Content-Type: multipart/mixed; boundary="-------------0308140733350" This is a multi-part message in MIME format. ---------------0308140733350 Content-Type: text/plain; name="03-368.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-368.comments" to appear in Markov chains and related fields ---------------0308140733350 Content-Type: text/plain; name="03-368.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-368.keywords" Lifshitz tails ---------------0308140733350 Content-Type: application/postscript; name="lifshitz.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lifshitz.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: lifshitz.dvi %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMMI12 CMR12 CMR10 CMBX10 CMMI10 CMBX12 CMMI8 %%+ CMSY8 CMR8 MSBM10 CMSY10 CMEX10 MSBM7 CMMI6 CMR6 CMTI12 CMSY6 CMSL12 %%EndComments 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 525003F3DBE5BF07B2E83E66B7F97DDD7CE0EEB75A78BD9227BF359D002B6ADB 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 6EC187E4B05182126BB0D0323D943170B795255260F9FD25F2248D04F45DFBFB DEF7FF8B19BFEF637B210018AE02572B389B3F76282BEB29CC301905D388C721 59616893E774413F48DE0B408BC66DCE3FE17CB9F84D205839D58014D6A88823 D9320AE93AF96D97A02C4D5A2BB2B8C7925C4578003959C46E3CE1A2F0EAC4BF 8B9B325E46435BDE60BC54D72BC8ACB5C0A34413AC87045DC7B84646A324B808 6FD8E34217213E131C3B1510415CE45420688ED9C1D27890EC68BD7C1235FAF9 1DAB3A369DD2FC3BE5CF9655C7B7EDA7361D7E05E5831B6B8E2EEC542A7B38EE 03BE4BAC6079D038ACB3C7C916279764547C2D51976BABA94BA9866D79F13909 95AA39B0F03103A07CBDF441B8C5669F729020AF284B7FF52A29C6255FCAACF1 74109050FBA2602E72593FBCBFC26E726EE4AEF97B7632BC4F5F353B5C67FED2 3EA752A4A57B8F7FEFF1D7341D895F0A3A0BE1D8E3391970457A967EFF84F6D8 47750B1145B8CC5BD96EE7AA99DDC9E06939E383BDA41175233D58AD263EBF19 AFC0E2F840512D321166547B306C592B8A01E1FA2564B9A26DAC14256414E4C8 42616728D918C74D13C349F4186EC7B9708B86467425A6FDB3A396562F7EE4D8 40B43621744CF8A23A6E532649B66C2A0002DD04F8F39618E4F572819DD34837 B5A08E643FDCA1505AF6A1FA3DDFD1FA758013CAED8ACDDBBB334D664DFF5B53 9560176676ABB71BBD0EE56B4CC492C0652750227CEC589B62962538637206C8 E9D4AE5031580E095B0989263584669928078E92CA026F4F4FD3598740539E0C 6F924A742EB0CB0EA3D23325B6CBB2A5188505D407310509BD37851E35572FCA 1E84E16AFB532EF89647676FC46B69B7FE9FBEE638446809D9745590E533F825 522F2D761C4F022F00393C0AE0C987B45D18DF0D199A981ACC6A72208E7244BD 8C55A240B637E37502972FD185486AA88E6A8F9836439CFF527E51A0E8014DFF 1D61781392F8957DFDECBC9193C54A5D4C763EC2C445A3AF19ADB7A395525986 A51DAB4D5F0EF6B799080E98EF048EEB4429CE058E05D76A296E34C80467C71F