Crossover from extensive to nonextensive behavior driven by long-range d=1 bond percolation

We present a Monte Carlo study of a linear chain (d=1) with long-range bonds whose occupancy probabilities are given by pij=p/rijα(0⩽p⩽1;α⩾0) where rij=1,2,… is the distance between sites. The α→∞(α=0) corresponds to the first-neighbor (“mean field”) particular case. We exhibit that the order parameter P∞ equals unity ∀p>0 for 0⩽α⩽1, presents a familiar behavior (i.e., 0 for p⩽pc(α) and finite otherwise) for 1<α<2, and vanishes ∀p<1 for α>2. Our results confirm recent conjecture, namely that the nonextensive region (0⩽α⩽1) can be meaningfully unfolded, as well as unified with the extensive region (α>1), by exhibiting P∞ as a function of p∗ where (1−p∗)=(1−p)N∗(N∗≡(N1−α/d−1)/(1−α/d),N being the number of sites of the chain). A corollary of this conjecture, now numerically verified, is that pc∝(α−1) in the α→1+0 limit.