= Percolation susceptibility exponent
{title2=$\gamma$}
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, then
$$
\chi(p)=(p_c-p)^{-\gamma+o(1)}\qquad(p\uparrow p_c).
$$
Equivalently, $\log\chi(p)/\log(1/(p_c-p))\to\gamma$. 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>, $d\ge3$, there are $d(d-1)^{n-1}$ vertices at distance $n$ from the root and a unique path to each. For $p<1/(d-1)$,
$$
\chi(p)=1+\sum_{n\ge1}d(d-1)^{n-1}p^n=1+\frac{dp}{1-(d-1)p}.
$$
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.<Tree graph inequalities and critical behavior in percolation models by Michael Aizenman and Charles M. Newman>[Abstract and susceptibility criterion]{ref}
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