From c4d8e66338c8930d632a0c0cfed6c3aa0eacd29a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 3 Oct 2018 22:59:46 -0400
Subject: updated exponents, added more explanation to performance section

---
 monte-carlo.tex | 18 +++++++++++++-----
 1 file changed, 13 insertions(+), 5 deletions(-)

(limited to 'monte-carlo.tex')

diff --git a/monte-carlo.tex b/monte-carlo.tex
index cf45bcd..fe6227d 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -547,7 +547,13 @@ large-field regime where the crossover happens the correlation length is on
 the scale of the lattice spacing and better algorithms exist, like
 Bortz--Kalos--Lebowitz for the Ising model \cite{bortz_new_1975}. Also plotted
 are lines proportional to $h^{-z\nu/\beta\delta}$, which match the behavior of
-the correlation times in the intermediate scaling region.
+the correlation times in the intermediate scaling region. Values of the
+critical exponents for the models were taken from the literature
+\cite{wu_potts_1982, el-showk_solving_2014, guida_critical_1998} with the
+exception of $z$ for the energy in the Wolff algorithm, which was determined
+for each model by making a power law fit to the constant low field behavior.
+These exponents are imprecise and are provided with only qualitative
+uncertainty.
 
 \begin{figure*}
   \include{fig_correlation-times}
@@ -557,7 +563,10 @@ the correlation times in the intermediate scaling region.
     various models of Table~\ref{table:models}. Critical exponents are
     model-dependent. Colored lines and points depict values as measured by the
     extended algorithm. Solid black lines show a plot proportional to
-    $h^{-z\nu/\beta\delta}$ for each model.
+    $h^{-z\nu/\beta\delta}$ for each model. The dynamic exponents $z$ are
+    roughly measured as \twodee Ising: 0.23(2), \threedee Ising: 0.28(2),
+    \twodee 3-State Potts: 0.55(1), \twodee 4-State Potts: 0.94(5),
+    \threedee O(2): 0.17(2), \threedee O(3): 0.13(2).
   }
   \label{fig:correlation_time-collapse}
 \end{figure*}
@@ -604,9 +613,8 @@ large argument. We further conjecture that this scaling behavior should hold
 for other models whose critical points correspond with the percolation
 transition of Wolff clusters.  This behavior is supported by our numeric work
 along the critical isotherm for various Ising, Potts, and $\mathrm O(n)$
-models, shown in Fig.~\ref{fig:cluster_scaling}. Fields for the Potts and
-$\mathrm O(n)$ models take the form $B(s)=(h/\beta)\sum_m\cos(2\pi(s-m)/q)$
-and $B(s)=(h/\beta)[1,0,\ldots,0]s$ respectively. As can be seen, the average
+models, shown in Fig.~\ref{fig:cluster_scaling}. Fields are the canonical ones
+referenced in Table~\ref{table:models}. As can be seen, the average
 cluster size collapses for each model according to the scaling hypothesis, and
 the large-field behavior likewise scales as we expect from the na\"ive Ising
 conjecture.
-- 
cgit v1.2.3-54-g00ecf

