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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]

For independent bond percolation on ${\mathbb{Z}}^d$, let $\tau (x,y)={\mathbb{P}}_{p_c}(x\leftrightarrow y)$ be the critical percolation two-point connection probability. The percolation triangle condition istranslation invariance makes this a threefold discrete convolution evaluated at zero. The factors describe connections between three positions; their product is a diagrammatic weight, rather than the probability of three independent events in a single configuration.

The condition controls the intersection corrections that separate percolation clusters from a simple branching approximation. In the homogeneous setting, it implies the mean-field percolation exponents value $\gamma =1$ for the divergence of percolation susceptibility. It must be established for the particular model; lace expansion provides one method in suitable high-dimensional models. The condition is therefore a useful bridge between spatial connectivity estimates and critical behaviour, rather than an assumption automatically satisfied at criticality.^[ref]