# Heuristics of Cohen-Lenstra-Martinet

Let $$h(d)$$ be the class numbers of the real quadratic field $$\mathbb{Q}(\sqrt{d})$$. There are some heuristics of Cohen-Lenstra-Martinet about divisibility of class numbers. Do they say anything about the density of primes $$p \equiv 1 \pmod{8}$$ for which $$h(2p)$$ is a multiple of $$4.$$?

The answer to the question as stated is "no". The original Cohen—Lenstra—Martinet heuristics say nothing about the $$2$$-Sylow subgroup of the class group of a quadratic field.
These heuristics were later modified by Gerth to take into account genus theory, see F. Gerth III, Densities for ranks of certain parts of $$p$$-class groups, Proc. AMS 99, and Gerth's modification does make predictions about the average behaviour of the $$2$$-Sylow of a "random" real quadratic number field. However, the fields you are considering form a density $$0$$ subfamily of the family of all real quadratic fields, so again, Gerth's heuristic says nothing about those.
However, a complete answer to the question "when is $$h(2p)$$ divisible by $$4$$" is known. This happens if and only if $$p$$ splits completely in $$K=\mathbb{Q}(\zeta_8,\sqrt{1+i})$$. One says that "$$K$$ is a governing field for the $$4$$-rank of class groups in this family". Since $$K$$ is a field of degree $$8$$, Chebotarev implies that the density of $$p$$ for which $$h(2p)$$ is divisible by $$4$$ is $$1/8$$.
For a summary of what is known about $$4$$- and $$8$$-ranks of these class groups and their governing fields, together with many references, see P. Koymans and D. Milovic, Spins of prime ideals and the negative Pell equation $$x^2 − 2py^2 = −1$$, Compos. Math. 155.