Added labels to some equations
This commit is contained in:
Ian Jauslin
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06e19c0db2
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@ -227,7 +227,7 @@ The strategy of the proof is very similar to the one of \cite{DG13}, in which a
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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.
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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
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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.
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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.
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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
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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
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@ -253,8 +253,8 @@ if $R_p\cap X\neq\emptyset$; $p$ is said to be {\it contained} in $X$ if $R_p\su
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\end{equation}
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where $\int_{\omega_\Lambda} dp$ is a shorthand for $\int_\Lambda dx\sum_{o\in\mathcal O}$, and
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\begin{equation}
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\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,\\
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0\ {\rm if}\ p\cap p'\neq \emptyset.\end{cases}.
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\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,\\
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0\ {\rm if}\ p\cap p'\neq \emptyset.\end{array}\right.
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\end{equation}
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As we shall see below, see the first remark after Theorem~\ref{theorem:nematic},
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fixing the activity is equivalent to fixing the densities, at least in the range of densities we are interested in.
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@ -440,9 +440,9 @@ The Mayer expansion allows us to estimate the partition function of uniformly ma
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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,
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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
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corresponding partition function can be evaluated by a Mayer expansion: for $q=1,2,3$ we have by~(\ref{mayer1}),
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\[
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\begin{equation}
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Z^q(\delta):=\int_{\Omega^q_\delta}dP\ \varphi(P)z^{|P|}=e^{\frac14zk^{3\alpha}(1+O(zk^{2}))}
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\]
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\end{equation}
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where we used the fact that the volume of the pebble is $|\delta|=k^{3\alpha }/8$.
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Given a configuration of plates in $\Delta$, we color each pebble according to the following.
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@ -541,7 +541,8 @@ terms in the sum over $\underline\Delta^{(t)}$ and $\underline\delta$: in fact,
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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
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each connected component. As observed above, $N'\le 6N$, so that the constant $C$ in (\ref{eq.3.13}) is smaller than $3^6$.
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From (\ref{eq.3.13}) we immediately get:
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\begin{equation}\begin{array}{>\displaystyle r>\displaystyle c>\displaystyle l}
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\begin{equation}
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\begin{array}{>\displaystyle r>\displaystyle c>\displaystyle l}
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\frac{Z^{\ge2}(\Delta)}{Z^q(\Delta)}
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& \le& e^{O(zk^3zk^2)} e^{-\frac1{32} k^{2(1-\alpha)}\cdot zk^{3\alpha}(1+O(zk^2))}
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\sum_{N=0}^{k^{3(1-\alpha)}}
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@ -553,7 +554,9 @@ From (\ref{eq.3.13}) we immediately get:
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\left(
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1+O(zk^{3-\alpha})+O(z^{-1}k^{1-4\alpha}e^{-\frac1{17}zk^{3\alpha}})\right),
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\right)
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\end{array}\end{equation}
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\end{array}
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\label{ineq_twodir_inproof}
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\end{equation}
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where the exponent $\frac1{17}$ in the last line may be replaced by any exponent smaller than $\frac1{16}$, for $zk^2$ sufficiently small.
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The last term can be bounded as follows $z^{-1}k^{1-4\alpha}e^{-\frac1{17}zk^{3\alpha}}=
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\frac{1}{zk^{3\alpha }}k^{1-\alpha } e^{-\frac1{17}zk^{3\alpha}}\ll k^{1-\alpha } e^{-\frac1{17}zk^{3\alpha}}.$
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@ -804,9 +807,12 @@ one of the plates in the contour (in this case we may have $|P|=1$). \medskip
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d\tilde P\ \varphi (\tilde P\cup P)z^{|\tilde P|},\end{equation}
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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,
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this expression is equivalent to
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\begin{equation}Z^{(\gamma)}(A|m)=\int_{\Omega^m_{\partial A}\setminus V_m(P_\gamma)}
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\begin{equation}
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Z^{(\gamma)}(A|m)=\int_{\Omega^m_{\partial A}\setminus V_m(P_\gamma)}
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dP\ \varphi (P)z^{|P|}\int_{\Omega_{A^{\circ}}}
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d\tilde P\ \varphi (\tilde P\cup P)z^{|\tilde P|}=: Z(A\setminus V_m(P_\gamma)\,|\,m),\end{equation}
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d\tilde P\ \varphi (\tilde P\cup P)z^{|\tilde P|}=: Z(A\setminus V_m(P_\gamma)\,|\,m),
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\label{Z_with_g}
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\end{equation}
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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
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\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}}
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\hskip-.5truecm\Delta_\xi\Bigg\}.\label{int'}\end{equation}
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@ -876,11 +882,12 @@ are $D$-connected with at least two contours in $\partial$.
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\indent The proof of this lemma is fairly straightforward, and virtually identical to~\cite[Lemma~2]{DG13}.
