Asymptotics of orbits on graphs












3












$begingroup$


Let $X$ be a connected, locally finite graph with vertex set $V(X)$ and $G$ a group acting freely on $X$ such that $X/G$ is a finite graph. Fix a vertex $x$ and for $kinmathbb N$ set
$$
N(k)=#{ gin G: d(gx,x)le k},
$$
where $d$ is the vertex distance in the graph $X$.
Further set
$$
A(k)=#{yin V(X):d(x,y)le k}.
$$
Is it true that, as $ktoinfty$, the number $N(k)/A(k)$ tends to $#V(X/G)^{-1}$? If so, what error term estimates are known?






graph-theory asymptotics







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$endgroup$












  • $begingroup$
    Very interesting! Can you please add the reference or the source of inspiration for this problem?
    $endgroup$
    – SeF
    13 hours ago










  • $begingroup$
    It's kind of a graph analogue of lattice point counting.
    $endgroup$
    – Zero
    13 hours ago
















3












$begingroup$


Let $X$ be a connected, locally finite graph with vertex set $V(X)$ and $G$ a group acting freely on $X$ such that $X/G$ is a finite graph. Fix a vertex $x$ and for $kinmathbb N$ set
$$
N(k)=#{ gin G: d(gx,x)le k},
$$
where $d$ is the vertex distance in the graph $X$.
Further set
$$
A(k)=#{yin V(X):d(x,y)le k}.
$$
Is it true that, as $ktoinfty$, the number $N(k)/A(k)$ tends to $#V(X/G)^{-1}$? If so, what error term estimates are known?






graph-theory asymptotics







share|cite|improve this question











$endgroup$












  • $begingroup$
    Very interesting! Can you please add the reference or the source of inspiration for this problem?
    $endgroup$
    – SeF
    13 hours ago










  • $begingroup$
    It's kind of a graph analogue of lattice point counting.
    $endgroup$
    – Zero
    13 hours ago














3












3








3





$begingroup$


Let $X$ be a connected, locally finite graph with vertex set $V(X)$ and $G$ a group acting freely on $X$ such that $X/G$ is a finite graph. Fix a vertex $x$ and for $kinmathbb N$ set
$$
N(k)=#{ gin G: d(gx,x)le k},
$$
where $d$ is the vertex distance in the graph $X$.
Further set
$$
A(k)=#{yin V(X):d(x,y)le k}.
$$
Is it true that, as $ktoinfty$, the number $N(k)/A(k)$ tends to $#V(X/G)^{-1}$? If so, what error term estimates are known?






graph-theory asymptotics







share|cite|improve this question











$endgroup$




Let $X$ be a connected, locally finite graph with vertex set $V(X)$ and $G$ a group acting freely on $X$ such that $X/G$ is a finite graph. Fix a vertex $x$ and for $kinmathbb N$ set
$$
N(k)=#{ gin G: d(gx,x)le k},
$$
where $d$ is the vertex distance in the graph $X$.
Further set
$$
A(k)=#{yin V(X):d(x,y)le k}.
$$
Is it true that, as $ktoinfty$, the number $N(k)/A(k)$ tends to $#V(X/G)^{-1}$? If so, what error term estimates are known?






graph-theory asymptotics




graph-theory asymptotics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 13 hours ago







Zero

















asked 15 hours ago









ZeroZero

2567




2567












  • $begingroup$
    Very interesting! Can you please add the reference or the source of inspiration for this problem?
    $endgroup$
    – SeF
    13 hours ago










  • $begingroup$
    It's kind of a graph analogue of lattice point counting.
    $endgroup$
    – Zero
    13 hours ago


















  • $begingroup$
    Very interesting! Can you please add the reference or the source of inspiration for this problem?
    $endgroup$
    – SeF
    13 hours ago










  • $begingroup$
    It's kind of a graph analogue of lattice point counting.
    $endgroup$
    – Zero
    13 hours ago
















$begingroup$
Very interesting! Can you please add the reference or the source of inspiration for this problem?
$endgroup$
– SeF
13 hours ago




$begingroup$
Very interesting! Can you please add the reference or the source of inspiration for this problem?
$endgroup$
– SeF
13 hours ago












$begingroup$
It's kind of a graph analogue of lattice point counting.
$endgroup$
– Zero
13 hours ago




$begingroup$
It's kind of a graph analogue of lattice point counting.
$endgroup$
– Zero
13 hours ago










1 Answer
1






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5












$begingroup$

It is possible that the limit does not exist at all: Consider the free group on two generators acting on the $(4,2)$-biregular tree in the obvious way. This action is free and has 3 orbits (one containing all vertices of degree 4, and the other two containing "half" of the vertices of degree 2).



