Added labels to some equations

Disertori_Giuliani_Jauslin_2018.tex

@@ -227,7 +227,7 @@ The strategy of the proof is very similar to the one of \cite{DG13}, in which a
 probability in the big parameter $\rho k^{2+\alpha}$. Once this is proved, the rest of the proof follows closely the one in \cite{DG13} and, therefore, we will not spell out all the details of the proofs, and, instead, refer the reader to~\cite{DG13} in which very similar arguments are expounded. 
 
 As far as we know, our result is the first rigorous one for the onset of a nematic-like phase in systems of finite-size particles, with finite-range interactions, in the three dimensional 
-continuum. For previous results, see \cite{AZ82,BKL84,HL79,IVZ06,Ru71,Za96}. We refer to the introduction of \cite{DG13} for a thorough, comparative, discussion of previous results. See also \cite{JL17} for a recent proof of the existence of nematic-like order in a monomer-dimer system with attractive interactions. 
+continuum. For previous results, see \cite{AZ82,BKL84,HL79,IVZ06,Ru71,Za96}. We refer to the introduction of \cite{DG13} for a thorough, comparative, discussion of previous results. See also \cite{JL17c} for a recent proof of the existence of nematic-like order in a monomer-dimer system with attractive interactions. 
 
 Our inability to rigorously control the bi-axial nematic phase, as well as the optimal range of densities where uni-axial nematic is expected, is related to the highly anisotropic 
 shape of the excluded regions created by the hard core interaction around any given plate. For instance, consider the range of densities between $k^{-2}$ and $k^{-1-\alpha}$, where
@@ -253,8 +253,8 @@ if $R_p\cap X\neq\emptyset$; $p$ is said to be {\it contained} in $X$ if $R_p\su
 \end{equation}
 where $\int_{\omega_\Lambda} dp$ is a shorthand for $\int_\Lambda dx\sum_{o\in\mathcal O}$, and 
 \begin{equation}
-  \varphi(p_1,\ldots p_n)=\prod_{i<j}\varphi(p_i,p_j),\qquad \varphi(p,p')=\begin{cases}1\ {\rm if}\  p\cap p'= \emptyset,\\
+  \varphi(p_1,\ldots p_n)=\prod_{i<j}\varphi(p_i,p_j),\qquad \varphi(p,p')=\left\{\begin{array}{>\displaystyle l}1\ {\rm if}\  p\cap p'= \emptyset,\\
-0\ {\rm if}\  p\cap p'\neq \emptyset.\end{cases}.
+0\ {\rm if}\  p\cap p'\neq \emptyset.\end{array}\right.
 \end{equation}
 As we shall see below, see the first remark after Theorem~\ref{theorem:nematic}, 
 fixing the activity is equivalent to fixing the densities, at least in the range of densities we are interested in. 
@@ -440,9 +440,9 @@ The Mayer expansion allows us to estimate the partition function of uniformly ma
 We split the block $\Delta$ into smaller $k^\alpha/2\times k^\alpha/2\times k^\alpha/2$ cubes, which we call {\it pebbles}. Because of the hard core interaction between plates,
 each pebble may only contain plates of a single type. Since $zk^{3\alpha }\gg 1$ each pebble $\delta$ still contains many plates, and the
 corresponding partition function can be evaluated by a Mayer expansion: for  $q=1,2,3$ we have by~(\ref{mayer1}),
-\[
+\begin{equation}
   Z^q(\delta):=\int_{\Omega^q_\delta}dP\ \varphi(P)z^{|P|}=e^{\frac14zk^{3\alpha}(1+O(zk^{2}))} 
-\]
+\end{equation}
 where we used the fact that the volume of the pebble is $|\delta|=k^{3\alpha }/8$.
 
