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Let $G_0(x)=G(x,0)$ be the Green's function of the simple symmetric random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G_0$ must always vary locally, i.e., whether

$$ \sum_{\substack{y\in\mathbb Z^d:\\ |y-x|=1}} |G_0(x)-G_0(y)| >0 $$

holds for all $x\in\mathbb Z^d$.

The claim seems intuitive in view of the random walk interpretation of $G_0$, i.e., $G_0(x)=\mathbb P(\exists n: S_n=x) G_0(0)$.

For large $x$, the claim can be verified from the asymptotics of $G_0$.

Let $G_0(x)=G(x,0)$ be the Green's function of the simple random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G_0$ must always vary locally, i.e., whether

$$ \sum_{\substack{y\in\mathbb Z^d:\\ |y-x|=1}} |G_0(x)-G_0(y)| >0 $$

holds for all $x\in\mathbb Z^d$.

The claim seems intuitive in view of the random walk interpretation of $G_0$, i.e., $G_0(x)=\mathbb P(\exists n: S_n=x) G_0(0)$.

For large $x$, the claim can be verified from the asymptotics of $G_0$.

Let $G_0(x)=G(x,0)$ be the Green's function of the simple symmetric random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G_0$ must always vary locally, i.e., whether

$$ \sum_{\substack{y\in\mathbb Z^d:\\ |y-x|=1}} |G_0(x)-G_0(y)| >0 $$

holds for all $x\in\mathbb Z^d$.

The claim seems intuitive in view of the random walk interpretation of $G_0$, i.e., $G_0(x)=\mathbb P(\exists n: S_n=x) G_0(0)$.

For large $x$, the claim can be verified from the asymptotics of $G_0$.

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Francesco Polizzi
Francesco Polizzi
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Let $G_0(x)=G(x,0)$ be the Green's function of the simple random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G_0$ must always vary locally, i.e., whether

$$ \sum_{\substack{y\in\mathbb Z^d:\\ |y-x|=1}} |G_0(x)-G_0(y)| >0. $$$$ \sum_{\substack{y\in\mathbb Z^d:\\ |y-x|=1}} |G_0(x)-G_0(y)| >0 $$

holds for all $x\in\mathbb Z^d$.

The claim seems intuitive in view of the random walk interpretation of $G_0$, i.e., $G_0(x)=\mathbb P(\exists n: S_n=x) G_0(0)$.

For large $x$, the claim can be verified from the asymptotics of $G_0$.

Let $G_0(x)=G(x,0)$ be the Green's function of the simple random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G_0$ must always vary locally, i.e., whether

$$ \sum_{\substack{y\in\mathbb Z^d:\\ |y-x|=1}} |G_0(x)-G_0(y)| >0. $$

holds for all $x\in\mathbb Z^d$.

The claim seems intuitive in view of the random walk interpretation of $G_0$, i.e., $G_0(x)=\mathbb P(\exists n: S_n=x) G_0(0)$.

For large $x$, the claim can be verified from the asymptotics of $G_0$.

Let $G_0(x)=G(x,0)$ be the Green's function of the simple random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G_0$ must always vary locally, i.e., whether

$$ \sum_{\substack{y\in\mathbb Z^d:\\ |y-x|=1}} |G_0(x)-G_0(y)| >0 $$

holds for all $x\in\mathbb Z^d$.

The claim seems intuitive in view of the random walk interpretation of $G_0$, i.e., $G_0(x)=\mathbb P(\exists n: S_n=x) G_0(0)$.

For large $x$, the claim can be verified from the asymptotics of $G_0$.

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Does the Green's function of the simple random walk on $\mathbb Z^d$ always vary locally?

Let $G_0(x)=G(x,0)$ be the Green's function of the simple random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G_0$ must always vary locally, i.e., whether

$$ \sum_{\substack{y\in\mathbb Z^d:\\ |y-x|=1}} |G_0(x)-G_0(y)| >0. $$

holds for all $x\in\mathbb Z^d$.

The claim seems intuitive in view of the random walk interpretation of $G_0$, i.e., $G_0(x)=\mathbb P(\exists n: S_n=x) G_0(0)$.

For large $x$, the claim can be verified from the asymptotics of $G_0$.