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The key identity is
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\[
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\begin{equation}
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e^{W^{(\Lambda )} (\partial)} = e^{\sum_{n\geq 2} \frac{(-1)^{n}}{n!} \sum^*_{\gamma_1,\cdots,\gamma_n\in\partial}
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\int_{\Omega^q_\Lambda}dP\ \varphi^T(P)z^{|P|}F_{\gamma_1}(P)\cdots F_{\gamma_n}(P)}=
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\prod_{Y\in \mathfrak B^{T}(\Lambda)} \left[ (e^{ \mathcal F_\partial(Y)} -1)+1\right].
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\]
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\label{mayer_id}
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\end{equation}
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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
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denoted by $\overline Y_i$). The details are left to the reader.
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@ -1003,29 +1010,32 @@ it must contain at least $1+c_0|Y'|$ plates, for a suitable constant $c_0$, whic
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dist is the Euclidean distance.
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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 $Y$,
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$\Delta_{\xi_1}$ 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 $\Delta_{\xi_1}$,
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\begin{equation}\begin{array}{>\displaystyle r@{\ }>\displaystyle l}
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\begin{equation}
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\begin{array}{>\displaystyle r@{\ }>\displaystyle l}
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|\mathcal F_\partial(Y)|&\le
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\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)
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\mathds 1_{\mathrm{dist}(Y,X_0)=0}
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\label{boundeF}\\
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\\
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& \leq 2^{N} zk^{3} (Czk^2)^{\max(2,c_0|Y'|)}\mathds 1_{\mathrm{dist}(Y,X_0)=0} \leq
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zk^3(C'zk^2)^{\max(2, c_0|Y'|)}\mathds 1_{\mathrm{dist}(Y,X_0)=0}
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\end{array}\end{equation}
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\end{array}
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\label{boundeF}
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\end{equation}
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for some constants $C,C'>0$, where we used (\ref{eqcvce_plate}) and, in the final bound, we used $N\leq |Y'|$.
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Moreover we have
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\[
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\begin{equation}
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\prod_{j=1}^p\left |e^{\mathcal F_\partial(Y_j)}-1\right | \leq e^{\sum_{j=1}^{p} |\mathcal F_\partial(Y_j) | } \prod_{j=1}^p |\mathcal F_\partial(Y_j) |,
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\]
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\end{equation}
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where
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\[
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\begin{equation}
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\sum_{j=1}^{p} |\mathcal F_\partial(Y_j) |\leq \sum_{\displaystyle\mathop{\scriptstyle Y\in\mathfrak B^{T}(X_{1})}_{\mathrm{dist}(Y,X_0)=0}} |\mathcal F_\partial(Y) |
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\leq C'' zk^3(zk^2)^2|X_0'|
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\]
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\end{equation}
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and, using $\sum_{j}|Y_{j}'|\geq |X_{1}'|,$
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\[
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\begin{equation}
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\prod_{j=1}^p |\mathcal F_\partial(Y_j) |\leq (zk^2)^{\frac{c_0}2|X_1'|}
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\prod_{j=1}^p |\mathcal F_\partial(Y_j) | (zk^2)^{-\frac{1}{2}\max(2,c_0|Y'_{j}|)}
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\]
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\end{equation}
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Inserting these estimates in the sum over $p$
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\begin{equation}\begin{array}{>\displaystyle c}
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\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}|)} \\
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@ -1163,6 +1173,7 @@ We also let $\mathfrak N_\gamma$ be the set of blocks in $\mathcal P$ that are n
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\left(\prod_{n\in\mathfrak N_\gamma}\frac{Z^{\sigma_n}(n)}{Z^q(n)}\right)
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e^{O(zk^3zk^2)|\Gamma'_\gamma|}
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\end{largearray}
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\label{ineqF1}
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\end{equation}
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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},
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and note that $Z^{\sigma_n}(n)=Z^q(n)$, thus getting
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@ -1193,6 +1204,7 @@ hold also in this case with the natural modifications, mostly of notational natu
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\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}=
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z\left.\frac{\delta}{\delta\tilde z(p_0)}\log Z^q(\Lambda)\right|_{\tilde z(p)\equiv z} +
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z\left.\frac{\delta}{\delta\tilde z(p_0)}\log \frac{Z(\Lambda|q)}{Z^{q} (\Lambda )}\right|_{\tilde z(p)\equiv z}
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\label{rho1_inproof}
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\end{equation}
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The Mayer expansion of the plate model implies that
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\begin{equation}\label{eq:derivlnZ}
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@ -1217,12 +1229,14 @@ If $\bar A$ is so small that $A$ cannot contain any contours, then
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$Z(A|q)=Z^{q} (A)$ and (\ref{bound}) is trivially true.
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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}).
