THE QUANTUM SYMMETRIC EXCLUSION PROCESS AND FREE PROBABILITY

Work package:
The quantum symmetric exclusion process has been introduced by Bernard and Jin in [BJ21] as a quantum analogue to the classical symmetric simple exclusion process. The model is given by Hamiltonian 𝑑𝐻𝑡 = ∑ 𝑐𝑗+1∗ 𝑐𝑗𝑁−1𝑗=0𝑑𝑊𝑡𝑗 + 𝑐𝑗∗𝑐𝑗+1𝑑𝑊𝑡𝑗̅̅̅̅ where 𝑊𝑡𝑗, 0 ≤ 𝑗 ≤ 𝑁 − 1 are independent complex Brownian motions. The Hamiltonian describes the motion of quantum particles which can jump from their site to one of the adjacent sites. The system has a boundary at the points 0 and N-1, where the system is in contact with a reservoir, with which it can exchange particles. Above, 𝑐𝑖∗ and 𝑐𝑖 are a family of fermionic creation and annihilation operators, satisfying the usual anti- commutation relations 𝑐𝑖𝑐𝑗∗ + 𝑐𝑗∗𝑐𝑖 = δ𝑖𝑗 . When studying the fermionic2-point correlation function of the density matrix describing the state of the system at large time 𝐺𝑖𝑗 =lim𝑡→∞ 𝑇𝑟(𝑐𝑗∗𝑐𝑖𝜌𝑡), it has been observed that free-cumulant-like quantities appear [NS06]. Interestingly, the connection between the quantum symmetric exclusion processes and free probability have been very recently [BBBH23] used to give a new description of the large deviation function of the classical symmetric simple exclusion process [Lig99]. The candidate will study the model of the QSEP and its relation with the theory of free probability. It is quite remarkable that combinatorial objects which are intimately related to the non-commutative notion of independence called freeness appear in this situation. Usually, free probability is connected to quantum mechanics via the use of random matrices; this new occurrence is quite surprising and deserves a better explanation from first principles.