EE0D41A366CA38B6FA3C5EB7ABCF854FA7ADC4D6EC565D7D8C39D192FB3B06BB 93518623A1B1C56CFE095354EF43FE73C1DE5541A7AF68C51BA01019D7685596 D730BEE70268 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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(C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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b(asso)s(ciated)f(to) f(a)h(co)m(v)-5 b(arian)m(t)32 b(family)g(of)f(1)p Fo(D)j Fp(op)s(erators)d(are)h(the)g(IDS)f Fi(N)-24 2078 y Fp(and)26 b(the)g(Ly)m(apuno)m(v)h(exp)s(onen)m(t)g([PF],)g(the)f(fo)s(cus)f (here)h(b)s(eing)g(only)g(on)f(the)h(former.)41 b(One)26 b(of)f(the)h(equiv)-5 b(alen)m(t)-24 2198 y(de\014nitions)39 b(of)e(the)g(Stieltjes)h(function)g Fi(N)51 b Fp(is)38 b(form)m(ula)f(\(5\))g(in)g(Section)h(2.)57 b(Its)38 b(supp)s(ort)f(is)h(sp)s(ec)q(\()p Fo(H)3779 2213 y Fm(!)3829 2198 y Fp(\).)57 b(If)-24 2318 y Fg(p)39 b Fp(=)f Fo(\016)234 2333 y Fm(\033)281 2318 y Fp(,)i(then)f(it)g(is)g(straigh)m(t-forw)m (ard)g(to)f(calculate)h(the)g(asso)s(ciated)h(IDS,)f(denoted)g Fi(N)3377 2333 y Fm(\033)3424 2318 y Fp(,)h(b)m(y)f(means)h(of)-24 2439 y(Blo)s(c)m(h-Flo)s(quet)33 b(theory)-8 b(.)123 2606 y(Let)28 b(no)m(w)g Fo(E)563 2621 y Fm(\027)635 2606 y Fi(2)g Fj(R)g Fp(b)s(e)g(a)f(b)s(oundary)i(p)s(oin)m(t)f(of)f (sp)s(ec)q(\()p Fo(H)2125 2621 y Fm(\027)2168 2606 y Fp(\))h(and)g(sp)s(ec)q(\()p Fo(H)2720 2621 y Fm(!)2771 2606 y Fp(\).)41 b(Hence)30 b Fo(E)3235 2621 y Fm(\027)3306 2606 y Fp(can)e(b)s(e)g(either)h(the)-24 2727 y(b)s(ottom)35 b(or)g(the)h(top)f(of)g(the)g(sp)s(ectrum)i(or)e(the)g(b)s(oundary)h(p) s(oin)m(t)f(of)g(an)g Fc(internal)g Fp(sp)s(ectral)h(gap.)51 b(Within)-24 2847 y(an)m(y)34 b(gap,)e(the)h(IDS)f Fi(N)15 b Fp(\()p Fo(E)943 2862 y Fm(\027)986 2847 y Fp(\))33 b(is)g(constan)m(t)g(and)g(therefore)g(it)g(is)g(natural)f(to)g(study) 1235 3088 y Fo(\016)t Fi(N)15 b Fp(\()p Fo(\017)p Fp(\))55 b(=)g Fi(\006)17 b(N)e Fp(\()p Fo(E)1981 3103 y Fm(\027)2046 3088 y Fi(\006)22 b Fo(\017)p Fp(\))39 b Fi(\007)h(N)15 b Fp(\()p Fo(E)2585 3103 y Fm(\027)2627 3088 y Fp(\))33 b(,)-24 3262 y(where)38 b(the)f(upp)s(er)g(signs)h(are)e(c)m(hosen)i (if)f Fo(E)1609 3277 y Fm(\027)1689 3262 y Fp(is)g(a)f(lo)m(w)m(er)i (band)f(edge)g(and)f(the)h(lo)m(w)m(er)h(signs)g(for)e(an)g(upp)s(er) -24 3383 y(band)c(edge.)43 b(As)32 b(ab)s(o)m(v)m(e,)g(one)f(also)h(in) m(tro)s(duces)g Fo(\016)t Fi(N)1894 3398 y Fm(\027)1937 3383 y Fp(\()p Fo(\017)p Fp(\).)43 b(Close)32 b(to)f Fo(E)2571 3398 y Fm(\027)2614 3383 y Fp(,)h(the)f(IDS)g Fi(N)45 b Fp(is)32 b(v)m(ery)g(small)g(and)f(its)-24 3503 y(scaling)37 b(is)g(univ)m(ersal)g(and)g(called)f(a)g(Lifshitz)h (tail,)g(honoring)f(the)g(original)g(con)m(tribution)h(of)e(L.)h(P)m (astur's)-24 3623 y(teac)m(her.)