
\newcommand*{\ds}[1]{\displaystyle{#1}}

\newcommand*{\bfidx}[1]{\index{#1}\textbf{#1}}
\newcommand*{\bfidxtwo}[2]{\textbf{#1}\index{#2}}
\newcommand*{\bfidxaux}[2]{\index{#1}\index{#2}\textbf{#1}}
\newcommand*{\emphidx}[1]{\index{#1}\emph{#1}}
\newcommand*{\plainidx}[1]{\index{#1}#1}

\newcommand*{\bra}[1]{\langle \, #1 \mid}
\newcommand*{\ket}[1]{\mid #1 \, \rangle}
\newcommand*{\braket}[2]{\langle #1 \mid #2 \rangle}
\newcommand*{\braCket}[3]{\langle #1 \mid #2 \mid #3 \rangle}
\newcommand*{\BraCKet}[3]
	{\Bigg\langle #1 \,\Bigg|\, #2 \,\Bigg|\, #3 \Bigg\rangle}
\newcommand*{\expn}[1]{\langle \, #1 \, \rangle}
\newcommand*{\ketbra}[1]{\ket{ #1 } \bra{ #1 }}
\newcommand*{\TrebH}{\Tr\left(e^{-\beta H}\right)}
\newcommand*{\TrXebH}[1]{\Tr\left(#1 e^{-\beta H}\right)}
\newcommand*{\Trsym}{\Tr_{L^2_{\mathrm{sym}}}}
\newcommand*{\intRdntimes}[2]{\underbrace{
	\int_{\R^{#1}} \int_{\R^{#1}} \cdots \int_{\R^{#1}}}_{#2 \textrm{ times}}}

\newcommand{\lmax}{\ell_{\textrm{max}}}

\newcommand{\ellx}{\ell_{\vecx}}
\newcommand{\elly}{\ell_{\vecy}}

\newcommand{\ellxpi}{\ell_{\vecx}(\pi)}
\newcommand{\ellypi}{\ell_{\vecy}(\pi)}

\newcommand{\ellxpiprime}{\ell_{\vecx}(\pi')}
\newcommand{\ellypiprime}{\ell_{\vecy}(\pi')}

\newcommand{\ellxypi}{\ell_{\vecx,\vecy}(\pi)}
\newcommand{\ellyxpi}{\ell_{\vecy,\vecx}(\pi)}

\newcommand{\essx}{s_{\vecx}}

\newcommand{\essxpi}{s_{\vecx}(\pi)}

\newcommand{\ellxf}{\ell^f_{\vecx}}
\newcommand{\essxf}{s^f_{\vecx}}
\newcommand{\ellmax}{\ell_{\textrm{max}}}
\newcommand{\fM}{f_{\textrm{max}}}
\newcommand{\ellbar}{\overline{\ell}}
\newcommand{\ellfbar}{\overline{\ell}_f}
\newcommand{\essbar}{\overline{s}}
\newcommand{\essfbar}{\overline{s}_f}

\newcommand{\rhocz}{\rho_c^{(0)}}
\newcommand{\rhocal}{\rho_c^{(\alpha)}}
\newcommand{\betacz}{\beta_c^{(0)}}
\newcommand{\betacal}{\beta_c^{(\alpha)}}
\newcommand{\Tcz}{T_c^{(0)}}
\newcommand{\Tcal}{T_c^{(\alpha)}}

\newcommand{\Xt}{X_t}
\newcommand{\Yt}{Y_t}
\newcommand{\Xtk}{X_{t+k}}
\newcommand{\Ytk}{Y_{t+k}}

\newcommand{\cmhat}{\hat{c}_m}
\newcommand{\chat}{\hat{c}}
\newcommand{\tauint}{{\tau_{\textrm{int}}}}
\newcommand{\hattauint}{\hat{\tau}_{\textrm{int}}}
\newcommand{\tauexp}{{\tau_{\textrm{exp}}}}
\newcommand{\stilde}{{\tilde{s}}}
\newcommand{\ttilde}{{\tilde{t}}}

