Next nearest neighbour Ising models on random graphs

CONNECTIVITY OF RANDOM k-NEAREST-NEIGHBOUR GRAPHS

LetP be a Poisson process of intensity one in a squareSn of arean. We construct a random geometric graph Gn,k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn,k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that Gn,k is connected tends to 1 as n → ∞. They conject...

On the Connectivity of Random k-th Nearest Neighbour Graphs

Continuous- and First-order Phase Transitions in Ising Antiferromagnets with Next-nearest- Neighbour Interactions

Both the square and triangular Ising antiferromagnetic systems with next-nearest neighbour interactions are studied by a new efficient entropic scheme. Using the energy distribution and order parameter’s fourth order cumulant it is illustrated that the two models behave quite differently. For the square model we establish strong numerical evidence showing a continuous transition scaling, wherea...

Crossovers in the two-dimensional Ising spin glass with ferromagnetic next-nearest-neighbour interactions

By means of extensive computer simulations we analyse in detail the twodimensional ±J Ising spin glass with ferromagnetic next-nearest-neighbour interactions. We found a crossover from ferromagnetic to ‘spin glass’-like order both from numerical simulations and analytical arguments. We also present proof of a second crossover from the ‘spin glass’ behaviour to a paramagnetic phase for the large...

Interface tension of the square lattice Ising model with next-nearest-neighbour interactions

In a recent letter, Zandvliet (Europhys. Lett., 73 (2006) 747) presented a simple derivation of an analytical expression for the interface free energy in the (10) direction of the Ising model on a square lattice with nearestand next-nearest-neighbour couplings, reproducing the famous exact Onsager formula in the special case of only nearest-neighbour interactions. By comparing the resulting tra...