-24 3815 y Fg(Theorem)41 b Fc(L)-5 b(et)42 b Fo(E)701 3830 y Fm(\027)786 3815 y Fc(b)-5 b(e)41 b(a)g(b)-5 b(and)41 b(e)-5 b(dge)41 b(of)g Fo(H)1662 3830 y Fm(\027)1746 3815 y Fc(and)g Fo(H)2023 3830 y Fm(!)2074 3815 y Fc(,)i(but)f(not)f(of)g Fo(H)2698 3830 y Fm(\033)2787 3815 y Fc(if)g Fo(\033)j Fi(6)p Fp(=)c Fo(\027)6 b Fc(.)65 b(Supp)-5 b(ose)41 b(mor)-5 b(e)g(over)-24 3935 y(that)36 b(the)g(eigenve)-5 b(ctor)35 b(of)g Fo(T)1033 3899 y Fm(E)1085 3907 y Fe(\027)1019 3960 y Fm(\027)1162 3935 y Fc(is)h(not)f(an)h(eigenve)-5 b(ctor)34 b(of)h Fo(T)2274 3899 y Fm(E)2326 3907 y Fe(\027)2260 3960 y Fm(\033)2404 3935 y Fc(for)g(any)h(other)f Fo(\033)e Fi(6)p Fp(=)c Fo(\027)6 b Fc(.)48 b(Then)34 b(ther)-5 b(e)36 b(exist)-24 4055 y(c)-5 b(onstants)35 b Fo(C)f(>)28 b Fp(0)35 b Fc(and)f Fo(\017)925 4070 y Fk(0)992 4055 y Fo(>)28 b Fp(0)35 b Fc(such)f(that)h(for)g(al)5 b(l)35 b Fo(\017)28 b Fi(2)g Fp([0)p Fo(;)17 b(\017)2216 4070 y Fk(0)2255 4055 y Fp(])861 4249 y(1)p 861 4293 49 4 v 861 4384 a(2)947 4316 y Fo(\016)t Fi(N)1076 4331 y Fm(\027)1119 4316 y Fp(\()p Fo(\017)p Fp(\))28 b Fo(p)1428 4216 y Fd(1)p 1321 4228 246 3 v 1321 4269 a Fe(L)10 b(\016)r Fb(N)1453 4277 y Fe(\027)1492 4269 y Fd(\()p Fe(\017)p Fd(\))1576 4243 y Fk(+1)1311 4326 y Fm(\027)1726 4316 y Fi(\024)55 b Fo(\016)t Fi(N)15 b Fp(\()p Fo(\017)p Fp(\))55 b Fi(\024)h Fo(C)24 b(\016)t Fi(N)2528 4331 y Fm(\027)2571 4316 y Fp(\()p Fo(\017)p Fp(\))k Fo(p)2880 4216 y Fd(1)p 2773 4228 V 2773 4269 a Fe(L)10 b(\016)r Fb(N)2905 4277 y Fe(\027)2944 4269 y Fd(\()p Fe(\017)p Fd(\))2763 4326 y Fm(\027)3067 4316 y Fc(.)794 b Fp(\(1\))123 4682 y(Because)34 b(of)e(the)h(v)-5 b(an)32 b(Ho)m(v)m(e)i (singularities,)g(this)g(result)f(allo)m(ws)h(to)e(read)h(o\013)f(the)h (Lifshitz)g(exp)s(onen)m(t)1355 4952 y(lim)1356 5012 y Fm(\017)p Fl(!)p Fk(0)1534 4885 y Fp(log)q(\()p Fi(j)17 b Fp(log\()p Fo(\016)t Fi(N)e Fp(\()p Fo(\017)p Fp(\)\))p Fi(j)p Fp(\))p 1534 4930 735 4 v 1745 5021 a Fi(j)i Fp(log\()p Fo(\017)p Fp(\))p Fi(j)2334 4952 y Fp(=)2475 4885 y(1)p 2475 4930 49 4 v 2475 5021 a(2)2566 4952 y(.)-24 5177 y(Moreo)m(v)m(er)24 b(it)d(giv)m(es)j(precise)f(b)s(ounds)g(on)e(the)h (IDS)g(and)g(hence)h(b)s(ounds)f(on)g(the)g(so-called)h(Lifshitz)f (constan)m(ts.)