Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in 𝐙 d
Journées équations aux dérivées partielles (1999), article no. 18, 16 p.

By using a supersymmetric gaussian representation, we transform the averaged Green's function for random walks in random potentials into a 2-point correlation function of a corresponding lattice field theory. We study the resulting lattice field theory using the Witten laplacian formulation. We obtain the asymptotics for the directional Lyapunov exponents.

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     title = {Supersymmetry, {Witten} complex and asymptotics for directional {Lyapunov} exponents in $\mathbf {Z}^d$},
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     publisher = {Universit\'e de Nantes},
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Wei-Min Wang. Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in $\mathbf {Z}^d$. Journées équations aux dérivées partielles (1999), article  no. 18, 16 p. https://proceedings.centre-mersenne.org/item/JEDP_1999____A18_0/

[BJS] V. Bach, T. Jecko AND J. Sjöstrand, Correlation asymptotics of classical lattice spin systems with nonconvex Hamilton function at low temperature, Preprint, 1998. | Zbl

[Be] F. A. Berezin, The Method of Second Quantization, 1966, Academic Press, New York. | MR | Zbl

[BGV] N. Berline, E. Getzler AND M. Vergne, Heat kernels and Dirac operators, Springer-Verlag, 1992. | MR | Zbl

[BCKP] A. Bovier, M. Campanino, A. Klein, AND F. Perez, Smoothness of the density of states in the Anderson model at high disorder, 1988 114, 439-461, Commun. Math. Phys. | MR | Zbl

[BF1] J. Bricmont, J. Fröhlich, Statistical mechanics methods in particle structure analysis of lattice field theory [I] General theory, 1985 251 [FS13], 517-552, Nuclear Phys. B.

[BF2] J. Bricmont, J. Fröhlich, Statistical mechanics methods in particle structure analysis of lattice field theory [II] Scalar and surface models, 1985 98, 553-578, Commun. Math. Phys.

[CC] J. T. Chayes AND L. Chayes, Ornstein-Zernike behavior for self-Avoiding walks at all noncritical temperature, 1986 105, Commun. Math. Phys., 221-238. | MR

[CFKS] H. L. Cycon, R. G. Froese, W. Kirsch AND B. Simon, Schrödinger Operators, Springer-Verlag, 1987. | Zbl

[EW1] J.-P Eckmann AND E. C. Wayne, Liapunov spectra for infinite chaines of non-linear oscillators, J. Stat. Phys. 50, 853-878, 1987. | Zbl

[EW2] J.-P Eckmann AND E. C. Wayne, The largest Liapunov exponent for random matrices and directed polymers in a random environment, Commun. Math. Phys. 121, 147-175, 1989. | MR | Zbl

[Ef] K. B. Effetov, Supersymmetry and the theory of disordered metals, Adv. Phys. 32, 1983, 53-127. | MR

[F] W. Feller, An Introduction to Probability Theory and Its Applications, 1966, John-Wiley and Sons. | MR | Zbl

[FFS] R. Fernandez, J. Fröhlich, A. Sokal, Random Walks, Critical phenomena and Triviality in Quantum Field Theory, 1992, Springer-Verlag. | Zbl

[FKG] C. M. Fortuin, P. W. Kasteleyn, J. Ginibre, Correlation inequalities on some partially ordered sets, Commun. Math. Phys., 89-103 22, 1971. | MR | Zbl

[Fr] M. Friedlin, Functional Integration and Partial Differential Equations, Ann. of Math. Studies 109, 1985, Princeton University Press. | MR | Zbl

[H] L. Hörmander, Introduction to Complex Analysis in Several Variables, Elsevier (North-Holland Mathematical Library, Vol. 7), Amsterdam, 3rd edition, 1990. | MR | Zbl

[HS1] B. Helffer AND J. Sjöstrand, Multiple wells in the semi-classical limit I, Commun. PDE 9 (4), 337-408, 1984. | Zbl

