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Charles M. Newman studied critical behavior in percolation with Michael Aizenman. Their tree-diagram inequalities bound multipoint connection probabilities using two-point functions. A diagrammatic convergence criterion gives mean-field behavior for the percolation susceptibility exponent, connecting the geometry of large clusters with quantitative critical scaling.^[ref]

Michael Aizenman coauthored tree-diagram inequalities for percolation with Charles M. Newman. Their work controls multipoint connectivity and gives a condition for the percolation susceptibility exponent to take its mean-field value. The method relates large-scale critical behavior to convergence properties of diagrams built from pairwise connection probabilities.^[ref]

The percolation susceptibility exponent $\gamma$ describes how the expected finite-cluster size diverges on approaching the percolation critical probability from below. If $\chi (p)$ is the percolation susceptibility and a power law holds, thenEquivalently, $\log \chi (p)/\log (1/(p_c-p))\to \gamma$. A stronger statement $\chi (p)\sim A(p_c-p{)}^{-\gamma }$ also specifies an amplitude $A>0$; existence of either form needs justification for the model in question.

For bond percolation on an infinite $d$-regular tree, $d\ge 3$, there are $d(d-1{)}^{n-1}$ vertices at distance $n$ from the root and a unique path to each. For $p<1/(d-1)$,Thus $p_c=1/(d-1)$ and $\gamma =1$. This solvable geometry explains the simplest divergence, while more complicated lattices need not share its exponent. The tree-graph moment bound for a percolation cluster gives a route to proving the value one under suitable diagrammatic conditions.^[ref]