 Given a configuration of plates in $\Delta$, we color each pebble according to the following.
@@ -541,7 +541,8 @@ terms in the sum over $\underline\Delta^{(t)}$ and $\underline\delta$: in fact,
 most a factor $3^{N'}$, where $N'$ is the number of connected components of $\Delta^{(t)}$, and $3$ is the number of `colors' (that is, $1,2$ or $3$) that we can attach to 
 each connected component. As observed above, $N'\le 6N$, so that the constant $C$ in (\ref{eq.3.13}) is smaller than $3^6$. 
 From (\ref{eq.3.13}) we immediately get:
- \begin{equation}\begin{array}{>\displaystyle r>\displaystyle c>\displaystyle l}   
+\begin{equation}
+  \begin{array}{>\displaystyle r>\displaystyle c>\displaystyle l}   
     \frac{Z^{\ge2}(\Delta)}{Z^q(\Delta)}
   &  \le&     e^{O(zk^3zk^2)}  e^{-\frac1{32} k^{2(1-\alpha)}\cdot zk^{3\alpha}(1+O(zk^2))}
     \sum_{N=0}^{k^{3(1-\alpha)}}
@@ -553,7 +554,9 @@ From (\ref{eq.3.13}) we immediately get:
       \left(
 	1+O(zk^{3-\alpha})+O(z^{-1}k^{1-4\alpha}e^{-\frac1{17}zk^{3\alpha}})\right),
     \right)
-\end{array}\end{equation}
+  \end{array}
+  \label{ineq_twodir_inproof}
+\end{equation}
 where the exponent $\frac1{17}$ in the last line may be replaced by any exponent smaller than $\frac1{16}$, for $zk^2$ sufficiently small. 
 The last term can be bounded as follows $z^{-1}k^{1-4\alpha}e^{-\frac1{17}zk^{3\alpha}}=
 \frac{1}{zk^{3\alpha }}k^{1-\alpha } e^{-\frac1{17}zk^{3\alpha}}\ll k^{1-\alpha } e^{-\frac1{17}zk^{3\alpha}}.$
@@ -804,9 +807,12 @@ one of the plates in the contour (in this case we may have $|P|=1$). \medskip
   d\tilde P\ \varphi (\tilde P\cup P)z^{|\tilde P|},\end{equation}
 where $\partial A=\cup_{\xi\in A': d_\infty'(\xi,({\rm Ext}A)')\le 8}\Delta_\xi$ is the layer of blocks that are uniformly magnetized by the boundary conditions, and $A^\circ=A\setminus \partial A$. On the other hand, 
 this expression is equivalent to 
-\begin{equation}Z^{(\gamma)}(A|m)=\int_{\Omega^m_{\partial A}\setminus V_m(P_\gamma)} 
+\begin{equation}
+  Z^{(\gamma)}(A|m)=\int_{\Omega^m_{\partial A}\setminus V_m(P_\gamma)} 
   dP\ \varphi (P)z^{|P|}\int_{\Omega_{A^{\circ}}} 
-  d\tilde P\ \varphi (\tilde P\cup P)z^{|\tilde P|}=: Z(A\setminus V_m(P_\gamma)\,|\,m),\end{equation}
+  d\tilde P\ \varphi (\tilde P\cup P)z^{|\tilde P|}=: Z(A\setminus V_m(P_\gamma)\,|\,m),
+  \label{Z_with_g}
+\end{equation}
 where $V_m(P_\gamma)$ is the excluded volume created by the plates in $P_\gamma$ on those in $P$. Note that $A\setminus V_m(P_\gamma)$ is an element of $\mathfrak{Int}'$, where
 \begin{equation} \mathfrak{Int}':=\Bigg\{A\setminus V:\ A\in\mathfrak{Int}\ {\rm and}\ V\subset \mathbb R^3\ {\rm such}\ {\rm that}\ V\subset\hskip-.5truecm \bigcup\limits_{\displaystyle\mathop{\scriptstyle xi\in A':}_{ d_\infty'(\xi,({\rm Ext}A)')\le 2}}
 \hskip-.5truecm\Delta_\xi\Bigg\}.\label{int'}\end{equation}
@@ -876,11 +882,12 @@ are $D$-connected with at least two contours in $\partial$.
 