\newcommand{\sumiB}{\sum_{i=0}^{B-1}}
\newcommand{\sumjB}{\sum_{j=0}^{B-1}}
\newcommand{\sumiM}{\sum_{i=0}^{M-1}}
\newcommand{\sumjM}{\sum_{j=0}^{M-1}}
\newcommand{\sumiN}{\sum_{i=0}^{N-1}}
\newcommand{\sumjN}{\sum_{j=0}^{N-1}}

%% ----------------------------------------------------------------
\def\ve{\varepsilon}
\def\veps{\varepsilon}
\def\bbB{\mathbb{B}}
\def\bbE{\mathbb{E}}
\def\bbP{\mathbb{P}}
\def\muhat{\hat{\mu}}
\def\psihat{\hat{\psi}}
\def\fhat{\hat{f}}
\def\ghat{\hat{g}}
\def\Ahat{\hat{A}}
\def\Fhat{\hat{F}}
\def\Hhat{\hat{H}}
\def\What{\hat{W}}
\def\Yhat{\hat{Y}}
\def\Zhat{\hat{Z}}
\def\nuhat{\hat{\nu}}
\def\rhohat{\hat{\rho}}
\def\Xihat{\hat{Xi}}
\def\xbar{\overline{x}}
%\def\torusdiff{\tilde{\vecd}}
\def\torusdiff{\vecd_\Lambda}

\def\muX{\mu_{X}}
\def\muY{\mu_{Y}}
\def\muXNbar{\mu_{\overline{X}_N}}
\def\muYNbar{\mu_{\overline{Y}_N}}
\def\sigmaX{\sigma_{X}}
\def\sigmaY{\sigma_{Y}}
\def\sigmaXNbar{\sigma_{\overline{X}_N}}
\def\sigmaYNbar{\sigma_{\overline{Y}_N}}

\def\ANbar{{\overline{A}_N}}

\def\QMbar{{\overline{Q}_M}}

\def\Xbar{\overline{X}}
\def\XNbar{{\overline{X}_N}}
\def\XMbar{{\overline{X}_M}}
\def\XNBbar{{\overline{X}_{N,B}}}

\def\Ybar{\overline{Y}}
\def\YNbar{\overline{Y}_N}
\def\YNBbar{\overline{Y}_{N,B}}

\def\Z{\mathbb{Z}}

\def\Htilde{\tilde{H}}
\def\Halpha{H^{(\alpha)}}
\def\HP{H_P}
\def\HPzero{H_P^{(0)}}
\def\HPone{H_P^{(1)}}

\def\bfH{\mathbf{H}}
\def\bfx{\mathbf{x}}

\def\bsH{\mathbf{H}}
\def\bsx{\mathbf{x}}

\def\caS{\mathcal{S}}
\def\mcS{\mathcal{S}}

\def\mcF{\mathcal{F}}
\def\mcK{\mathcal{K}}
\def\mcL{\mathcal{L}}
\def\mcH{\mathcal{H}}
\def\mcN{\mathcal{N}}
\def\mcO{\mathcal{O}}

\def\msH{\mathscr{H}}
\newcommand*{\pig}[1]{\langle #1 \rangle}
\newcommand*{\pigm}[1]{\langle #1 \rangle_M}

%% ----------------------------------------------------------------
\newcommand{\Sn}{\mathcal{S}_n}
\newcommand{\SN}{\mathcal{S}_N}
\newcommand{\SNp}{\mathcal{S}_{N+1}}
\newcommand{\SNpc}{{\SNp \setminus \SN}}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\e}{\mathrm{e}}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Corr}{\mathrm{Corr}}
\newcommand{\Cov}{\mathrm{Cov}}