[HS2] B. Helffer AND J. Sjöstrand, On the correlation for Kac like models in the convex case, J. Stat. Phys., 349-409, 1994 74 (1,2). | MR | Zbl

[IS] J. Z. Imbrie AND T. Spencer, Diffusion of directed polymers in a random environment, J of Stat. Phys., 609-626, 1988. | MR | Zbl

[Jo] J. Johnsen, On spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb inequality, Aalborg University preprint, 1998. | Zbl

[K] A. Klein, The supersymmetric replica trick and smoothness of the density of states for the random Schrödinger operators, 1990 51, Proceedings of Symposium in Pure Mathematics. | MR | Zbl

[KS] A. Klein, A. Spies, Smmothness of the density of states in the Anderson model on a one-dimensional strip, Ann. of Phys. 183, 1988, 352-398. | Zbl

[Kom] T. Komorowski Brownian motion in a Poissonian obstacle field, Séminaire N. Bourbaki, 853, 1998. | Numdam | Zbl

[La] Intersection of Random Walks G. F. Lawler, Birkhäuser, 1991.

[Li] T. M. Liggett, Interacting Particle Systems, 1985, Springer-Verlag. | MR | Zbl

[MS] The self-Avoiding Walk N. Madras, G. Slade, Birkhäuser, 1993. | MR | Zbl

[PF] Spectra of Random and Almost Periodic Operators L. Pastur AND A. Figotin, Springer, 1992. | MR | Zbl

[Po] T. Povel, The one dimensional annealed δ-Lyapunov exponent, Ann. IHP, Probab. et Stat., 61-72, 1998 34. | Numdam | MR | Zbl

[P-L] P. J. Paes-Leme, Ornstein-Zernike and analyticity properties for classical lattice spin systems, Ann. Phys. 115, 367, 1978. | MR | Zbl

[Sch] R. S. Schor, The particle structure of v-dimensional Ising models at low temperature, Commun. Math. Phys. 59, 213, 1978. | MR

[Sin] Ya. G. Sinai, A remark concerning random walk with random potentials, Fund. Math 147, 173-180, 1995. | MR | Zbl

[Sj1] J. Sjöstrand, Correlation asymptotics and Witten Laplacians, Algebra and Analysis, 1996 8. | Zbl

[Sj2] J. Sjöstrand, (In preparation).

[Spi] F. Spitzer, Principles of Random Walks, D. Van Nostrand Company, Inc, 1964. | MR | Zbl

[Spe] T. Spencer, (Private conversations).

[Sz1] A. S. Sznitman, Brownian motion, Obstacles and Random media, Springer Monograph in Mathematics, 1998. | MR | Zbl

[Sz2] A. S. Sznitman, Shape theorem, Lyapunov exponents and large deviations for Brownian motion in a Poissonian potential, Commun. Pure Appl. Math, 1994. | Zbl

[SW1] J. Sjöstrand AND W. M. Wang, Supersymmetric measures and maximum priciples in the complex domain-decay of Green's functions, Ann. Scient. Éc. Norm. Sup. 32, 1999. | Numdam | Zbl

[SW2] J. Sjöstrand AND W. M. Wang, Decay of averaged Green's functions-a direct approach, Ann. Scient. Éc. Norm. Sup. 32, 1999. | Numdam | Zbl

[V] T. Voronov, Geometric Integration Theory on Supermanifolds, Mathematical Physics Review, USSR Academy of Sciences, 1993, Moscow. | Zbl

[Wa] W-M Wang, Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in Zd, Orsay Preprint, 1999.

[W] E. Witten, Supersymmetry and Morse theory, J. Diff. Geom 17, 661-692, 1982. | MR | Zbl

[Z] M. P. W. Zerner, Directional decay of the Green's function for a random nonnegative potential on Zd, The Annals of Applied Probability, 1998, 246-280. | MR | Zbl