 \indent The proof of this lemma is fairly straightforward, and virtually identical to~\cite[Lemma~2]{DG13}.
 The key identity is
-\[
+\begin{equation}
   e^{W^{(\Lambda )} (\partial)} = e^{\sum_{n\geq 2} \frac{(-1)^{n}}{n!} \sum^*_{\gamma_1,\cdots,\gamma_n\in\partial} 
   \int_{\Omega^q_\Lambda}dP\ \varphi^T(P)z^{|P|}F_{\gamma_1}(P)\cdots F_{\gamma_n}(P)}=
   \prod_{Y\in \mathfrak B^{T}(\Lambda)} \left[ (e^{ \mathcal F_\partial(Y)} -1)+1\right].
-\]
+  \label{mayer_id}
+\end{equation}
 The only real difference is that the sets $Y_i$ cover all the plates responsible for the interaction between contours, whereas in~\cite{DG13}, only the extremal blocks are kept (in~\cite{DG13}, the analog of the sets $Y_i$ are 
 denoted by $\overline Y_i$). The details are left to the reader. 
 
@@ -1003,29 +1010,32 @@ it must contain at least $1+c_0|Y'|$ plates, for a suitable constant $c_0$, whic
 dist is the Euclidean distance. 
 Therefore, letting: $l_Y:=1+\max(2,c_0|Y'|)$, $N$ be the number of contours in $\partial$ that are $D$-connected to the set $Y$,
 $\Delta_{\xi_1}$ be the `first' block of $Y$ (with respect to any given order of its blocks) and $S_Y$ the union of the sampling cubes intersecting $\Delta_{\xi_1}$, 
-\begin{equation}\begin{array}{>\displaystyle r@{\ }>\displaystyle l}
+\begin{equation}
+  \begin{array}{>\displaystyle r@{\ }>\displaystyle l}
     |\mathcal F_\partial(Y)|&\le 
     \sum_{n=2}^{N}\frac{1}{n!}\sum_{\gamma_1,\cdots,\gamma_n\subset\partial}^*\int_{\Omega^{\ge l_Y,q}_\Lambda}dP\ z^{|P|}|\varphi^T(P)|\mathds{1}(p_1\ {\rm belongs}\ {\rm to}\ S_Y)
     \mathds 1_{\mathrm{dist}(Y,X_0)=0}
-\label{boundeF}\\
+    \\
     & \leq 2^{N} zk^{3} (Czk^2)^{\max(2,c_0|Y'|)}\mathds 1_{\mathrm{dist}(Y,X_0)=0} \leq 
   zk^3(C'zk^2)^{\max(2, c_0|Y'|)}\mathds 1_{\mathrm{dist}(Y,X_0)=0}
-\end{array}\end{equation}
+  \end{array}
+  \label{boundeF}
+\end{equation}
 for some constants $C,C'>0$, where we used (\ref{eqcvce_plate}) and, in the final bound, we used $N\leq |Y'|$.
 Moreover we have
-\[
+\begin{equation}
 \prod_{j=1}^p\left |e^{\mathcal F_\partial(Y_j)}-1\right | \leq e^{\sum_{j=1}^{p}  |\mathcal F_\partial(Y_j) | } \prod_{j=1}^p |\mathcal F_\partial(Y_j) |,
-\]
+\end{equation}
 where
-\[
+\begin{equation}
 \sum_{j=1}^{p}  |\mathcal F_\partial(Y_j) |\leq \sum_{\displaystyle\mathop{\scriptstyle Y\in\mathfrak B^{T}(X_{1})}_{\mathrm{dist}(Y,X_0)=0}} |\mathcal F_\partial(Y) |
 \leq  C'' zk^3(zk^2)^2|X_0'|
-\]
+\end{equation}
 and, using $\sum_{j}|Y_{j}'|\geq |X_{1}'|,$
-\[
+\begin{equation}
 \prod_{j=1}^p |\mathcal F_\partial(Y_j) |\leq  (zk^2)^{\frac{c_0}2|X_1'|}
 \prod_{j=1}^p  |\mathcal F_\partial(Y_j) | (zk^2)^{-\frac{1}{2}\max(2,c_0|Y'_{j}|)}
-\]
+\end{equation}
 Inserting these estimates in the sum over $p$
 \begin{equation}\begin{array}{>\displaystyle c}
 \sum_{p\ge1}\frac{1}{p!}\sum_{\displaystyle\mathop{\scriptstyle Y_1,\cdots,Y_p\subset\mathfrak B^T(X)}_{Y_1\cup\cdots\cup Y_p=X_1}}^* \prod_{j=1}^p  |\mathcal F_\partial(Y_j) | (zk^2)^{-\frac{1}{2}\max(2,c_0|Y'_{j}|)} \\