%% ----------------------------------------------------------------
\def\veca{\mathbf{a}}
\def\vecb{\mathbf{b}}
\def\vecc{\mathbf{c}}
\def\vecd{\mathbf{d}}
\def\vecg{\mathbf{g}}
\def\veck{\mathbf{k}}
\def\vecm{\mathbf{m}}
\def\vecn{\mathbf{n}}
\def\vecr{\mathbf{r}}
\def\vecu{\mathbf{u}}
\def\vecv{\mathbf{v}}
\def\vecw{\mathbf{w}}
\def\vecx{\mathbf{x}}
\def\vecy{\mathbf{y}}
\def\vecz{\mathbf{z}}

\def\vecK{\mathbf{K}}
\def\vecM{\mathbf{M}}
\def\vecP{\mathbf{P}}
%\def\vecP{\mathcal{P}}
\def\vecV{\mathbf{V}}
\def\vecW{\mathbf{W}}
\def\vecWhat{\hat{\mathbf{W}}}
\def\vecX{\mathbf{X}}
\def\vecY{\mathbf{Y}}
\def\vecZ{\mathbf{Z}}

\def\vecPpiz{\vecP^{(\pi_0)}}
\def\Ppiz{P^{(\pi_0)}}
\def\P{P_{\mathrm{Gibbs}}}
\def\Phat{\hat{P}_{\mathrm{Gibbs}}}

%\def\mkvM{\mathbf{M}}
%\def\mkvM{R}
\def\mkvM{A}
%\def\mkvM{\mathcal{M}}

\newcommand{\vecomega}{{\boldsymbol{\omega}}}
\newcommand{\vecmu}{{\boldsymbol{\mu}}}
\newcommand{\vecnu}{{\boldsymbol{\nu}}}

\def\vecxi{\mathbf{x}_{i}}
\def\vecxj{\mathbf{x}_{j}}
\def\vecxpii{\mathbf{x}_{\pi(i)}}
\def\vecxpij{\mathbf{x}_{\pi(j)}}
\def\vecxpik{\mathbf{x}_{\pi(k)}}
\def\veckpii{\mathbf{k}_{\pi(i)}}

\def\pii{\pi^{-1}}
\def\piinv{\pi^{-1}}

\def\pix{\pi(\vecx)}
\def\piy{\pi(\vecy)}

\def\piprimex{\pi'(\vecx)}
\def\piprimey{\pi'(\vecy)}

\def\piv{\pi(\vecv)}
\def\piw{\pi(w)}
\def\piiw{\pi^{-1}(w)}
\def\piix{\pi^{-1}(\vecx)}
\def\piiy{\pi^{-1}(\vecy)}
\def\edh{e^{-\Delta H}}

\def\vecwi{\mathbf{w}^{(i)}}
\def\vecwj{\mathbf{w}^{(j)}}
\def\vecwk{\mathbf{w}^{(k)}}
\def\vecwij{\mathbf{w}^{(ij)}}

\def\vecwil{\mathbf{w}^{(i_\ell)}}
\def\vecwjl{\mathbf{w}^{(j_\ell)}}

\def\xhat{\hat{x}}
\def\yhat{\hat{y}}

\DeclareMathOperator*{\ctdto}{\circ\hspace{-0.2mm}--\hspace{-0.3mm}\circ}
\DeclareMathOperator*{\notctdto}{\circ\hspace{-0.2mm}--\hspace{-2.5mm}\not--\hspace{-0.3mm}\circ}
\def\maybeeq{\stackrel{?}{=}}
\DeclareMathOperator*{\twocyc}{\circ\hspace{-0.2mm}-\pi-\hspace{-0.4mm}\circ}
\DeclareMathOperator*{\ntwocyc}{\circ\hspace{-0.2mm}-\not\pi-\hspace{-0.4mm}\circ}


%% ----------------------------------------------------------------
\newcommand{\tand}{\textrm{and}}
\newcommand{\qand}{\quad\textrm{and}\quad}
\newcommand{\qqand}{\qquad\textrm{and}\qquad}
\newcommand{\qor}{\quad\textrm{or}\quad}
\newcommand{\qqor}{\qquad\textrm{or}\qquad}
\newcommand{\qie}{\quad\textrm{i.e.}\quad}
\newcommand{\qqie}{\qquad\textrm{i.e.}\qquad}
\newcommand{\qqwith}{\qquad\textrm{with}\qquad}

