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A Griffiths phase has recently been observed by Monte Carlo simulations inthe 2D $q$-state Potts model with strongly correlated quenched randomcouplings. In particular, the magnetic susceptibility was shown to divergealgebraically with the lattice size in a broad range of temperatures. However,only relatively small lattice sizes could be considered so one can wonderwhether this Griffiths phase will not shrink and collapse into a single point,the critical point, as the lattice size is increased to much larger values. Inthis paper, the 2D eight-state Potts model is numerically studied for fourdifferent disorder correlations. It is shown that the Griffiths phase cannot beexplained as a simple spreading of local transition temperatures caused bydisorder fluctuations. As a consequence, the vanishing of the latter in thethermodynamic limit does not necessarily imply the collapse of the Griffithsphase into a single point. In contrast, the width of the Griffiths phase iscontrolled by the disorder strength. However, for disorder correlationsdecaying slower than $1/r$, no cross-over to a more usual critical behaviorcould be observed as this strength is tuned to weaker values.

Stability of the Griffiths phase in the 2D Potts model with correlated disorder