The fractal dimensions of the largest clusters of like spins connected by nearest-neighbor bonds in the critical two-dimensional Q-state Potts model have been determined via Monte Carlo simulations to be
df=1.946(3), 1.922(5), and 1.909(8) for Q=2, 3, and 4, respectively. These dimensions are close to the magnetic tricritical exponents of the (Q-1)-state dilute Potts model (d-x=187/96=1.947..., 77/40=1.925, and 40/21=1.904... for Q-1=1, 2, and 3, respectively). The critical Q-state model and the tricritical (Q-1)-state model are related to a two-component Potts lattice gas; this model generates percolative susceptibilities of critical Potts clusters, and suggests a renormalization-group argument for
df=d-x.
Using formal arguments based on conformal invariance and on the connection between correlated-site percolation and the q-state Potts model with vacancies, we show that the exponents describing Ising clusters at Onsager's critical point are those of the tricritical q=1 Potts model. This implies, in particular, a fractal dimension d\ifmmode\bar\else\textasciimacron\fi{}=(187/96 and a percolative susceptibility exponent \ensuremath{\gamma}=(91/48, in good agreement with existing numerical estimates. This d\ifmmode\bar\else\textasciimacron\fi{} is also clearly supported by a new very accurate Monte Carlo finite-size scaling determination. We also conjecture an exponent
yJ=(13/24 controlling the crossover between clusters and droplets.