%% ================================================================
%\newcommand{\Suto}{S\"ut\H{o}}
\def\Suto{S\"ut\H{o}}

\newcommand{\grad}{\nabla}
\newcommand{\tensor}{\otimes}
\newcommand{\into}{\hookrightarrow}
\newcommand{\longto}{\longrightarrow}
\def\upto{\nearrow}
\newcommand{\ix}{\pi(x)}
\newcommand{\iy}{\pi^{-1}(y)}
\newcommand{\iu}{\pi(u)}
\newcommand{\tat}{\textasciitilde}

\newcommand*{\boxalign}[1]{
	\begin{equation*}
	\addtolength{\fboxsep}{5pt}
	\boxed{
	\begin{aligned}
	#1
	\end{aligned}
	}
	\end{equation*}
}

\newcommand*{\rowvectwo}[2]{\begin{pmatrix}#1 & #2 \end{pmatrix}}
\newcommand*{\rowvecthree}[3]{\begin{pmatrix}#1 & #2 & #3\end{pmatrix}}

\newcommand*{\colvecone}[1]{\left(\begin{array}{r} #1 \end{array}\right)}
\newcommand*{\colvectwo}[2]{\left(\begin{array}{r} #1 \\ #2 \end{array}\right)}
\newcommand*{\colvecthree}[3]{\left(\begin{array}{r} #1 \\ #2 \\ #3\end{array}\right)}
\newcommand*{\colvecfour}[4]{\left(\begin{array}{r} #1 \\ #2 \\ #3 \\ #4\end{array}\right)}

\newcommand*{\collvecone}[1]{\left(\begin{array}{l} #1 \end{array}\right)}
\newcommand*{\collvectwo}[2]{\left(\begin{array}{l} #1 \\ #2 \end{array}\right)}
\newcommand*{\collvecthree}[3]{\left(\begin{array}{l} #1 \\ #2 \\ #3\end{array}\right)}
\newcommand*{\collvecfour}[4]{\left(\begin{array}{l} #1 \\ #2 \\ #3 \\ #4\end{array}\right)}

\newcommand*{\colcvecone}[1]{\left(\begin{array}{c} #1 \end{array}\right)}
\newcommand*{\colcvectwo}[2]{\left(\begin{array}{c} #1 \\ #2 \end{array}\right)}
\newcommand*{\colcvecthree}[3]{\left(\begin{array}{c} #1 \\ #2 \\ #3\end{array}\right)}
\newcommand*{\colcvecfour}[4]{\left(\begin{array}{c} #1 \\ #2 \\ #3 \\ #4\end{array}\right)}

\newcommand*{\colrvecone}[1]{\left(\begin{array}{r} #1 \end{array}\right)}
\newcommand*{\colrvectwo}[2]{\left(\begin{array}{r} #1 \\ #2 \end{array}\right)}
\newcommand*{\colrvecthree}[3]{\left(\begin{array}{r} #1 \\ #2 \\ #3\end{array}\right)}
\newcommand*{\colrvecfour}[4]{\left(\begin{array}{r} #1 \\ #2 \\ #3 \\ #4\end{array}\right)}

%% ----------------------------------------------------------------
\newcommand{\D}{\partial}
\newcommand{\DA}{\partial A}
\newcommand{\DB}{\partial B}
\newcommand{\DC}{\partial C}
\newcommand{\DX}{\partial X}
\newcommand{\Dc}{\partial c}
\newcommand{\Df}{\partial f}
\newcommand{\Dg}{\partial g}
\newcommand{\DG}{\partial G}
\newcommand{\Dh}{\partial h}
\newcommand{\DM}{\partial M}
\newcommand{\Dr}{\partial r}
\newcommand{\Du}{\partial u}
\newcommand{\Dx}{\partial x}
\newcommand{\Dy}{\partial y}
\newcommand{\Dz}{\partial z}
\newcommand{\Ds}{\partial s}
\newcommand{\Dt}{\partial t}

