Dilution-induced criticality is discussed using exact and scaling descriptions. A simple exact argument is used to obtain the scaling form for the density of states in dilute chains. For higher-dimensional systems it is shown that the condition for the existence of a 'fraction' edge is equivalent to the violation of a scaling relation between the fracton dimension of the infinite cluster, d and various static exponents. In addition the result dd/d f is obtained for the fracton dimensionality of the finite clusters near the percolation threshold; d f is the fractal dimension of the infinite cluster.
The equality φ = γ between the exponent φ, which describes crossover from zero to finite-random-field critical behaviour, and the order parameter susceptibility exponent, γ, is shown to be replaced by φ > γ when the random exchange dominates. Rough renormalization group estimates yield φ/γ ≃ 1.1 for three-dimensional random Ising models. The implications to experiments on dilute antiferromagnets in uniform fields are discussed. Similarly to spin glasses, the nonlinear susceptibility of the latter is shown to diverge as ( T - T N ) -0.7 .
A general method is given whereby
m-connectedness correlation functions can be studied in the percolation problem. The method for
m=1 involves calculating the kth power of the correlation function $ 〈\ensuremath{\sigma}(x)\ensuremath{\sigma}({x}^{\ensuremath{'}})〉 $ for a randomly dilute Ising model at nonzero temperature and subsequently averaging over configurations. The final step is to take the limit $k\ensuremath{\rightarrow}0$. This method is tested by reproducing the standard results for the percolation problem from an extension of the calculation of Stephen and Grest. For
m>1 an additional "color" index is introduced and a Hamiltonian is constructed in which different colors repel one another, thereby giving an exact prescription for
m-connectedness. Order parameters for
m-connectedness are identified. The
m=2 order parameter couples through a trilinear term to the
m=1 order parameter. The main result is that ${\ensuremath{\beta}}^{(m)}$ the exponent for
m-connectedness is given by ${\ensuremath{\beta}}^{(m)}=m\ensuremath{\beta}+\ensuremath{\nu}{\ensuremath{\psi}}^{(m)}$, where $\ensuremath{\beta}$ and $\ensuremath{\nu}$ are the usual exponents for percolation and ${\ensuremath{\psi}}^{(m)}$ is a new crossover exponent which, to lowest order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, is given by ${\ensuremath{\psi}}^{(m)}=\frac{m(m\ensuremath{-}1){\ensuremath{\epsilon}}^{2}}{49}$. This result implies that the fractal dimensionality of the biconnected part of the critically percolating cluster is given in terms of the percolation critical exponents as $\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}\ensuremath{-}{\ensuremath{\psi}}^{(2)}$. If "nodes" are defined as triconnected points, then ${\ensuremath{\beta}}^{(3)}$ is the critical exponent associated with their density in the infinite cluster. We also discuss evidence that the "node-link" model of Skal-Shklovskii-de Gennes breaks down for
d less than some critical value $\stackrel{^}{d}$. Numerically we adduce evidence that $\stackrel{^}{d}$ may be larger than 3.
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.
The scaling laws derived by Grinstein (1976) for the random-field Ising model (RFIM) are rederived on the assumption that the transition is second order and that the critical behaviour is controlled by a zero-temperature fixed point. The scaling laws involve three independent exponents nu , eta and gamma , the last appearing in a modified hyperscaling relation, 2- alpha =(d-y) nu . It is argued that such hyperscaling modifications are a general feature of phase transitions controlled by zero-temperature fixed points. Explicit evaluation of the RFIM exponents in d=2+ epsilon dimensions, yields, to order epsilon , 1/ nu = epsilon , eta =1- epsilon /2 and y=1+ epsilon /2. The exponent nu is different from that of the pure model in (d-y) dimensions implying that no exact 'dimensional reduction' is possible near two dimensions.
The crossover between the surface and bulk values of the order-parameter critical exponent (\ensuremath{\beta}) is investigated for the site-percolation model in a rectangular geometry (L\ifmmode\times\else\texttimes\fi{}M, L\ensuremath{\ll}M) with free boundary conditions. Due to the reduction in the connectivity, when the space is cut by the surfaces of the sample which cannot be penetrated by the percolating clusters, \ensuremath{\beta} decays exponentially from the surface value (${\mathrm{\ensuremath{\beta}}}_{\mathit{s}}$\ensuremath{\simeq}0.40) to the bulk value (${\mathrm{\ensuremath{\beta}}}_{\mathit{b}}$=5/36). This decay is interpreted in terms of multiscaling which implies that each row of the percolating cluster (taken in the M direction) has its own scaling and consequently a different fractal dimension.