First let me define what is the "connective constant" of a two dimensional lattice. Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. Then the limit (which can be shown to exist) $$ \mu := \lim_{n \rightarrow \infty } c_{n}^{\frac{1}{n}} $$ is the connective constant of the lattice.

My question is the following: does this definition also make sense on any periodic tiling of the plane? To clarify the difference, a lattice is a discrete subgroup of $\mathbb{R}^2$. So I think a lattice does give a tiling of the plane, but not the other way round.

My second question is the following: If $\mu$ does make sense for any tiling, then are there any conjectures for what this $\mu$ could be for any of the tilings? Presently $\mu$ is known for the Hexagonal Lattice (Smirnov) and conjectured for a few other lattices. Are there any conjectures for tilings that are not lattices?

$\textbf{Note}:$ For simplicity I was talking about $2$ dimensions, but I believe my question makes sense in any dimension.

First let me define what is the "connective constant" of a two dimensional lattice. Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. Then the limit (which can be shown to exist) $$ \mu := \lim_{n \rightarrow \infty } c_{n}^{\frac{1}{n}} $$ is the connective constant of the lattice.

My question is the following: does this definition also make sense on any periodic tiling of the plane? To clarify the difference, a lattice is a discrete subgroup of $\mathbb{R}^2$. So I think a lattice does give a tiling of the plane, but not the other way round.

$\textbf{Edit:}$ It seems from the comments I received that $\mu$ does make sense for any tiling. Is there any reason why $\mu$ has been studied only for a lattice and not for other tilings? Is there a reason why this is a "natural" question for lattices and an "artificial" question for other tilings.

Of course I could be wrong, i.e. this question has been studied for other lattices. In which case can someone point out any reference for this? Presently $\mu$ is known for the Hexagonal Lattice (Smirnov) and conjectured for a few other lattices. I am looking for conjectures/numerical simulations for tilings that are not lattices.

For simplicity I was talking about $2$ dimensions, but the question makes sense in any dimension.

First let me define what is the "connective constant" of a two dimensional lattice. Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. Then the limit (which can be shown to exist) $$ \mu := \lim_{n \rightarrow \infty } c_{n}^{\frac{1}{n}} $$ is the connective constant of the lattice.

My question is the following: does this definition also make sense on any periodic tiling of the plane? To clarify the difference, a lattice is a discrete subgroup of $\mathbb{R}^2$. So I think a lattice does give a tiling of the plane, but not the other way round.

As an example, consider the tiling of $\mathbb{R}^2$ by $4g$-gon (which after identification gives a genus $g$ surface).

My second question is the following: If $\mu$ does make sense for any tiling, then are there any conjectures for what this $\mu$ could be for any of the tilings? Presently $\mu$ is known for the Hexagonal Lattice (Smirnov) and conjectured for a few other lattices. Are there any conjectures for tilings that are not lattices?

$\textbf{Note}:$ For simplicity I was talking about $2$ dimensions, but I believe my question makes sense in any dimension.

First let me define what is the "connective constant" of a two dimensional lattice. Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. Then the limit (which can be shown to exist) $$ \mu := \lim_{n \rightarrow \infty } c_{n}^{\frac{1}{n}} $$ is the connective constant of the lattice.

My question is the following: does this definition also make sense on any periodic tiling of the plane? To clarify the difference, a lattice is a discrete subgroup of $\mathbb{R}^2$. So I think a lattice does give a tiling of the plane, but not the other way round.

My second question is the following: If $\mu$ does make sense for any tiling, then are there any conjectures for what this $\mu$ could be for any of the tilings? Presently $\mu$ is known for the Hexagonal Lattice (Smirnov) and conjectured for a few other lattices. Are there any conjectures for tilings that are not lattices?

$\textbf{Note}:$ For simplicity I was talking about $2$ dimensions, but I believe my question makes sense in any dimension.