Consider a set of lattice sites, each with a set of adjacent sites (e.g. a graph) forming a -dimensional lattice. For … See more

The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a d-dimensional lattice, namely, Λ = Z , Jij = 1, h = 0.
In his 1924 PhD thesis, Ising solved the model for t… See more

One of Democritus' arguments in support of atomism was that atoms naturally explain the sharp phase boundaries observed in materials , as when ice melts to water or water turns to steam. His idea was that small chan… See more

The Ising model can often be difficult to evaluate numerically if there are many states in the system. Consider an Ising model with
L = |Λ|: the total number of sites on the lattice, σj … See more

The thermodynamic limit exists as long as the interaction decay is with α > 1.
• In the case of ferromagnetic interaction with 1 < α < 2, Dyson proved, by comparison with the hierarchical case, that there is phase tran… See more

• In the ferromagnetic case there is a phase transition. At low temperature, the Peierls argument proves positive magnetization for the nearest neighbor case and then, by the Griffiths inequality, also when longer range interaction… See more

In three as in two dimensions, the most studied case of the Ising model is the translation-invariant model on a cubic lattice with nearest-neighbor coupling in the zero magnetic field. Many theoreticians searched fo… See more