Motivated by the prospect of attaining Majorana modes at the ends of
nanowires, we analyze interacting Majorana systems on general networks and
lattices in an arbitrary number of dimensions, and derive various universal
spin duals. Such general complex Majorana architectures (other than those of
simple square or other crystalline arrangements) might be of empirical
relevance. As these systems display low-dimensional symmetries, they are
candidates for realizing topological quantum order. We prove that (a) these
Majorana systems, (b) quantum Ising gauge theories, and (c) transverse-field
Ising models with annealed bimodal disorder are all dual to one another on
general graphs. As any Dirac fermion (including electronic) operator can be
expressed as a linear combination of two Majorana fermion operators, our
results further lead to dualities between interacting Dirac fermionic systems.
The spin duals allow us to predict the feasibility of various standard
transitions as well as spin-glass type behavior in {\it interacting} Majorana
fermion or electronic systems. Several new systems that can be simulated by
arrays of Majorana wires are further introduced and investigated: (1) the {\it
XXZ honeycomb compass} model (intermediate between the classical Ising model on
the honeycomb lattice and Kitaev's honeycomb model), (2) a checkerboard lattice
realization of the model of Xu and Moore for superconducting (p+ip) arrays,
and a (3) compass type two-flavor Hubbard model with both pairing and hopping
terms. By the use of dualities, we show that all of these systems lie in the 3D
Ising universality class. We discuss how the existence of topological orders
and bounds on autocorrelation times can be inferred by the use of symmetries
and also propose to engineer {\it quantum simulators} out of these Majorana
networks.Comment: v3,19 pages, 18 figures, submitted to Physical Review B. 11 new
figures, new section on simulating the Hubbard model with nanowire systems,
and two new appendice