-24 5298 y(F)-8 b(or)30 b(suc)m(h)i(asymptotics)g(to)e (hold,)h(it)f(is)h(crucial)g(that)f(the)h(transfer)g(matrices)g Fo(T)2960 5261 y Fm(E)2946 5322 y(\033)3050 5298 y Fp(b)s(e)g (uniformly)g(b)s(ounded.)1972 5564 y(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop -24 171 a Fp(Ho)m(w)m(ev)m(er,)33 b(it)c(should)i(b)s(e)f(p)s(ossible)g(to)g(relax)g(the)g(condition)g (that)f(\(\006)p Fo(;)17 b Fg(p)p Fp(\))30 b(b)s(e)g(discrete.)44 b(By)30 b(appro)m(ximating)-24 291 y(a)25 b(con)m(tin)m(uous)h(measure) g Fg(p)e Fp(b)m(y)i(a)e(discrete)i(one,)g(a)f(straigh)m(t-forw)m(ard)f (adoption)g(of)g(the)h(presen)m(ted)i(argumen)m(t)-24 411 y(allo)m(ws)40 b(to)e(obtain)g(at)g(least)h(the)g(lo)m(w)m(er)g(b)s (ound.)61 b(Finally)-8 b(,)40 b(the)f(\(generic\))g(conditions)g(that)g Fo(E)3562 426 y Fm(\027)3643 411 y Fp(is)g(only)g(a)-24 532 y(band)d(edge)g(of)f Fo(H)643 547 y Fm(\027)721 532 y Fp(as)g(w)m(ell)i(as)e(the)h(condition)g(on)f(the)g(eigen)m(v)m (ectors)j(of)d Fo(T)2809 496 y Fm(E)2861 504 y Fe(\027)2795 556 y Fm(\033)2938 532 y Fp(are)h(not)f(essen)m(tial,)j(but)d(the)-24 652 y(statemen)m(t)f(of)f(the)g(result)g(w)m(ould)h(b)s(e)e(a)h(bit)f (more)h(in)m(v)m(olv)m(ed.)123 820 y(As)25 b(will)g(b)s(e)g(discussed)i (in)e(Section)g(4,)h(the)f(estimates)h(\(1\))e(are)h(in)f(a)g(certain)i (sense)g(based)f(on)g(a)f(determin-)-24 940 y(istic)j(argumen)m(t,)h (alb)s(eit)e(taking)h(place)f(in)g(a)g(random)g(mo)s(del.)42 b(The)27 b(Lifshitz)g(constan)m(ts)g(c)m(haracterizing)g(the)-24 1061 y(true)j(b)s(eha)m(vior)g(of)f(the)h(IDS)f(are)g(mo)s(del-dep)s (enden)m(t.)44 b(One)30 b(approac)m(h)g(to)f(calculate)h(them)g(is)g(p) s(erturbation)-24 1181 y(theory)-8 b(.)44 b(As)34 b(will)f(b)s(e)g(sk)m (etc)m(hed)i(in)e(Section)g(4,)g(this)g(also)f(allo)m(ws)i(to)e(go)g(b) s(ey)m(ond)i(the)f(Lifshitz)g(tail)g(regime.)-24 1349 y Fg(Ac)m(kno)m(wledgemen)m(ts:)45 b Fp(This)33 b(w)m(ork)h(w)m(as)f (supp)s(orted)h(b)m(y)f(the)g(SFB)g(288.)-24 1681 y Fn(2)161 b(Setup)-24 1900 y Fp(First)41 b(let)h(us)f(recall)h(some)g(of)e(the)h (analysis)i(of)d(the)i(p)s(erio)s(dic)f(op)s(erators)g Fo(H)2897 1915 y Fm(\033)2943 1900 y Fp(.)69 b(The)42 b(eigen)m(v)-5 b(alues)43 b(of)d(the)-24 2043 y(transfer)27 b(matrix)g Fo(T)713 2006 y Fm(E)699 2067 y(\033)798 2043 y Fp(are)f Fo(\025)i Fp(=)1152 2003 y Fk(1)p 1152 2020 36 4 v 1152 2077 a(2)1214 1932 y Fh(\020)1274 2043 y Fp(T)-8 b(r\()p Fo(T)1483 2006 y Fm(E)1469 2067 y(\033)1542 2043 y Fp(\))22 b(+)1700 1957 y Fh(p)p 1800 1957 517 4 v 86 x Fp(T)-8 b(r\()p Fo(T)2009 2014 y Fm(E)1995 2067 y(\033)2068 2043 y Fp(\))2106 2014 y Fk(2)2168 2043 y Fi(\000)22 b Fp(4)2316 1932 y Fh(\021)2402 2043 y Fp(and)k(1)p Fo(=\025)p Fp(.)41 b(Hence,)29 b(if)d Fi(j)p Fp(T)-8 b(r\()p Fo(T)3440 2006 y Fm(E)3426 2067 y(\033)3499 