In this ongoing joint work with Axel Saenz (Oregon), we are interested in the following interacting particle system. On a periodic ring of size L, there are N particles jumping according to some specific rules. At rate p ≥ 0, each particle tries to jump to the right by 1 and the jump happens only if the target site is not occupied (called TASEP dynamics). At rate q ≥ 0, each particle jumps to the left by 1, by propagating its jump if the target site is occupied (called Push dynamics). We want to understand the asymptotic behavior of such a system for large L and N with the ratio N/L fixed. 

It is an informal talk and all the necessary notions will be introduced gradually during the talk.