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Is a random walk on an isoradial graph transient?

0

$begingroup$

Is the random walk on an isoradial graph defined in Chelkak and Smirnov (2011) (in pages 3-4) transient?

Let us define the random walk on an isoradial graph $Gamma$ starting from $x$ by,
beginequation
X_t = X_0 + sum_j=0^t-1 xi_X_j^(j),
endequation
where $X_0 = x$ and the increments $xi_u^(t)$ are independently and identically distributed for fixed vertex $u in Gamma$ and for all $t in mathbbN$. These are distributed according to,
beginequation
mathbbP(xi_u = u_k - u) = fractantheta_ksum_s=1^ntantheta_s,
endequation
for $u_k$ $sim$ $u$ are adjacent vertices in $Gamma$. The half-angle of the rhombus is denoted by $theta_k$. These are uniformly bounded from $0$ and $pi/2$.

I am hesitating whether $(X_t)$ is transient? For fixed $u_0 in Gamma$ and given the mesh size $delta > 0$, the free Green's function is finite: $G(u_0,u_0) = frac12pi(log delta - gamma_Euler -log 2) < infty$. Does this imply that $(X_t)$ is transient? In the light of Pólya's theorem this feels counter-intuitive. Since increments of $(X_t)$ have zero expectations, that is $mathbbE(Re xi_u) = 0 $, $mathbbE(Im xi_u) = 0 $, and $Gamma$ is embedded in $mathbbC simeq mathbbR^2$, one may expect that $(X_t)$ is recurrent.

Thank you for your time.

random-walk planar-graph

share|cite|improve this question

edited Mar 30 at 16:21

Marius

asked Mar 30 at 11:29

MariusMarius

13

$endgroup$

add a comment | 

0

$begingroup$

Is the random walk on an isoradial graph defined in Chelkak and Smirnov (2011) (in pages 3-4) transient?

Let us define the random walk on an isoradial graph $Gamma$ starting from $x$ by,
beginequation
X_t = X_0 + sum_j=0^t-1 xi_X_j^(j),
endequation
where $X_0 = x$ and the increments $xi_u^(t)$ are independently and identically distributed for fixed vertex $u in Gamma$ and for all $t in mathbbN$. These are distributed according to,
beginequation
mathbbP(xi_u = u_k - u) = fractantheta_ksum_s=1^ntantheta_s,
endequation
for $u_k$ $sim$ $u$ are adjacent vertices in $Gamma$. The half-angle of the rhombus is denoted by $theta_k$. These are uniformly bounded from $0$ and $pi/2$.

I am hesitating whether $(X_t)$ is transient? For fixed $u_0 in Gamma$ and given the mesh size $delta > 0$, the free Green's function is finite: $G(u_0,u_0) = frac12pi(log delta - gamma_Euler -log 2) < infty$. Does this imply that $(X_t)$ is transient? In the light of Pólya's theorem this feels counter-intuitive. Since increments of $(X_t)$ have zero expectations, that is $mathbbE(Re xi_u) = 0 $, $mathbbE(Im xi_u) = 0 $, and $Gamma$ is embedded in $mathbbC simeq mathbbR^2$, one may expect that $(X_t)$ is recurrent.

Thank you for your time.

random-walk planar-graph

share|cite|improve this question

edited Mar 30 at 16:21

Marius

asked Mar 30 at 11:29

MariusMarius

13

$endgroup$

add a comment | 

$begingroup$

Is the random walk on an isoradial graph defined in Chelkak and Smirnov (2011) (in pages 3-4) transient?

Let us define the random walk on an isoradial graph $Gamma$ starting from $x$ by,
beginequation
X_t = X_0 + sum_j=0^t-1 xi_X_j^(j),
endequation
where $X_0 = x$ and the increments $xi_u^(t)$ are independently and identically distributed for fixed vertex $u in Gamma$ and for all $t in mathbbN$. These are distributed according to,
beginequation
mathbbP(xi_u = u_k - u) = fractantheta_ksum_s=1^ntantheta_s,
endequation
for $u_k$ $sim$ $u$ are adjacent vertices in $Gamma$. The half-angle of the rhombus is denoted by $theta_k$. These are uniformly bounded from $0$ and $pi/2$.

I am hesitating whether $(X_t)$ is transient? For fixed $u_0 in Gamma$ and given the mesh size $delta > 0$, the free Green's function is finite: $G(u_0,u_0) = frac12pi(log delta - gamma_Euler -log 2) < infty$. Does this imply that $(X_t)$ is transient? In the light of Pólya's theorem this feels counter-intuitive. Since increments of $(X_t)$ have zero expectations, that is $mathbbE(Re xi_u) = 0 $, $mathbbE(Im xi_u) = 0 $, and $Gamma$ is embedded in $mathbbC simeq mathbbR^2$, one may expect that $(X_t)$ is recurrent.

Thank you for your time.

random-walk planar-graph

share|cite|improve this question

edited Mar 30 at 16:21

Marius

asked Mar 30 at 11:29

MariusMarius

13

$endgroup$

share|cite|improve this question

edited Mar 30 at 16:21

Marius

asked Mar 30 at 11:29

MariusMarius

13

share|cite|improve this question

edited Mar 30 at 16:21

Marius

asked Mar 30 at 11:29

MariusMarius

13

asked Mar 30 at 11:29

MariusMarius

13

asked Mar 30 at 11:29

MariusMarius

13