Vwwxrf

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?

%latex
documentclass[reqno]amsart
usepackageamsmath
usepackageamssymb
usepackagehyperref
usepackagepgfplots

usepgfplotslibraryfillbetween

usepackagetikz
usetikzlibraryautomata
usetikzlibrarypositioning % ...positioning nodes
usetikzlibraryarrows % ...customizing arrows
tikzsetnode 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

begindocument

beginfigure[htb]
centering
begintikzpicture
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);

endtikzpicture
endfigure

enddocument

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 Note also that the arrows library got superseded by arrows.meta but I kept arrows for now.)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
usepackagetikz
usetikzlibraryautomata
usetikzlibrarypositioning % ...positioning nodes
usetikzlibraryarrows % ...customizing arrows
tikzsetnode 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

begindocument

beginfigure[htb]
centering
begintikzpicture
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);

endtikzpicture
endfigure
enddocument

you can reduce default value of bend angle. just add bend angle=15 to your tikzset (similarly @marmoth change it locally for two arrows bend).

off topic:

• for labeling of arrows is handy to use quotes library and than wrote it as for example ... (s1) edge["label",bend left] (s2).


• 
  

  package hyperref had to be load last in preamble (except in rare exception)

  
  
  

  documentclass[reqno]amsart
  usepackageamsmath, amssymb
  
  usepackagepgfplots % it load tikz too
  pgfplotssetcompat=1.16
  usetikzlibraryautomata,
   arrows.meta, % ...customizing arrows
   positioning, % ...positioning nodes
   quotes % For edge labels
  usepgfplotslibraryfillbetween
  tikzsetnode distance=4.5cm, % Minimum distance between 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,
   semithick,
   -Stealth, % Makes edges directed with bold arrowheads
   auto,
   bend angle=15 % Reduce default bend angle
   
  
  usepackagehyperref % had to be last in preamble
  
  begindocument
   beginfigure[htb]
   centering
  begintikzpicture[]
  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)
   (s1) edge[bend left] (s2)
   (s1) edge[bend right] (s3)
  %
   (s2) edge[bend left] (s1)
   (s2) edge[loop right] (s2)
   (s2) edge[bend right] (s3)
  %
   (s3) edge[bend right] (s1)
   (s3) edge[bend right] (s2)
   (s3) edge[loop left] (s3);
  endtikzpicture
   endfigure
  enddocument