Melonic field theories are defined over the $p$-adic numbers with the help of a sign character. Our construction works over the reals as well as the $p$-adics, and it includes the fermionic and bosonic Klebanov-Tarnopolsky models as special cases; de
pending on the sign character, the symmetry group of the field theory can be either orthogonal or symplectic. Analysis of the Schwinger-Dyson equation for the two-point function in the leading melonic limit shows that power law scaling behavior in the infrared arises for fermionic theories when the sign character is non-trivial, and for bosonic theories when the sign character is trivial. In certain cases, the Schwinger-Dyson equation can be solved exactly using a quartic polynomial equation, and the solution interpolates between the ultraviolet scaling controlled by the spectral parameter and the universal infrared scaling. As a by-product of our analysis, we see that melonic field theories defined over the real numbers can be modified by replacing the time derivative by a bilocal kinetic term with a continuously variable spectral parameter. The infrared scaling of the resulting two-point function is universal, independent of the spectral parameter of the ultraviolet theory.
We formulate a Euclidean theory of edge length dynamics based on a notion of Ricci curvature on graphs with variable edge lengths. In order to write an explicit form for the discrete analog of the Einstein-Hilbert action, we require that the graph sh
ould either be a tree or that all its cycles should be sufficiently long. The infinite regular tree with all edge lengths equal is an example of a graph with constant negative curvature, providing a connection with $p$-adic AdS/CFT, where such a tree takes the place of anti-de Sitter space. We compute simple correlators of the operator holographically dual to edge length fluctuations. This operator has dimension equal to the dimension of the boundary, and it has some features in common with the stress tensor.
