A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice
نویسندگان
چکیده
In 1982, Tsallis derived a formula which proposed an exact value of 0.522372078 . . . for the bond percolation threshold of the kagome lattice. We use the substitution method, which is based on stochastic ordering, to compare the probability distribution of connections in the homogeneous bond percolation model on the kagome lattice to those of an exactly-solved inhomogeneous bond percolation model on the martini lattice. The bounds obtained are 0.522394 < pc(kagome) < 0.526750, where the lower bound shows that the value conjectured by Tsallis is incorrect.
منابع مشابه
Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis.
In the preceding paper, one of us (F. Y. Wu) considered the Potts model and bond and site percolation on two general classes of two-dimensional lattices, the triangular-type and kagome-type lattices, and obtained closed-form expressions for the critical frontier with applications to various lattice models. For the triangular-type lattices Wu's result is exact, and for the kagome-type lattices W...
متن کاملDetermination of the bond percolation threshold for the Kagomé lattice
The hull-gradient method is used to determine the critical threshold for bond percolation on the two-dimensional Kagomé lattice (and its dual, the dice lattice). For this system, the hull walk is represented as a self-avoiding trail, or mirror-model trajectory, on the (3,4,6,4)-Archimedean tiling lattice. The result pc = 0.524 4053 ± 0.000 0003 (one standard deviation of error) is not consisten...
متن کاملRigorous bounds relating bond percolation thresholds of two three-dimensional lattices
A percolation model is an infinite random graph model for phase transitions and critical phenomena. The percolation threshold corresponds to a phase transition point, such as a melting or freezing temperature. The exact value of the percolation threshold is not known for any three-dimensional percolation models, which are important for physical applications. Furthermore, rigorous bounds for the...
متن کاملUniversal crossing probabilities and incipient spanning clusters in directed percolation
Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. In a dynamical interpretation, the crossing probability is the probability that, on a system with size L, an epidemic spreading without immunization remains active at time t. Since the sys...
متن کاملCritical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. I. Closed-form expressions.
We consider the Potts model and the related bond, site, and mixed site-bond percolation problems on triangular-type and kagome-type lattices, and derive closed-form expressions for the critical frontier. For triangular-type lattices the critical frontier is known, usually derived from a duality consideration in conjunction with the assumption of a unique transition. Our analysis, however, is ri...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electr. J. Comb.
دوره 22 شماره
صفحات -
تاریخ انتشار 2015