2043 y Fp(\))p Fi(j)27 b Fo(<)h Fp(2,)f(there)-24 2197 y(are)33 b(complex)i(conjugate)e(eigen)m(v)-5 b(alues)34 b Fo(\025)28 b Fp(=)h Fo(e)1703 2161 y Fm({\021)1801 2197 y Fp(and)k Fo(e)2036 2161 y Fl(\000)p Fm({\021)2189 2197 y Fp(and)g(the)g (transfer)g(matrix)h Fo(T)3297 2161 y Fm(E)3283 2222 y(\033)3389 2197 y Fp(is)f(conjugate)g(to)-24 2318 y(a)g(rotation)f(b)m (y)h(the)g(phase)h Fo(\021)t Fp(.)43 b(This)34 b(phase)f(is)g(also)g (called)h(the)f(rotation)f(n)m(um)m(b)s(er)i(and)e(one)h(sp)s(eaks)h (of)e(the)-24 2438 y(elliptic)k(case.)50 b(Then)36 b Fo(E)h Fi(2)32 b Fp(sp)s(ec)q(\()p Fo(H)1313 2453 y Fm(\033)1359 2438 y Fp(\))j(and)g(the)g(IDS)f Fi(N)2074 2453 y Fm(\033)2155 2438 y Fp(at)g Fo(E)41 b Fp(is)35 b(equal)g(to)g Fo(\021)t(=)p Fp(\()p Fo(L\031)t Fp(\))f(up)h(to)f(a)g(m)m(ultiple)i(of)-24 2558 y(1)p Fo(=L)f Fp(coming)f(from)h(the)f(gap)h(lab)s(el)f([JM)q(,)g (Bel)q(].)49 b(On)34 b(the)h(other)f(hand,)i(if)e Fi(j)p Fp(T)-8 b(r\()p Fo(T)3001 2522 y Fm(E)2987 2583 y(\033)3060 2558 y Fp(\))p Fi(j)31 b Fo(>)f Fp(2,)35 b(the)g(eigen)m(v)-5 b(alues)-24 2679 y(are)37 b(b)s(oth)f(real.)56 b(One)37 b(of)g(them)g(has)g(a)f(mo)s(dulus)i(bigger)f(than)f(1)h(and)f(one)h (smaller)h(than)f(1.)55 b(This)38 b(is)f(the)-24 2852 y(h)m(yp)s(erb)s(olic)29 b(case,)h Fo(E)45 b(=)-61 b Fi(2)28 b Fp(sp)s(ec)r(\()p Fo(H)1179 2867 y Fm(\033)1225 2852 y Fp(\))f(and)h(the)g(transfer)g(matrix)g(is)g(conjugate)f(to)g (the)h(dilation)3465 2711 y Fh(\022)3579 2791 y Fo(\025)136 b Fp(0)3583 2911 y(0)87 b(1)p Fo(=\025)3915 2711 y Fh(\023)3989 2852 y Fp(.)-24 3030 y(Again)33 b(due)g(to)f(the)h(gap)g(lab)s(elling,) g(one)f(has)h Fi(N)1771 3045 y Fm(\033)1818 3030 y Fp(\()p Fo(E)6 b Fp(\))28 b(=)f Fo(m=L)33 b Fp(for)f(some)i(p)s(ositiv)m(e)g (in)m(teger)f Fo(m)28 b Fi(\024)g Fo(L)p Fp(.)123 3198 y(Let)f(no)m(w)h Fo(E)562 3213 y Fm(\027)633 3198 y Fp(b)s(e)g(a)f (band)h(edge)g(of)e(the)i(op)s(erator)f Fo(H)2030 3213 y Fm(\027)2073 3198 y Fp(.)42 b(One)28 b(then)g(has)f Fi(j)p Fp(T)-8 b(r\()p Fo(T)2965 3162 y Fm(E)3017 3170 y Fe(\027)2951 3222 y Fm(\027)3059 3198 y Fp(\))p Fi(j)28 b Fp(=)f(2)g(and)h Fo(\025)f Fp(=)h Fi(\006)p Fp(1)g(and)-24 3318 y(is)38 b(therefore)f(in)g(the)g(so-called)h(parab)s(olic)e(case.) 57 b(The)38 b(transfer)f(matrix)g Fo(T)2831 3282 y Fm(E)2883 3290 y Fe(\027)2817 3343 y Fm(\027)2962 3318 y Fp(has)g(only)h(one)f (eigen)m(v)m(ector)-24 3439 y(denoted)32 b Fo(~)-45 b(v)38 b Fp(as)d(w)m(ell)h(as)f(a)f(principal)i(v)m(ector)48 b Fo(~)-62 b(w)37 b Fp(satisfying)f(\()p Fo(T)2315 3402 y 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