
interpretation below.\footnote{\label{foot:Burger}M.~Pugh told us that
Equation~\eqref{eq:sythesis} is a variant of ``Burger's equation'',
and that it's relationship with the heat equation~\eqref{eq:heat} is a
variant of the ``Cole-Hopf transformation''.}

\Needspace{24mm} % 23mm is not enough.
\parpic[r]{\raisebox{-0mm}{\def\F{$\lambda F$}\input{figs/FDCuts.pdf_t}}}
\begin{interpretation} \label{int:masterequation}
For the initiated, we cannot resist including a Feynman-diagram
interpretation of Equation~\eqref{eq:sythesis}. With $E$ ``the
vertices'' and $F$ ``the contraction tensor'' (roughly, ``the
propagator''), $\calZ_\lambda=\langle\lambda F\colon\bbe^E\rangle$
is the sum of all Feynman diagrams that can be made with
vertices in $E$ and contractions as dictated by $F$, with each
contraction multiplied by an additional factor of $\lambda$. Then
$Z_\lambda=\log\calZ_\lambda$ is the same, except restricting to
connected Feynman diagrams. And then $\partial_\lambda Z_\lambda$
picks out one contraction in $Z_\lambda$. If it is ``separating'',
it contributes an $F$-weighted product of two connected diagrams ---
the term $(\partial_{z_i}Z_\lambda)(\partial_{z_j}Z_\lambda)$. If it not
separating, it can be seen to contribute the $\partial_{z_iz_j}Z_\lambda$
term. See the picture on the right. \endpar{\ref{int:masterequation}}
\end{interpretation}

{\blue
\noindent{\bf Lemma 1.} $\displaystyle \left\langle F\colon\calE\,\bbe^{\sum_{i\in B}y_iz_i}\right\rangle_B
  = \bbe^{\frac12\sum_{i,j\in B}F_{ij}y_iy_j}
    \left\langle F\colon \left.\calE\right|_{z_B\to z_B+Fy_B}\right\rangle_B$ and
\begin{multline*} \left[F\colon\calE\,\bbe^{\sum_{i\in B}y_iz_i}\right]_B
  = \bbe^{\frac12\sum_{i,j\in B}F_{ij}y_iy_j+\sum_{i\in B}y_iz_i}
    \left[F\colon \left.\calE\right|_{z_B\to z_B+Fy_B}\right]_B \\
  = \bbe^{\frac12\sum_{i,j\in B}F_{ij}y_iy_j+\sum_{i\in B}y_iz_i}
    \left(\left.\left[F\colon\calE\right]_B\right|_{z_B\to z_B+Fy_B}\right).
\end{multline*}

\noindent{\bf Lemma 2.} With convergences left to the reader,
\[ \left\langle F\colon\calE\,\bbe^{\frac12\sum_{i,j\in B}G_{ij}z_iz_j}\right\rangle_B
  = \det(1-GF)^{-1/2}\left\langle F(1-GF)^{-1} \colon \calE \right\rangle_B,
\]
and
\begin{multline*} \left[F\colon\calE\,\bbe^{\frac12\sum_{i,j\in B}G_{ij}z_iz_j}\right]_B
  = \det(1-GF)^{-1/2}\bbe^{\frac12\sum_{i,j\in B}(G(I-FG)^{-1})_{ij}z_iz_j} \\
    \cdot\left(\left[F(1-GF)^{-1} \colon \calE \right]_B\right)_{z_B\to(I-FG)^{-1}z_B}.
\end{multline*}

\[ \input{figs/Lemmas.pdf_t} \]
}

{\red MORE.}
