I will introduce the (double) random current model, and briefly talk about its history
and more recent applications. I will then focus on some newly discovered properties of the two-dimensional model.
This includes the fact that probabilities of certain non-local events are given by determinants of local observables —
an analogous property of random groves and double dimers was discovered by Kenyon and Wilson in 2006.
I will also describe a dimer and spanning tree representation of double random currents and its connection to the conjecture of Wilson
which states that the contour lines of the critical XOR Ising model converge to the level lines of the Gaussian free field in the scaling limit.