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We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.

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We show that for every g ≥ 2 g\geq 2 there exists a number c = c ( g ) > 0 c=c(g)>0 such that the smallest positive eigenvalue of a random closed 3-manifold M M of Heegaard genus g g is at most c ( g ) / v o l ( M ) 2 c(g)/{\mathrm {vol}}(M)^2 .