Ian Jauslin
summaryrefslogtreecommitdiff
blob: 7609b3910910f2fdea8127445bebefd3a69cb9e9 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
\documentclass{ian-presentation}

\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
\usepackage{dsfont}

\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil Liquid crystals\par
\smallskip
\hfil and the Heilmann-Lieb model\par
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Elliott H. Lieb}\par
\vfil
arXiv: {\tt \href{http://arxiv.org/abs/1709.05297}{1709.05297}}
\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject

\setcounter{page}1
\pagestyle{plain}

\title{Liquid crystals}
\begin{itemize}
  \item Orientational order and positional disorder.
\end{itemize}
\hfil\includegraphics[width=5cm]{nematic.png}
\hfil\includegraphics[width=5cm]{chiral.png}
\vfill
\eject

\title{History}
\begin{itemize}
  \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: mean field model for hard needles in $\mathbb R^3$.
  \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: interacting dimers.
\vphantom{
  \item \href{http://dx.doi.org/10.1007/BF00535264}{[Bricmont, Kuroda, Lebowitz, 1984]}: hard needles in $\mathbb R^2$ with a {\it finite} number of orientations.
  \item \href{http://dx.doi.org/10.1007/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}: hard rods in $\mathbb Z^2$ (variable length).
  \item \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: hard rods in $\mathbb Z^2$.
}
\end{itemize}
\vfill
\eject

\title{Heilmann-Lieb model}
\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}
\vfil
\hfil\includegraphics[width=5cm]{grid.pdf}
\vfil\eject

\addtocounter{page}{-1}
\title{Heilmann-Lieb model}
\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}
\vfil
\hfil\includegraphics[width=5cm]{dimers.pdf}
\vfil\eject

\addtocounter{page}{-1}
\title{Heilmann-Lieb model}
\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}
\vfil
\hfil\includegraphics[width=5cm]{interaction.pdf}
\vfil\eject

\title{History}
\begin{itemize}
  \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: mean field model for hard needles in $\mathbb R^3$.
  \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: interacting dimers.
  \item \href{http://dx.doi.org/10.1007/BF00535264}{[Bricmont, Kuroda, Lebowitz, 1984]}: hard needles in $\mathbb R^2$.
  \item \href{http://dx.doi.org/10.1007/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}: hard rods in $\mathbb Z^2$ (variable length).
  \item \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: hard rods in $\mathbb Z^2$.
\end{itemize}
\vfill
\eject

\title{Heilmann-Lieb conjecture}
\vfill
\begin{itemize}
  \item \href{http://dx.doi.org/10.1007/s10955-015-1421-8}{[Alberici, 2016]}: different fugacities for horizontal and vertical dimers.
  \vfill
  \item \href{http://dx.doi.org/10.1103/PhysRevB.89.035128}{[Papanikolaou, Charrier, Fradkin, 2014]}: numerics.
\end{itemize}
\vfill
\eject

\title{Heilmann-Lieb model}
\begin{itemize}
  \item Grand-canonical Gibbs measure:
  $$
    \left<A\right>_{\mathrm v}
    :=
    \lim_{\Lambda\to\mathbb Z^2}
    \frac1{\Xi_{\Lambda,\mathrm v}(z)}
    \sum_{\underline\delta\in\Omega_{\mathrm v}(\Lambda)}A(\underline\delta)z^{|\underline\delta|}\prod_{\delta\neq \delta'\in \underline\delta}e^{\frac12J\mathds 1_{\delta\sim\delta'}}
  $$
  \vskip-15pt
  \begin{itemize}
    \item $\Lambda$: finite box.
    \item $\Omega_{\mathrm v}(\Lambda)$: non-overlapping dimer configurations satisfying the boundary condition.
    \item $z\geqslant 0$: fugacity.
    \item $J\geqslant 0$: interaction strength.
    \item $\mathds 1_{\delta\sim\delta'}$ indicator that dimers are adjacent and aligned.
  \end{itemize}
\end{itemize}
\vfill
\eject

\title{Theorem}
For $1\ll z\ll J$, $\|(x,y)\|_{\mathrm{HL}}:=J|x|+e^{-\frac32J}z^{-\frac12}|y|$,
\begin{itemize}
  \item Given two vertical edges $e_{\mathrm v},f_{\mathrm v}$, $\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm v}$ and
  $$
    \begin{array}{c}
      \left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}=\frac12(1+O(e^{-\frac12J}z^{-\frac12}))
      \\[0.3cm]
      \left<\mathds 1_{e_{\mathrm v}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
      -\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}
      \left<\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
      =O(e^{-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm v},f_{\mathrm v})})
    \end{array}
  $$
  \vskip-5pt
  \item Given two horizontal edges $e_{\mathrm h},f_{\mathrm h}$, $\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm h}$ and
  $$
    \begin{array}{c}
      \left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}=O(e^{-3J})
      \\[0.3cm]
      \left<\mathds 1_{e_{\mathrm h}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
      -\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}
      \left<\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v}
      =O(e^{-3J-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm v},f_{\mathrm v})})
    \end{array}
  $$
\end{itemize}
\vfill
\eject

\title{1D system}
\begin{itemize}
  \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: mostly vertical dimers.
  \item {\it Only} vertical dimers: integrable.
  \item Given two vertical edges $e_{\mathrm v},f_{\mathrm v}$, $\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm v}$ and
  $$
    \begin{array}{c}
      \left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}=\frac12(1+O(e^{-\frac12J}z^{-\frac12}))
      \\[0.3cm]
      \left<\mathds 1_{e_{\mathrm v}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
      -\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}
      \left<\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
      =O(e^{-c\ \mathrm{dist}_{\mathrm{1D}}(e_{\mathrm v},f_{\mathrm v})})
    \end{array}
  $$
  with $\|(x,y)\|_{\mathrm{1D}}:=e^{-\frac32J}z^{-\frac12}|y|$.
\end{itemize}
\vfill
\eject

\title{Peierls argument}
\vfill
\hfil\includegraphics[width=5cm]{dimer_contour.pdf}
\vfill

\end{document}