\newcommand{\DD}{\partial/\partial}
\newcommand{\DDA}{\partial/\partial A}
\newcommand{\DDB}{\partial/\partial B}
\newcommand{\DDC}{\partial/\partial C}
\newcommand{\DDX}{\partial/\partial X}
\newcommand{\DDc}{\partial/\partial c}
\newcommand{\DDf}{\partial/\partial f}
\newcommand{\DDg}{\partial/\partial g}
\newcommand{\DDG}{\partial/\partial G}
\newcommand{\DDh}{\partial/\partial h}
\newcommand{\DDM}{\partial/\partial M}
\newcommand{\DDr}{\partial/\partial r}
\newcommand{\DDx}{\partial/\partial x}
\newcommand{\DDy}{\partial/\partial y}
\newcommand{\DDz}{\partial/\partial z}
\newcommand{\DDs}{\partial/\partial s}
\newcommand{\DDt}{\partial/\partial t}

%% ----------------------------------------------------------------
\newcommand*{\slashD}[1]{\partial/\partial #1}
\newcommand*{\fracD}[1]{\frac{\partial}{\partial #1}}
\newcommand*{\slashDtwo}[2]{\partial #1/\partial #2}
\newcommand*{\fracDtwo}[2]{\frac{\partial #1}{\partial #2}}

\newcommand{\slashDx}{\partial/\partial x}
\newcommand{\slashDy}{\partial/\partial y}
\newcommand{\slashDz}{\partial/\partial z}

\newcommand{\fracDs}{\frac{\partial}{\partial s}}
\newcommand{\fracDt}{\frac{\partial}{\partial t}}
\newcommand{\fracDx}{\frac{\partial}{\partial x}}
\newcommand{\fracDy}{\frac{\partial}{\partial y}}
\newcommand{\fracDz}{\frac{\partial}{\partial z}}
\newcommand{\fracDxx}{\frac{\partial^2}{\partial x^2}}

\newcommand{\fracDfx}{\frac{\partial f}{\partial x}}
\newcommand{\fracDfy}{\frac{\partial f}{\partial y}}
\newcommand{\fracDfz}{\frac{\partial f}{\partial z}}

\newcommand{\fracDgt}{\frac{\partial g}{\partial t}}
\newcommand{\fracDgu}{\frac{\partial g}{\partial u}}
\newcommand{\fracDguu}{\frac{\partial^2 g}{\partial u^2}}
\newcommand{\fracDgv}{\frac{\partial g}{\partial v}}
\newcommand{\fracDgvv}{\frac{\partial^2 g}{\partial v^2}}
\newcommand{\fracDguv}{\frac{\partial^2 g}{\partial u \partial v}}

\newcommand{\fracDht}{\frac{\partial h}{\partial t}}
\newcommand{\fracDhu}{\frac{\partial h}{\partial u}}
\newcommand{\fracDhuu}{\frac{\partial^2 h}{\partial u^2}}
\newcommand{\fracDhv}{\frac{\partial h}{\partial v}}
\newcommand{\fracDhvv}{\frac{\partial^2 h}{\partial v^2}}
\newcommand{\fracDhuv}{\frac{\partial^2 h}{\partial u \partial v}}

\newcommand{\fracDgx}{\frac{\partial g}{\partial x}}
\newcommand{\fracDgxx}{\frac{\partial^2 g}{\partial x^2}}
\newcommand{\fracDgy}{\frac{\partial g}{\partial y}}
\newcommand{\fracDgyy}{\frac{\partial^2 g}{\partial y^2}}
\newcommand{\fracDgz}{\frac{\partial g}{\partial z}}
\newcommand{\fracDgxy}{\frac{\partial^2 g}{\partial x \partial y}}

\newcommand{\fracDhx}{\frac{\partial h}{\partial x}}
\newcommand{\fracDhxx}{\frac{\partial^2 h}{\partial x^2}}
\newcommand{\fracDhy}{\frac{\partial h}{\partial y}}
\newcommand{\fracDhyy}{\frac{\partial^2 h}{\partial y^2}}
\newcommand{\fracDhz}{\frac{\partial h}{\partial z}}
\newcommand{\fracDhxy}{\frac{\partial^2 h}{\partial x \partial y}}