@@ -1163,6 +1173,7 @@ We also let $\mathfrak N_\gamma$ be the set of blocks in $\mathcal P$ that are n
     \left(\prod_{n\in\mathfrak N_\gamma}\frac{Z^{\sigma_n}(n)}{Z^q(n)}\right)
     e^{O(zk^3zk^2)|\Gamma'_\gamma|}
   \end{largearray}
+  \label{ineqF1}
 \end{equation}
 where the $2$ in  the first factor in the right side is due to the activity associated with spin 0, see (\ref{eq:2.8}), and the factor $e^{O(zk^3zk^2)|\Gamma'_\gamma|}$ comes from splitting $Z^q$ into blocks and dipoles, as per~(\ref{mayer1}). We now use Lemma~\ref{lemma:twodircubes} and Corollary~\ref{corollary:twodircubes}, 
 and note that $Z^{\sigma_n}(n)=Z^q(n)$, thus getting 
@@ -1193,6 +1204,7 @@ hold also in this case with the natural modifications, mostly of notational natu
   \rho_1^{(q,\Lambda)} (p_0) =z\left.\frac{\delta}{\delta\tilde z(p_0)}\log Z(\Lambda|q)\right|_{\tilde z(p)\equiv z}=
   z\left.\frac{\delta}{\delta\tilde z(p_0)}\log Z^q(\Lambda)\right|_{\tilde z(p)\equiv z} +
   z\left.\frac{\delta}{\delta\tilde z(p_0)}\log \frac{Z(\Lambda|q)}{Z^{q} (\Lambda )}\right|_{\tilde z(p)\equiv z}
+  \label{rho1_inproof}
 \end{equation}
  The Mayer expansion of the plate model implies that
 \begin{equation}\label{eq:derivlnZ}
@@ -1217,12 +1229,14 @@ If $\bar A$ is so small that $A$ cannot contain any contours, then
 $Z(A|q)=Z^{q} (A)$ and (\ref{bound}) is trivially true.
 Assume now by induction that  (\ref{bound}) holds for all $a\in\mathfrak{Int}'$ such that $|\bar a|< |\bar A|$, and let us prove (\ref{bound}). 
 By the analogue of Theorem~\ref{theorem:cluster} with $\Lambda$ replaced by $A\in{\mathfrak{Int}'}$ and plate-dependent activities,
-\begin{equation}\begin{array}{>\displaystyle r>\displaystyle c>\displaystyle l}
+\begin{equation}
-&&   z\left.\frac{\delta}{\delta\tilde z(p_0)}\log \frac{Z(A|q)}{Z^{q} (A)}\right|_{\tilde z(p)\equiv z} = \label{eq:6.4}\\
+  \begin{array}{>\displaystyle r>\displaystyle c>\displaystyle l}
+  &&   z\left.\frac{\delta}{\delta\tilde z(p_0)}\log \frac{Z(A|q)}{Z^{q} (A)}\right|_{\tilde z(p)\equiv z} = \\
   &&\qquad =\sum_{n\ge 0}\frac1{n!}\sum_{X_0,\ldots, X_n\in \mathfrak{B}^T(\bar A)}\phi^T(X_0,\ldots,X_n)
       z\left.\frac{\delta}{\delta\tilde z(p_0)}K_q^{(A)}(X_0)\right|_{\tilde z(p)\equiv z}\prod_{i=1}^nK_q^{(A)}(X_i).
-  
+  \end{array}
-\end{array}\end{equation}
+  \label{eq:6.4}
+\end{equation}
 We claim that $\left.\frac{\delta}{\delta\tilde z(p_0)}K_q^{(A)}(X)\right|_{\tilde z(p)\equiv z}$ admits a bound similar to the one for $K_q^{(A)}(X)$, namely
 \begin{equation} \big|\left.\frac{\delta}{\delta\tilde z(p_0)}K_q^{(A)}(X)\right|_{\tilde z(p)\equiv z}\big|\le \bar\epsilon^{\, c|X'|}e^{-m\,{\rm dist}'(X',\xi_{x})},\label{eq:6.5}\end{equation}
 for some constants $c,m>0$, where $\xi_x$ is the center of the block containing $x$. Inserting (\ref{eq:6.5}) in (\ref{eq:6.4}), together with $|K_q^{(A)}(X)|\le \bar\epsilon^{\,|X'|}$, the result follows. We are left with proving 

