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We investigate the geometric properties displayed by the magnetic patternsdeveloping on a two-dimensional Ising system, when a diffusive thermal dynamicsis adopted. Such a dynamics is generated by a random walker which diffusesthroughout the sites of the lattice, updating the relevant spins. Since thewalker is biased towards borders between clusters, the border-sites are morelikely to be updated with respect to a non-diffusive dynamics and therefore, weexpect the spin configurations to be affected. In particular, by means of thebox-counting technique, we measure the fractal dimension of magnetic patternsemerging on the lattice, as the temperature is varied. Interestingly, ourresults provide a geometric signature of the phase transition and they alsohighlight some non-trivial, quantitative differences between the behaviorspertaining to the diffusive and non-diffusive dynamics.

Fractal geometry of Ising magnetic patterns: signatures of criticality and diffusive dynamics