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By the analogue of Theorem~\ref{theorem:cluster} with $\Lambda$ replaced by $A\in{\mathfrak{Int}'}$ and plate-dependent activities,
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\begin{equation}\begin{array}{>\displaystyle r>\displaystyle c>\displaystyle l}
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&& 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}\\
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\begin{equation}
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\begin{array}{>\displaystyle r>\displaystyle c>\displaystyle l}
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&& z\left.\frac{\delta}{\delta\tilde z(p_0)}\log \frac{Z(A|q)}{Z^{q} (A)}\right|_{\tilde z(p)\equiv z} = \\
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&&\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)
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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).
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\end{array}\end{equation}
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\end{array}
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\label{eq:6.4}
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\end{equation}
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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
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\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}
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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
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120
libs/symbols.sty
120
libs/symbols.sty
@ -1,120 +0,0 @@
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\let\a=\alpha
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\let\b=\beta
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\let\g=\gamma
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\let\d=\delta
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\let\e=\varepsilon
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\let\z=\zeta
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\let\h=\eta
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\let\th=\theta
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\let\k=\kappa
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\let\l=\lambda
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\let\m=\mu
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\let\n=\nu
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\let\x=\xi
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\let\p=\pi
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\let\r=\rho
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\let\s=\sigma
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\let\t=\tau
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\let\f=\varphi
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\let\ph=\varphi
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\let\c=\chi
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\let\ps=\psi
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\let\y=\upsilon
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\let\o=\omega
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\let\si=\varsigma
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\let\G=\Gamma
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\let\D=\Delta
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\let\Th=\Theta
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\let\L=\Lambda
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\let\X=\Xi
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\let\P=\Pi
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\let\Si=\Sigma
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\let\F=\Phi
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\let\Ps=\Psi
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\let\O=\Omega
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\let\Y=\Upsilon
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\def\AAA{\mathcal A}
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\def\XXX{\mathcal X}
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\def\PPP{\mathcal P}
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\def\HHH{\mathcal H}
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\def\BBB{\mathcal B}
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\def\III{\mathcal I}
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\def\EE{\mathcal E}
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\def\MM{\mathcal M}
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\def\VV{\mathcal V}
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\def\CC{\mathcal C}
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\def\FF{\mathcal F}
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\def\WW{\mathcal W}
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\def\TT{\mathcal T}
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\def\NN{\mathcal N}
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\def\RR{\mathcal R}
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\def\LL{\mathcal L}
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\def\JJ{\mathcal J}
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\def\OO{\mathcal O}
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\def\DD{\mathcal D}
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\def\GG{\mathcal G}
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\def\SS{\mathcal S}
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\def\KK{\mathcal K}
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\def\UU{\mathcal U}
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\def\QQ{\mathcal Q}
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\def\aaa{\mathbf a}
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\def\bbb{\mathbf b}
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\def\hhh{\mathbf h}
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\def\hh{\mathbf h}
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\def\HH{\mathbf H}
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\def\AA{\mathbf A}
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\def\qq{\mathbf q}
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\def\BB{\mathbf B}
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\def\YY{\mathbf Y}
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\def\XX{\mathbf X}
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\def\PP{\mathbf P}
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\def\pp{\mathbf p}
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\def\vv{\mathbf v}
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\def\xx{\mathbf x}
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\def\yy{\mathbf y}
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\def\zz{\mathbf z}
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\def\II{\mathbf I}
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\def\ii{\mathbf i}
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\def\jj{\mathbf j}
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\def\kk{\mathbf k}
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\def\bS{\mathbf S}
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\def\mm{\mathbf m}
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\def\Vn{\mathbf n}
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\def\ch{\chi}
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\def\Exp{\mathrm exp}
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\def\Log{\mathrm log}
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\def\Ft{\varphi}
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\def\E{H}
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\def\RRR{\mathbb R}
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\def\ZZZ{\mathbb Z}
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\def\ss{\underline{\sigma}}
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\let\==\equiv
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\let\io=\infty
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\let\0=\noindent
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\def\media#1{\left<#1\right>}
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\let\dpr=\partial
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\def\sign{\mathrm{sign}}
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\def\const{\mathrm{const}}
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\def\wt{\widetilde}
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\def\wh{\widehat}
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\def\Val{\mathrm{Val}}
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\let\ul=\underline
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\def\lis{\overline}
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\let\V=\mathbf
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\def\be{\begin{equation}}
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\def\ee{\end{equation}}
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\def\bea{\begin{eqnarray}}
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\def\eea{\end{eqnarray}}
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\def\nn{\nonumber}
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\def\supp{\mathrm{supp}}
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\def\dist{\mathrm{dist}}
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\def\Ext{\mathrm{Ext}}
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\def\Int{\mathrm{Int}}
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\def\diam{\mathrm{diam}}
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