[Tex/LaTex] Arrows in tikz Markov chain diagram overlap

diagramstikz-pgf

I am trying to draw a Markov chain using tikz. The diagram is in the correct setup except the arrow going from State 2 and 3 overlaps two other arrows. I tried repositioning the states using node distance but that did not seem to work. How can I force the arrows not to overlap?
Markov Chain 

%latex
\documentclass[reqno]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage{pgfplots}

\usepgfplotslibrary{fillbetween}

\usepackage{tikz}
\usetikzlibrary{automata}
\usetikzlibrary{positioning}  %                 ...positioning nodes
\usetikzlibrary{arrows}       %                 ...customizing arrows
\tikzset{node distance=4.5cm, % Minimum distance between two nodes. Change if necessary.
         every state/.style={ % Sets the properties for each state
           semithick,
           fill=gray!10},
         initial text={},     % No label on start arrow
         double distance=4pt, % Adjust appearance of accept states
         every edge/.style={  % Sets the properties for each transition
         draw,
           ->,>=stealth',     % Makes edges directed with bold arrowheads
           auto,
           semithick}}

\begin{document}

\begin{figure}[htb]
\centering
\begin{tikzpicture}
\node[state] (s1) {State 1};
\node[state, below right of=s1] (s2) {State 2};
\node[state, below left of=s1] (s3) {State 3};

\draw (s1) edge[loop above] node {} (s1);
\draw (s1) edge[bend left] node {} (s2);
\draw (s1) edge[bend right] node {} (s3);

\draw (s2) edge[bend left] node {} (s1);
\draw (s2) edge[loop right] node {} (s2);
\draw (s2) edge[bend right] node {} (s3);

\draw (s3) edge[bend right] node {} (s1);
\draw (s3) edge[bend right] node {} (s2);
\draw (s3) edge[loop left] node {} (s3);

\end{tikzpicture}
\end{figure}

\end{document}

Best Answer

bend left and bend right come with parameters, the bending angles. Adjusting them allows you to avoid the intersections. (BTW, I also removed packages that were not used. Note also that the arrows library got superseded by arrows.meta but I kept arrows for now.)

\documentclass[reqno]{amsart}
\usepackage{tikz}
\usetikzlibrary{automata}
\usetikzlibrary{positioning}  %                 ...positioning nodes
\usetikzlibrary{arrows}       %                 ...customizing arrows
\tikzset{node distance=4.5cm, % Minimum distance between two nodes. Change if necessary.
         every state/.style={ % Sets the properties for each state
           semithick,
           fill=gray!10},
         initial text={},     % No label on start arrow
         double distance=4pt, % Adjust appearance of accept states
         every edge/.style={  % Sets the properties for each transition
         draw,
           ->,>=stealth',     % Makes edges directed with bold arrowheads
           auto,
           semithick}}

\begin{document}

\begin{figure}[htb]
\centering
\begin{tikzpicture}
\node[state] (s1) {State 1};
\node[state, below right of=s1] (s2) {State 2};
\node[state, below left of=s1] (s3) {State 3};

\draw (s1) edge[loop above]  (s1);
\draw (s1) edge[bend left]  (s2);
\draw (s1) edge[bend right]  (s3);

\draw (s2) edge[bend left=12]  (s1);
\draw (s2) edge[loop right]  (s2);
\draw (s2) edge[bend right=12]  (s3);

\draw (s3) edge[bend right=12]  (s1);
\draw (s3) edge[bend right]  (s2);
\draw (s3) edge[loop left]  (s3);

\end{tikzpicture}
\end{figure}
\end{document}

enter image description here

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