From bb10258c700c70936eb1918b57e89a913bb3476e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 8 Mar 2018 18:25:06 -0500
Subject: new figures

---
 aps_mm_2018.html | 33 +++++++++++++++++++++++++--------
 1 file changed, 25 insertions(+), 8 deletions(-)

(limited to 'aps_mm_2018.html')

diff --git a/aps_mm_2018.html b/aps_mm_2018.html
index a3f6d91..3be9597 100644
--- a/aps_mm_2018.html
+++ b/aps_mm_2018.html
@@ -39,14 +39,14 @@ class: center, middle
 
 Described by Hamiltonians whose degrees of freedom are spins `\(s\in X\)`
 
-$$\mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iH(s_i)$$
+$$\mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iB(s_i)$$
 
 for \\(Z\\) invariant under the action of rotations \\(r\in R\\), or \\(Z(r\cdot s,r\cdot t)=Z(s,t)\\).
 
 <table style="border-collapse: collapse; width: 100%; margin: auto;">
   <thead style="border-bottom: 2px solid #000;">
     <tr>
-      <td class="first"></td><td><i>X</i><td><i>R</i></td><td>Group Action</td><td>\(Z(s,t)\)</td><td>\(H(s)\)</td>
+      <td class="first"></td><td><i>X</i><td><i>R</i></td><td>Group Action</td><td>\(Z(s,t)\)</td><td>\(B(s)\)</td>
     </tr>
   </thead>
   <tbody>
@@ -54,13 +54,13 @@ for \\(Z\\) invariant under the action of rotations \\(r\in R\\), or \\(Z(r\cdot
       <td class="first">Ising model</td><td>\(\{-1,1\}\)</td><td>ℤ/2ℤ</td><td>\(0\cdot s\mapsto s\)<br>\(1\cdot s\mapsto-s\)</td><td>\(s\times t\)</td><td>\(Hs\)</td>
     </tr>
     <tr>
-      <td class="first">Order-\(n\) model</td><td>\(S^n\)</td><td>\(\mathop{\mathrm{O}}(n)\)</td><td>\(O\cdot s\mapsto Os\)</td><td>\(s^{\mathrm T}t\)</td><td>\(H\cdot s\)</td>
+      <td class="first">Order-\(n\) model</td><td>\(S^n\)</td><td>\(\mathop{\mathrm{O}}(n)\)</td><td>\(O\cdot s\mapsto Os\)</td><td>\(s^{\mathrm T}t\)</td><td>\(H^{\mathrm T}s\)</td>
     </tr>
     <tr>
-      <td class="first">Potts model</td><td>ℤ/<i>q</i>ℤ</td><td>ℤ/<i>q</i>ℤ</td><td>\(m\cdot s\mapsto{(s+m)}\pmod q\)</td><td>\(\delta(s,t)\)</td><td>\(\sum_mH_m\delta(m,s)\)</td>
+      <td class="first">Potts model</td><td>{0,…,<i>q</i> – 1}</td><td><i>D<sub>q</sub></i></td><td>Rotation & reflection<br> of polygon vertices</td><td>\(\delta(s,t)\)</td><td>\(\sum_mH_m\delta(m,s)\)</td>
     </tr>
     <tr>
-      <td class="first">Clock model</td><td>ℤ/<i>q</i>ℤ</td><td>ℤ/<i>q</i>ℤ</td><td>\(m\cdot s\mapsto{(s+m)}\pmod q\)</td><td>\(\cos(2\pi\frac{s-t}q)\)</td><td>\(\sum_mH_m\cos(2\pi\frac{s-m}q)\)</td>
+      <td class="first">Clock model</td><td>{0,…,<i>q</i> – 1}</td><td><i>D<sub>q</sub></i></td><td>Rotation & reflection<br> of polygon vertices</td><td>\(\cos(2\pi\frac{s-t}q)\)</td><td>\(\sum_mH_m\cos(2\pi\frac{s-m}q)\)</td>
     </tr>
   </tbody>
 </table>
@@ -188,9 +188,9 @@ class: split-40
 Consider the clock Potts model with field
 `\[
 H(m)=0.01\times\begin{cases}
-0 & \text{if $m=0$ (black)}\\
-\cos(2\pi/6) & \text{if $m=1$ (grey}\\
--\cos(2\pi/6) & \text{if $m=2$ (white)}
+0 & \text{if $m=0$ (blue)}\\
+\cos(2\pi/6) & \text{if $m=1$ (green}\\
+-\cos(2\pi/6) & \text{if $m=2$ (yellow)}
 \end{cases}
 \]`
 
@@ -204,6 +204,23 @@ No rotation `\(r\in\mathbb Z/3\mathbb Z\)` that takes field to the identity!
 
 ---
 
+class: split-40
+
+.column[
+
+# Correlation time comparison
+
+The correlation time of this algorithm beats metropolis and hybrid Wolff/metropolis in whole phase space!
+
+Pictured: 2D Ising (`\(128\times128\)`) correlation time as a function of field for three algorithms in high-temperature phase (top), at `\(T_c\)` (middle), and in the low temperature phase (bottom).
+]
+
+.column[
+<img src="time_compare.png" style="float: right; width: 32%" />
+]
+
+---
+
 # Correlation time scaling
 
 Correlation time scales consistently in the whole phase space!
-- 
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