libs/symbols.sty

@@ -1,120 +0,0 @@
-\let\a=\alpha
-\let\b=\beta
-\let\g=\gamma
-\let\d=\delta
-\let\e=\varepsilon
-\let\z=\zeta
-\let\h=\eta
-\let\th=\theta
-\let\k=\kappa
-\let\l=\lambda
-\let\m=\mu
-\let\n=\nu
-\let\x=\xi
-\let\p=\pi
-\let\r=\rho
-\let\s=\sigma
-\let\t=\tau
-\let\f=\varphi
-\let\ph=\varphi
-\let\c=\chi
-\let\ps=\psi
-\let\y=\upsilon
-\let\o=\omega
-\let\si=\varsigma
-\let\G=\Gamma
-\let\D=\Delta
-\let\Th=\Theta
-\let\L=\Lambda
-\let\X=\Xi
-\let\P=\Pi
-\let\Si=\Sigma
-\let\F=\Phi
-\let\Ps=\Psi
-\let\O=\Omega
-\let\Y=\Upsilon
-
-\def\AAA{\mathcal A}
-\def\XXX{\mathcal X}
-\def\PPP{\mathcal P}
-\def\HHH{\mathcal H}
-\def\BBB{\mathcal B}
-\def\III{\mathcal I}
-\def\EE{\mathcal E}
-\def\MM{\mathcal M}
-\def\VV{\mathcal V}
-\def\CC{\mathcal C}
-\def\FF{\mathcal F}
-\def\WW{\mathcal W}
-\def\TT{\mathcal T}
-\def\NN{\mathcal N}
-\def\RR{\mathcal R}
-\def\LL{\mathcal L}
-\def\JJ{\mathcal J}
-\def\OO{\mathcal O}
-\def\DD{\mathcal D}
-\def\GG{\mathcal G}
-\def\SS{\mathcal S}
-\def\KK{\mathcal K}
-\def\UU{\mathcal U}
-\def\QQ{\mathcal Q}
-
-\def\aaa{\mathbf a}
-\def\bbb{\mathbf b}
-\def\hhh{\mathbf h}
-\def\hh{\mathbf h}
-\def\HH{\mathbf H}
-\def\AA{\mathbf A}
-\def\qq{\mathbf q}
-\def\BB{\mathbf B}
-\def\YY{\mathbf Y}
-\def\XX{\mathbf X}
-\def\PP{\mathbf P}
-\def\pp{\mathbf p}
-\def\vv{\mathbf v}
-\def\xx{\mathbf x}
-\def\yy{\mathbf y}
-\def\zz{\mathbf z}
-\def\II{\mathbf I}
-\def\ii{\mathbf i}
-\def\jj{\mathbf j}
-\def\kk{\mathbf k}
-\def\bS{\mathbf S}
-\def\mm{\mathbf m}
-\def\Vn{\mathbf n}
-
-\def\ch{\chi}
-\def\Exp{\mathrm exp}
-\def\Log{\mathrm log}
-\def\Ft{\varphi}
-\def\E{H}
-
-\def\RRR{\mathbb R}
-\def\ZZZ{\mathbb Z}
-
-\def\ss{\underline{\sigma}}
-
-\let\==\equiv
-\let\io=\infty 
-\let\0=\noindent
-\def\media#1{\left<#1\right>}
-\let\dpr=\partial
-\def\sign{\mathrm{sign}}
-\def\const{\mathrm{const}}
-\def\wt{\widetilde}
-\def\wh{\widehat}
-\def\Val{\mathrm{Val}}
-\let\ul=\underline
-\def\lis{\overline}
-\let\V=\mathbf
-\def\be{\begin{equation}}
-\def\ee{\end{equation}}
-\def\bea{\begin{eqnarray}}
-\def\eea{\end{eqnarray}}
-\def\nn{\nonumber}
-
-\def\supp{\mathrm{supp}}
-\def\dist{\mathrm{dist}}
-\def\Ext{\mathrm{Ext}}
-\def\Int{\mathrm{Int}}
-\def\diam{\mathrm{diam}}