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We consider the recent calculation gr-qc/0508124 of the graviton propagatorin the spinfoam formalism. Within the 3d toy model introduced in gr-qc/0512102,we test how the spinfoam formalism can be used to construct the perturbativeexpansion of graviton amplitudes. Although the 3d graviton is a pure gauge, onecan choose to work in a gauge where it is not zero and thus reproduce thestructure of the 4d perturbative calculations. We compute explicitly the nextto leading and next to next to leading orders, corresponding to one-loop andtwo-loop corrections. We show that while the first arises entirely from theexpansion of the Regge action around the flat background, the latter receivescontributions from the microscopic, non Regge-like, quantum geometry.Surprisingly, this new contribution reduces the magnitude of the next to nextto leading order. It thus appears that the spinfoam formalism is likely tosubstantially modify the conventional perturbative expansion at higher orders.  This result supports the interest in this approach. We then address a numberof open issues in the rest of the paper. First, we discuss the boundary stateansatz, which is a key ingredient in the whole construction. We propose a wayto enhance the ansatz in order to make the edge lengths and dihedral anglesconjugate variables in a mathematically well-defined way. Second, we show thatthe leading order is stable against different choices of the face weights ofthe spinfoam model; the next to leading order, on the other hand, is changed ina simple way, and we show that the topological face weight minimizes it.Finally, we extend the leading order result to the case of a regular, but notequilateral, tetrahedron.Comment: 24 pages, many figure

Towards the graviton from spinfoams: Higher order corrections in the 3D toy model