Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in ${𝐙}^d$

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1. [BJS] V. BACH, T. JECKO AND J. SJÖSTRAND, Correlation asymptotics of classical lattice spin systems with nonconvex Hamilton function at low temperature, Preprint, 1998. Zbl1021.82002
2. [Be] F. A. BEREZIN, The Method of Second Quantization, 1966, Academic Press, New York. Zbl0151.44001MR34 #8738
3. [BGV] N. BERLINE, E. GETZLER AND M. VERGNE, Heat kernels and Dirac operators, Springer-Verlag, 1992. Zbl0744.58001MR94e:58130
4. [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. Zbl0644.60057MR89c:82050
5. [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. Zbl1223.81146
6. [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. Zbl1223.82010
7. [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. MR87k:82008
8. [CFKS] H. L. CYCON, R. G. FROESE, W. KIRSCH AND B. SIMON, Schrödinger Operators, Springer-Verlag, 1987. Zbl0619.47005
9. [EW1] J.-P ECKMANN AND E. C. WAYNE, Liapunov spectra for infinite chaines of non-linear oscillators, J. Stat. Phys. 50, 853-878, 1987. Zbl1084.37500
10. [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. Zbl0673.60066MR90h:60097
11. [Ef] K. B. EFFETOV, Supersymmetry and the theory of disordered metals, Adv. Phys. 32, 1983, 53-127. MR85b:82014
12. [F] W. FELLER, An Introduction to Probability Theory and Its Applications, 1966, John-Wiley and Sons. Zbl0138.10207MR35 #1048
13. [FFS] R. FERNANDEZ, J. FRÖHLICH, A. SOKAL, Random Walks, Critical phenomena and Triviality in Quantum Field Theory, 1992, Springer-Verlag. Zbl0761.60061
14. [FKG] C. M. FORTUIN, P. W. KASTELEYN, J. GINIBRE, Correlation inequalities on some partially ordered sets, Commun. Math. Phys., 89-103 22, 1971. Zbl0346.06011MR46 #8607
15. [Fr] M. FRIEDLIN, Functional Integration and Partial Differential Equations, Ann. of Math. Studies 109, 1985, Princeton University Press. Zbl0568.60057MR87g:60066
16. [H] L. HÖRMANDER, Introduction to Complex Analysis in Several Variables, Elsevier (North-Holland Mathematical Library, Vol. 7), Amsterdam, 3rd edition, 1990. Zbl0685.32001MR91a:32001
17. [HS1] B. HELFFER AND J. SJÖSTRAND, Multiple wells in the semi-classical limit I, Commun. PDE 9 (4), 337-408, 1984. Zbl0546.35053
18. [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). Zbl0946.35508MR95g:82022
19. [IS] J. Z. IMBRIE AND T. SPENCER, Diffusion of directed polymers in a random environment, J of Stat. Phys., 609-626, 1988. Zbl1084.82595MR90m:60122
20. [Jo] J. JOHNSEN, On spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb inequality, Aalborg University preprint, 1998. Zbl1023.58012
21. [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. Zbl0709.60105MR92b:82076
22. [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. Zbl0635.60077
23. [Kom] T. KOMOROWSKIBrownian motion in a Poissonian obstacle field, Séminaire N. Bourbaki, 853, 1998. Zbl0964.60091
24. [La] Intersection of Random WalksG. F. LAWLER, Birkhäuser, 1991. 
25. [Li] T. M. LIGGETT, Interacting Particle Systems, 1985, Springer-Verlag. Zbl0559.60078MR86e:60089
26. [MS] The self-Avoiding WalkN. MADRAS, G. SLADE, Birkhäuser, 1993. Zbl0780.60103MR94f:82002
27. [PF] Spectra of Random and Almost Periodic OperatorsL. PASTUR AND A. FIGOTIN, Springer, 1992. Zbl0752.47002MR94h:47068
28. [Po] T. POVEL, The one dimensional annealed δ-Lyapunov exponent, Ann. IHP, Probab. et Stat., 61-72, 1998 34. Zbl0903.60093MR2000c:60167
29. [P-L] P. J. PAES-LEME, Ornstein-Zernike and analyticity properties for classical lattice spin systems, Ann. Phys. 115, 367, 1978. Zbl0397.42011MR80a:81044
30. [Sch] R. S. SCHOR, The particle structure of v-dimensional Ising models at low temperature, Commun. Math. Phys. 59, 213, 1978. MR58 #4122
31. [Sin] YA. G. SINAI, A remark concerning random walk with random potentials, Fund. Math 147, 173-180, 1995. Zbl0835.60062MR96k:60185
32. [Sj1] J. SJÖSTRAND, Correlation asymptotics and Witten Laplacians, Algebra and Analysis, 1996 8. Zbl0877.35084
33. [Sj2] J. SJÖSTRAND, (In preparation). 
34. [Spi] F. SPITZER, Principles of Random Walks, D. Van Nostrand Company, Inc, 1964. Zbl0119.34304MR30 #1521
35. [Spe] T. SPENCER, (Private conversations). 
36. [Sz1] A. S. SZNITMAN, Brownian motion, Obstacles and Random media, Springer Monograph in Mathematics, 1998. Zbl0973.60003MR2001h:60147
37. [Sz2] A. S. SZNITMAN, Shape theorem, Lyapunov exponents and large deviations for Brownian motion in a Poissonian potential, Commun. Pure Appl. Math, 1994. Zbl0814.60022
38. [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. Zbl0941.47033
39. [SW2] J. SJÖSTRAND AND W. M. WANG, Decay of averaged Green's functions-a direct approach, Ann. Scient. Éc. Norm. Sup. 32, 1999. Zbl0934.35036
40. [V] T. VORONOV, Geometric Integration Theory on Supermanifolds, Mathematical Physics Review, USSR Academy of Sciences, 1993, Moscow. Zbl0839.58014
41. [Wa] W-M WANG, Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in Zd, Orsay Preprint, 1999. 
42. [W] E. WITTEN, Supersymmetry and Morse theory, J. Diff. Geom 17, 661-692, 1982. Zbl0499.53056MR84b:58111
43. [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. Zbl0938.60098MR99f:60172