
%% ================================================================
\section{Order parameters}

Parameters which will be used to determine $T_c(a)$:
\begin{itemize}
\item Fraction of sites in infinite cycles $f_I$.  This is an order parameter,
	non-zero for $T<T_c$ and zero for $T>T_c$.
\item Superfluid fraction $f_S$.  This is an order parameter,
	non-zero for $T<T_c$ and zero for $T>T_c$.
\item Correlation length $\xi$.  We expect it to be defined for $T>T_c$,
	and to blow up at $T=T_c$.  In an analogy to percolation above the critical
	bond density, can we extend the definition to $T<T_c$?
\item Specific heat $\partial H / \partial T$.  This doesn't have a sharp
	peak at $T_c$.  However, it's an extra check.
\end{itemize}

Ideas for computing the correlation length $\xi$:
\begin{itemize}

\item Length of the cycle containing the origin:
	$\sum_{\vecx \in \gamma_0} \|\pix - \vecx\|$.

\item Average length of cycles:
	$\frac{1}{N_\gamma} \sum_{\vecx \in \gamma} \|\pix - \vecx\|$.

\item Diameter of the cycle containing the origin:
	$\max_{\vecx \in \gamma_0}\{\|\vecx-0\|\}$.

\item Average diameter of cycles:
	$\frac{1}{N_\gamma} \max_{\vecx \in \gamma}\{\|\vecx-0\|\}$.

\item $G_{\Psi,\Psi}
	= \pig{\Psi(0)\Psi(\vecx)} - \pig{\Psi(0)} \pig {\Psi(\vecx)}$
	and then $G_{\Psi,\Psi} \sim \exp(-\|\vecx\|/\xi)$.
	What is a good choice of $\Psi$?
\end{itemize}

For $T<T_c$, exclude cycles $\gamma$ which are macroscopic.

Questions:
\begin{itemize}
\item Is the distribution of non-macroscopic cycle lengths independent of
temperature?  Ueltschi expects so.
\end{itemize}

%% ================================================================
\newpage
\section{Survey}

Bib notes for all theses, papers, books.
Esp. Caracciolo et al.

\begin{itemize}

\item Define
$$
	t = \frac{T-T_c}{T_c}.
$$
This \bfidx{reduced temperature} is the relative distance of $T$ from $T_c$.
The fundamental goal of our work here is to determine $T_c$ with error bars.

\item Two issues: (a) $T_c(L)$ drifts toward, but needs not necessarily equal,
the infinite-volume $T_c$.  [xxx illustrate this with specific-heat plots].
(b) One hopes that the different order parameters (e.g. $f_S$, $\xi$) have the
same $T_c$.

\item Let $S$ stand for any quantity such as $\xi$, $f_S$, $f_I$, etc.  There
is an \bfidx{infinite-volume quantity} $S_\infty(T)$, and a
\bfidx{finite-volume quantity} $S_L(T)$.

\item The \bfidx{correlation length} follows a \bfidx{power law}
$$
	\xi(T) \sim |t|^{-\nu}, \quad T \to T_c
$$
for positive $\nu$.  That is, the correlation length diverges at $T_c$.  This
is true only for $T$ ``close'' to $T_c$ --- but how close?  In the
computational physics literature, one sees $0.99 < t < 1.01$ used as the region
of validity of the power law.

\item For the infinite-volume quantity, we expect a power-law behavior
$$
	S_\infty(T) \sim t^{\rho}, (-t)^{\rho}, \qor |t|^\rho,
$$
respectively, if $S$ is zero left of $T_c$ (e.g. $\xi$ as currently defined),
zero right of $T_c$ (e.g.  $f_I, f_S$, or non-zero on both sides (e.g.  if we
can define a subcritical correlation length).  Again, we expect this power law
to be valid for $0.99 < t < 1.01$.

\item The \bfidx{finite-size scaling hypothesis} or \bfidx{ansatz} is that the
infinite-volume and finite-volume quantities, for $T$ near $T_c$, are related
by a \bfidx{universal function} $Q$ which depends only on the ratio $L/\xi$:
\begin{align*}
	S_L(T) &= L^{-\rho/\nu} Q(L^{1/\nu} t) \\
	&\sim L^{-\rho/\nu} Q((L t^\nu)^{1/\nu}) \\
	&=    L^{-\rho/\nu} Q((L/\xi)^{1/\nu}).
\end{align*}
[xxx To do: derive/justify the placement of the exponents.] xxx Properties of
$Q$ \ldots (1) $Q \to 1$ as $L \to \infty$, i.e.  $S_L(T)$ converges to
$S_\infty(T)$. (2) $t \to 0$ stuff \ldots.

