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Phase transition of the classical Ising model on the Sierpi{n}ski carpet, which has the fractal dimension $log_3^{~} 8 approx 1.8927$, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature $T_{rm c}^{~} = 1.4783(1)$. Position dependence of local functions is studied by means of impurity tensors, which are inserted at different locations on the fractal lattice. The critical exponent $beta$ associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas $T_{rm c}^{~}$ is not affected.
The feedback vertex number $tau(G)$ of a graph $G$ is the minimum number of vertices that can be deleted from $G$ such that the resultant graph does not contain a cycle. We show that $tau(S_p^n)=p^{n-1}(p-2)$ for the Sierpi{n}ski graph $S_p^n$ with $
In this paper, we investigate the existence of Sierpi{n}ski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpi{n}ski numbers and Riesel numbers of the form $binom{k}{
The Fokker--Planck equation describes the evolution of a probability distribution towards equilibrium--the flow parameter is the equilibration time. Assuming the distribution remains normalizable for all times, it is equivalent to an open hierarchy o
We study the ground state energy E_G(n) of N classical n-vector spins with the hamiltonian H = - sum_{i>j} J_ij S_i.S_j where S_i and S_j are n-vectors and the coupling constants J_ij are arbitrary. We prove that E_G(n) is independent of n for all n
The quantum O(N) model in the infinite $N$ limit is a paradigm for symmetry-breaking. Qualitatively, its phase diagram is an excellent guide to the equilibrium physics for more realistic values of $N$ in varying spatial dimensions ($d>1$). Here we in