\newcommand{\fracDphis}{\frac{\partial    \phi}{\partial s}}
\newcommand{\fracDphit}{\frac{\partial    \phi}{\partial t}}
\newcommand{\fracDphix}{\frac{\partial    \phi}{\partial x}}
\newcommand{\fracDphixx}{\frac{\partial^2 \phi}{\partial x^2}}
\newcommand{\fracDphiy}{\frac{\partial    \phi}{\partial y}}
\newcommand{\fracDphiyy}{\frac{\partial^2 \phi}{\partial y^2}}
\newcommand{\fracDphiz}{\frac{\partial    \phi}{\partial z}}
\newcommand{\fracDphixy}{\frac{\partial^2 \phi}{\partial x \partial y}}

\newcommand{\fracDxDy}{\frac{\partial x}{\partial y}}
\newcommand{\fracDxDz}{\frac{\partial x}{\partial z}}
\newcommand{\fracDyDx}{\frac{\partial y}{\partial x}}
\newcommand{\fracDyDz}{\frac{\partial y}{\partial z}}
\newcommand{\fracDzDx}{\frac{\partial z}{\partial x}}
\newcommand{\fracDzDy}{\frac{\partial z}{\partial y}}

\newcommand{\fracDfDx}{\frac{\partial f}{\partial x}}
\newcommand{\fracDfDy}{\frac{\partial f}{\partial y}}
\newcommand{\fracDfDz}{\frac{\partial f}{\partial z}}

\newcommand{\slashDxDy}{\partial x/\partial y}
\newcommand{\slashDxDz}{\partial x/\partial z}
\newcommand{\slashDyDx}{\partial y/\partial x}
\newcommand{\slashDyDz}{\partial y/\partial z}
\newcommand{\slashDzDx}{\partial z/\partial x}
\newcommand{\slashDzDy}{\partial z/\partial y}

\newcommand{\slashDfDx}{\partial f/\partial x}
\newcommand{\slashDfDy}{\partial f/\partial y}
\newcommand{\slashDfDz}{\partial f/\partial z}

%% ----------------------------------------------------------------
% Parenthesized matrix with left alignment.
\newcommand*{\plmatrix}[1]{
	\left(\begin{array}{llllllllllll}
		#1
	\end{array}\right)
}

% Parenthesized matrix with centered alignment.
\newcommand*{\pcmatrix}[1]{
	\left(\begin{array}{cccccccccccc}
		#1
	\end{array}\right)
}

% Parenthesized matrix with right alignment.
\newcommand*{\prmatrix}[1]{
	\left(\begin{array}{rrrrrrrrrrrr}
		#1
	\end{array}\right)
}

%% ----------------------------------------------------------------
% Bracketed matrix with left alignment.
\newcommand*{\blmatrix}[1]{
	\left[\begin{array}{llllllllllll}
		#1
	\end{array}\right]
}

% Bracketed matrix with centered alignment.
\newcommand*{\bcmatrix}[1]{
	\left[\begin{array}{cccccccccccc}
		#1
	\end{array}\right]
}

% Bracketed matrix with right alignment.
\newcommand*{\brmatrix}[1]{
	\left[\begin{array}{rrrrrrrrrrrr}
		#1
	\end{array}\right]
}

%% ----------------------------------------------------------------
% Unbracketed matrix with left alignment.
\newcommand*{\nlmatrix}[1]{
	\begin{array}{llllllllllll}
		#1
	\end{array}
}

% Unbracketed matrix with centered alignment.
\newcommand*{\ncmatrix}[1]{
	\begin{array}{cccccccccccc}
		#1
	\end{array}
}

% Unbracketed matrix with right alignment.
\newcommand*{\nrmatrix}[1]{
	\begin{array}{rrrrrrrrrrrr}
		#1
	\end{array}
}