\item Transpose plots \ldots these check the correlation length,
since for $L \gg \xi$, we should have $Q \to 1$.

\item Once experimental data have been collected for a range of $L$'s and a
range of $T$'s, \bfidx{testing}\index{hypothesis testing} of the FSS hypothesis
is done by computing (as described below) $\rho$, $\nu$, and $T_c$, then
plotting $L^{\rho/\nu} S_L(T)$ as a function of $L^{1/\nu}t$.  This is a plot
of the scaling function $Q$.  If the hypothesis is correct, the curves for all
$L$ should coincide, or \bfidx{collapse}.

\item If the hypothesis is not borne out by the collapse plots, one may wish
to attempt \bfidx{corrections to scaling}.  This means that $Q$ depends on
one or more length scales in addition to $L/\xi$, with their own critical
exponents.  E.g.
$$
	S_L(T) = L^{-\rho/\nu} Q((L/\xi)^{1/\nu},
	\zeta_1^{\Delta_1/\nu},
	\zeta_2^{\Delta_2/\nu}, \ldots
	).
$$
One obtains a new hypothesis to be tested by the collapse method; there are
more parameters to fit; one must figure out what the other length scale(s) are.

\end{itemize}

Techniques:

\begin{itemize}

\item \bfidxtwo{Guess}{guess} one or both exponents.

\item Use a \bfidx{linear approximation} $Q(x) \approx q_0 + q_1 x$;
use Levenburg-Marquart (generalized least squares) to fit\index{curve fit}
$$
	S_L(T) = L^{-\rho/\nu} (q_0 + q_1 L^{1/\nu} t)
$$
in the five parameters $\rho$, $\nu$, $q_0$, $q_1$, $T_c$.  Guessing or
otherwise obtaining values for $\rho$, $\nu$, $q_0$, and/or $q_1$ will reduce
the number of parameters in the fit.

\item One may do a linear regression if the parameter may be handled two at a
time.  See section \ref{sec:fss_lin_reg} for details.

\item \bfidxtwo{Crossing methods}{crossing methods}:  If $\rho=\nu$, or
if $\rho/\nu$ is known, then plot $L^{\rho/\nu} S_L(T)$ as a function of $T$.
Since at $T=T_c$ we have $t=0$ and
$$
	L^{\rho/\nu} S_L(T)  = Q(0)
$$
regardless of $L$, these curves will cross (approximately, due to sampling
variability) at $T=T_c$.  This also gives an estimate of the parameter
$q_0$ for the linearized curve-fit method.

\item \bfidxtwo{Quotient methods}{quotient methods}:  If $\rho/\nu$ is not
known, define
\begin{align*}
	R_{12}(T) &= \frac{S_{L_2}(T)}{S_{L_1}(T)} \\
	&= \frac{L_2^{-\rho/\nu} Q(L_2^{1/\nu} t)}
		{L_1^{-\rho/\nu} Q(L_1^{1/\nu} t)} \\
	&= \left(\frac{L_2}{L_1}\right)^{-\rho/\nu} 
		\frac{Q(L_2^{1/\nu} t)} {Q(L_1^{1/\nu} t)}.
\end{align*}
Do this for several $L_1,L_2$ pairs with the same ratio $L_2/L_1=r$:  for
example, $10,20$; $20,40$; $40,80$ where $r=2$.  Then $r^{-\rho/\nu}$
is a constant $c$, and
\begin{align*}
	R_{ij}(T) &= c \frac{Q(L_j^{1/\nu} t)} {Q(L_i^{1/\nu} t)}.
\end{align*}
These curves will also cross at $t=0$, i.e. $T=T_c$.  At $t=0$ we have
$R_{ij}(0) = c$, so this method also produces the ratio of critical exponents
$\rho/\nu$.

\item \bfidxtwo{Phenomenological renormalization}{phenomenological
renormalization}:
fix $L_1$, $L_2$, and $T_1$.  Determine $T_2$ such that
$$
	\frac{\xi_{L_2}(T_2)}{\xi_{L_1}(T_1)} = \frac{L_2}{L_1}.
$$
Then [xxx state why]
$$
	(T_2 - T_c)
	= \left(\frac{L_1}{L_2}\right)^{1/\nu} (T_1 - T_c).
$$
This may also be used iteratively.  [xxx explain.]