Let $x$ be a vertex of degree $4$. Then $N(k)$ is the number of vertices of degree 4 in $B_x(k)$, and $A(k)$ is the total number of vertices in $B_x(k)$. If we write $a_k$ and $b_k$ for the number of vertices at distance exactly $k$ from $x$ which have degree 4 or 2 respectively, we get $a_0 = 1$, and $b_{2l+1} = a_{2l+2} = 4cdot3^l$ and $b_{2l} = a_{2l+1} = 0$ for $l geq 0$. Note that
$$frac{N(k)}{A(k)} = frac{sum_{i leq k} a_i}{sum_{i leq k} a_i + b_i}$$
and if I'm not mistaken, plugging in the above values gives a limit of $frac 12$ for the subsequence of even $k$, and $frac 14$ for the subsequence of odd $k$.






share|cite|improve this answer









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    1 Answer
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    5












    $begingroup$

    It is possible that the limit does not exist at all: Consider the free group on two generators acting on the $(4,2)$-biregular tree in the obvious way. This action is free and has 3 orbits (one containing all vertices of degree 4, and the other two containing "half" of the vertices of degree 2).



    Let $x$ be a vertex of degree $4$. Then $N(k)$ is the number of vertices of degree 4 in $B_x(k)$, and $A(k)$ is the total number of vertices in $B_x(k)$. If we write $a_k$ and $b_k$ for the number of vertices at distance exactly $k$ from $x$ which have degree 4 or 2 respectively, we get $a_0 = 1$, and $b_{2l+1} = a_{2l+2} = 4cdot3^l$ and $b_{2l} = a_{2l+1} = 0$ for $l geq 0$. Note that
    $$frac{N(k)}{A(k)} = frac{sum_{i leq k} a_i}{sum_{i leq k} a_i + b_i}$$
    and if I'm not mistaken, plugging in the above values gives a limit of $frac 12$ for the subsequence of even $k$, and $frac 14$ for the subsequence of odd $k$.






    share|cite|improve this answer









    $endgroup$


















      5












      $begingroup$

      It is possible that the limit does not exist at all: Consider the free group on two generators acting on the $(4,2)$-biregular tree in the obvious way. This action is free and has 3 orbits (one containing all vertices of degree 4, and the other two containing "half" of the vertices of degree 2).



      Let $x$ be a vertex of degree $4$. Then $N(k)$ is the number of vertices of degree 4 in $B_x(k)$, and $A(k)$ is the total number of vertices in $B_x(k)$. If we write $a_k$ and $b_k$ for the number of vertices at distance exactly $k$ from $x$ which have degree 4 or 2 respectively, we get $a_0 = 1$, and $b_{2l+1} = a_{2l+2} = 4cdot3^l$ and $b_{2l} = a_{2l+1} = 0$ for $l geq 0$. Note that
      $$frac{N(k)}{A(k)} = frac{sum_{i leq k} a_i}{sum_{i leq k} a_i + b_i}$$
      and if I'm not mistaken, plugging in the above values gives a limit of $frac 12$ for the subsequence of even $k$, and $frac 14$ for the subsequence of odd $k$.






      share|cite|improve this answer









      $endgroup$
















        5












        5








        5





        $begingroup$

        It is possible that the limit does not exist at all: Consider the free group on two generators acting on the $(4,2)$-biregular tree in the obvious way. This action is free and has 3 orbits (one containing all vertices of degree 4, and the other two containing "half" of the vertices of degree 2).



        Let $x$ be a vertex of degree $4$. Then $N(k)$ is the number of vertices of degree 4 in $B_x(k)$, and $A(k)$ is the total number of vertices in $B_x(k)$. If we write $a_k$ and $b_k$ for the number of vertices at distance exactly $k$ from $x$ which have degree 4 or 2 respectively, we get $a_0 = 1$, and $b_{2l+1} = a_{2l+2} = 4cdot3^l$ and $b_{2l} = a_{2l+1} = 0$ for $l geq 0$. Note that
        $$frac{N(k)}{A(k)} = frac{sum_{i leq k} a_i}{sum_{i leq k} a_i + b_i}$$
        and if I'm not mistaken, plugging in the above values gives a limit of $frac 12$ for the subsequence of even $k$, and $frac 14$ for the subsequence of odd $k$.






        share|cite|improve this answer









        $endgroup$



        It is possible that the limit does not exist at all: Consider the free group on two generators acting on the $(4,2)$-biregular tree in the obvious way. This action is free and has 3 orbits (one containing all vertices of degree 4, and the other two containing "half" of the vertices of degree 2).



        Let $x$ be a vertex of degree $4$. Then $N(k)$ is the number of vertices of degree 4 in $B_x(k)$, and $A(k)$ is the total number of vertices in $B_x(k)$. If we write $a_k$ and $b_k$ for the number of vertices at distance exactly $k$ from $x$ which have degree 4 or 2 respectively, we get $a_0 = 1$, and $b_{2l+1} = a_{2l+2} = 4cdot3^l$ and $b_{2l} = a_{2l+1} = 0$ for $l geq 0$. Note that
        $$frac{N(k)}{A(k)} = frac{sum_{i leq k} a_i}{sum_{i leq k} a_i + b_i}$$
        and if I'm not mistaken, plugging in the above values gives a limit of $frac 12$ for the subsequence of even $k$, and $frac 14$ for the subsequence of odd $k$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 11 hours ago









        Florian LehnerFlorian Lehner

        54138




        54138






























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