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We call a finite set ${\mathcal{D}}\subset {\Bbb Z}^s$ a {\it (self-affine) tile digit set} with respect to an expanding integral matrix ${\bf A}$ if the self-affine set $T({\bf A}, \D)$ is a tile in ${\Bbb R}^s$. It has been a widely open problem to characterize the tile digit sets for a given ${\bf A}$. While there are substantial investigations on ${\Bbb R}$, there is no result on ${\Bbb R}^s$ other than the case where $|\det {\bf A}| =p$ with $p$ a prime. In this paper, we make an initiation to study a basic case ${\bf A} = p{\bf I}_2$ in ${\Bbb R}^2$. We characterize the tile digit sets by making use of the zeros of the mask polynomial of ${\mathcal{D}}$ associated with a tile criterion of Kenyon [K], together with a recent result of Iosevich {\it et al} on factorization of sets in ${\Bbb Z}_p \times {\Bbb Z}_p$ [IMP].

In a previous paper, we studied certain random walks on the hyperbolic graphs $X$ associated with the self-similar sets $K$, and showed that the discrete energy ${\mathcal E}_X$ on $X$ has an induced energy form ${\mathcal E}_K$ on $K$. The domain of ${\mathcal E}_K$ is a Besov space $\Lambda^{\alpha, \beta/2}_{2,2}$ where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ is a parameter determined by the ``return ratio" of the random walk. In this paper, we consider the functional relationship of ${\mathcal E}_X$ and ${\mathcal E}_K$. In particular, we investigate the critical exponents of the $\beta$ in the domain $\Lambda^{\alpha, \beta/2}_{2,2}$ in order for ${\mathcal E}_K$ to be a Dirichlet form. We provide some criteria to determine the critical exponents through the effective resistance of the random walk on $X$, and make use of certain electrical network techniques to calculate the exponents for some concrete examples.

Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich’s elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure $E$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_H X$ with the Gromov metric is H\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0 < \lambda < 1$ on $(X,E)$. We show that the Martin boundary ${\mathcal M}$ can be identified with $\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain precise estimates of the Martin kernel and the Na\"im kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate $\Theta (\xi,\eta) \asymp |\xi-\eta|^{-(\alpha+\beta)}$, where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ depends on $\lambda$. For suitable $\beta$, the kernel defines a regular non-local Dirichlet form on $K$. This extends the results of Kigami concerning random walks on certain trees with Cantor-type sets as boundaries.