Abstract
In this paper we discuss a family of models of particle and energy diffusion on a one-dimensional lattice, related to those studied previously in Sasamoto and Wadati (Phys Rev E 58:4181–4190, 1998), Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 2017) and Povolotsky (J Phys A 46(46):465205, 2013) in the context of KPZ universality class. We show that they may be mapped onto an integrable $${\mathfrak {sl}}(2)$$ Heisenberg spin chain whose Hamiltonian density in the bulk has been already studied in the AdS/CFT and the integrable system literature. Using the quantum inverse scattering method, we study various new aspects, in particular we identify boundary terms, modeling reservoirs in non-equilibrium statistical mechanics models, for which the spin chain (and thus also the stochastic process) continues to be integrable. We also show how the construction of a “dual model” of probability theory is possible and useful. The fluctuating hydrodynamics of our stochastic model corresponds to the semiclassical evolution of a string that derives from correlation functions of local gauge invariant operators of $$\mathcal {N}=4$$ super Yang–Mills theory (SYM), in imaginary-time. As any stochastic system, it has a supersymmetric completion that encodes for the thermal equilibrium theorems: we show that in this case it is equivalent to the $${\mathfrak {sl}}(2|1)$$ superstring that has been derived directly from $$\mathcal {N}=4$$ SYM.
Highlights
1.1 The settingStochastic systems may be described by a linear operator – called here the Hamiltonian H – that describes the infinitesimal evolution of the probability distribution
Seen as a stochastic operator the bulk Hamiltonian density is related to the q-Hahn Asymmetric Zero Range Process (AZRP) of Barraquand and Corwin [2], which in turn generalizes the q-Hahn Totally Asymmetric Zero Range Process (TAZRP) introduced in [3] by allowing jumps in both directions
In contrast to the stochastic R-matrix approach, where the particle process is described in terms of two commuting Hamiltonians which generate left and right moving particles separately, we find that the standard nearest-neighbor Hamiltonian of the non-compact spin chain yields immediately a process of particles hopping to the left and the right
Summary
Stochastic systems may be described by a linear operator – called here the Hamiltonian H – that describes the infinitesimal evolution of the probability distribution. Another popular system is the Kipnis-Marchioro-Presutti (KMP) model [4], where each pair of neighboring sites exchange randomly their energies It was realized years ago [5, 6] that it may be put in direct relation with a chain of sl(2) ‘spins’ H = i 2S0[i]S0[i+1] − S+[i]S−[i+1] − S−[i]S+[i+1] satisfying at each site of the spin chain the commutation relations [S0, S±] = ±S±, [S+, S−] = −2S0. Non-compact integrable spin chains were studied in theoretical physics, in relation to high energy QCD [8,9,10], N = 4 super Yang-Mills theory (N = 4 SYM) [11,12,13] and the AdS/CFT dual string theory limit considered in [14,15,16] Their interpretation as stochastic processes has not been studied.
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