\item \bfidxtwo{Fourth-order cumulants}{fourth-order cumulants}:
\begin{align*}
	U_{L_1} &= U_{L_2}? \\
	L_1U_{L_1} &= L_2U_{L_2}? \\
	L_1^{\rho/\nu}U_{L_1} &= L_2^{\rho/\nu}U_{L_2}?
\end{align*}

\item If $T_c$ is known (or estimated), then
$$
	S_L(T_c) = L^{-\rho/\nu} Q(0)
$$
for all $L$.  From this, $\rho/\nu$ can be determined.  If $T_c$ is unknown,
this might be used as a step in an iterative method.  Even though this method
requires known $T_c$ whereas our goal is to compute $T_c$, it serves as a
check:  once $T_c$, $\rho$, and $\nu$ have been found, this method provides a
verification of $\rho/\nu$.

\item Caracciolo:  fix $L_1$ and $L_2$.  Find $T$ so that
$$
	\frac{\xi_{L_1}(T)}{\xi_{L_2}(T)} = \frac{L_1}{L_2}.
$$
Then [xxx state why] get $\rho/\nu$ from
$$
	\frac{S_{L_1}(T)}{S_{L_2}(T)} = \left(\frac{L_1}{L_2}\right)^{\rho/\nu}.
$$
Take larger $L_1,L_2$.

\end{itemize}

xxx make a table of methods, inputs, and outputs.


%% ================================================================
\newpage
\section{xxx $\rho=\nu$ section}
\label{sec:fss_lin_reg}

%xxx see
%\cite{bib:gcl97,bib:hk99,bib:kps01,bib:nl04,bib:pc87,bib:pgp08,bib:pr92},
%especially \cite{bib:pgp08, bib:hk99}.
%
%xxx also barber v. 8 ...
%
%xxx also Fisher 1971 ...

Infinite-volume system:
\begin{align*}
	f_S(T) &= \left(\frac{T-T_c}{T_c}\right)^\nu
\end{align*}
$T \ge T_c$?
[xxx plot.]

On the lattice, $L=N^{1/3}$.

Finite system:

Analytic $Q$ on both sides:
\begin{align*}
	f_S(L) &= \frac{1}{L} \; Q\left(L^{1/\nu} \frac{T-T_c}{T_c} \right) \\
	L f_S(L) &= Q\left(L^{1/\nu} \frac{T-T_c}{T_c} \right) \\
\end{align*}
\begin{align*}
	L f_S(L) &=
	q_0 + q_1\; L^{1/\nu} \frac{T-T_c}{T_c} + \ldots
\end{align*}
Experimental data:  $(T, L, a, f_S(L))$.
There are four undetermined parameters above: $q_0$, $q_1$, $T_c$, and $\nu$.
We are primarily interested in $T_c$ and $\nu$.

Technique (xxx cite \ldots):  Find two parameters at a time.
xxx cite PGP08, HK99 (VA), GCL97 (vague), NL04 (vague), PR92 (vague).

\textbf{First step:}
At $T=T_c$, we have
\begin{align*}
	L f_S(L) &= q_0.
\end{align*}
xxx show a whisker plot.  Intersection has coordinates $(T_c, q_0)$.
Determine the error bars on each of these two quantities.
xxx also the synthetic-data method from Numerical Recipes.  [xxx into bib.]

\textbf{Second step:}
Now fit $q_1$ and $\nu$.  We still have
\begin{align*}
	L f_S(L) &=
	q_0 + q_1\; L^{1/\nu} \frac{T-T_c}{T_c} + \ldots
\end{align*}
but now $T_c$ and $q_0$ are known.  Re-write:
\begin{align*}
	N^{1/3} f_S(L) - q_0 &= q_1\; N^{1/3\nu} \frac{T-T_c}{T_c} \\
	t  &= \frac{T-T_c}{T_c} \\
	L f_S(L) - q_0 &= q_1\; L^{1/\nu} t \\
	\log(L f_S(L) - q_0) &= \log(q_1) + \frac{1}{\nu}\log(L) + \log(t) \\
	\log(L f_S(L) - q_0) - \log(t) &= \log(q_1) + \frac{1}{\nu}\log(L)
\end{align*}
Now we can do a linear regression of the form
\begin{align*}
	y_i &= mx_i + b
\end{align*}
where
\begin{align*}
	y_i &= \log(L f_S(L) - q_0) - \log(t) \\
	m   &= \frac{1}{\nu} \\
	x_i &= \log(L) \\
	b   &= \log(q_1).
\end{align*}

xxx question:  with $a>0$ there is the correction \ldots
(1) why does the VA trick work;
(2a) do we need to know the exponent $\omega$;
(2b) can we determine the exponent $\omega$?

Misc:

BEC slope is $3/2$ for $f_I$?  What about $f_S^\infty$?
