The Annals of Applied Probability

Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models

Pietro Caputo and Fabio Martinelli

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Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, we consider an anisotropic simple exclusion process with $N$ particles in a rectangle of $\bbZ^2$. Every particle at row $h$ tries to jump to an arbitrary empty site at row $h\pm 1$ with rate $q^{+ 1}$, where $q\in (0,1)$ is a measure of the drift driving the particles toward the bottom of the rectangle. We prove that the spectral gap of the generator is uniformly positive in $N$ and in the size of the rectangle. The proof is inspired by a recent interesting technique envisioned by E. Carlen, M. C. Carvalho and M. Loss to analyze the Kac model for the nonlinear Boltzmann equation. We then apply the result to prove precise upper and lower bounds on the energy gap for the spin-$S$, $S\in \ov2\bbN$, $\mbox{\textit{{XXZ}}}$ chain and for the 111 interface of the spin-$S$ $\mbox{\textit{{XXZ}}}$\vspace{-1pt} Heisenberg model, thus generalizing previous results valid only for spin $\ov2$.

Article information

Ann. Appl. Probab., Volume 13, Number 2 (2003), 691-721.

First available in Project Euclid: 18 April 2003

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Primary: 60K40: Other physical applications of random processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J27: Continuous-time Markov processes on discrete state spaces 82B10: Quantum equilibrium statistical mechanics (general) 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Asymmetric simple exclusion diffusion limited chemical reactions spectral gap $XXZ$ model equivalence of ensembles


Caputo, Pietro; Martinelli, Fabio. Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models. Ann. Appl. Probab. 13 (2003), no. 2, 691--721. doi:10.1214/aoap/1050689600.

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  • [1] ALCARAZ, F. C. (1994). Exact steady states of asy mmetric diffusion and two-species annihilation with back reaction from the ground state of quantum spin models. Internat. J. Modern Phy s. B 8 3449-3461.
  • [2] ALCARAZ, F. C., SALINAS, S. R. and WRESZINSKI, W. F. (1995). Anisotropic ferromagnetic quantum domains. Phy s. Rev. Lett. 75 930-933.
  • [3] BENJAMINI, I., BERGER, N., HOFFMAN, C. and MOSSEL, E. (2002). Mixing time for biased shuffling card. Preprint.
  • [4] BOLINA, O., CONTUCCI, P., NACHTERGAELE, B. and STARR, S. (2000). Finite volume excitations of the 111 interface in the quantum XXZ model. Comm. Math. Phy s. 212 63-91.
  • [5] BRAMSON, M. and GRIFFEATH, D. (1980). Clustering and dispersion rates for some interacting particle sy stems on Z. Ann. Probab. 8.
  • [6] BRAMSON, M. and LEBOWITZ, J. (2001). Spatial structure in low dimensions for diffusion limited two-particle reactions. Ann. Appl. Probab. 11 121-181.
  • [7] CAPUTO, P. and MARTINELLI, F. (2002). Asy mmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model. Comm. Math. Phy s. 226 323-375.
  • [8] CARLEN, E., CARVALHO, M. C. and LOSS, M. (2001). Many-body aspects of approach to equilibrium. Seminaire: Equations aux derives partielles, 2000-2001. Exp. XIX, Semin. Equ. Deriv. Partielles. Ecole Poly tech., Palaiseau.
  • [9] DIACONIS, P. and SALOFF-COSTE, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695-750.
  • [10] DIACONIS, P. and SHAHSHAHANI, M. (1987). Time to reach stationarity in the Bernoulli- Laplace diffusion model. SIAM J. Math. Anal. 18 208-218.
  • [11] KOMA, T. and NACHTERGAELE, B. (1997). The spectral gap of the ferromagnetic XXZ chain. Lett. Math. Phy s. 40 1-16.
  • [12] KOMA, T., NACHTERGAELE, B. and STARR, S. (2001). The spectral gap for the ferromagnetic spin-J XXZ chain. Preprint. Available at
  • [13] LU, S. T. and YAU, H.-T. (1993). Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dy namics. Comm. Math. Phy s. 156 399-433.
  • [14] NACHTERGAELE, B. (2000). Interfaces and droplets in quantum lattice models. Preprint. Available at
  • [15] STARR, S. (2001). Some properties of the low lying spectrum of the ferromagnetic, quantum XXZ Heisenberg model. Ph.D. dissertation, Univ. California, Davis. Available at