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Łs t r t rs tø r P r s tø PrØ rø rs tø P r s r t t r s t Ø t q s P r s tr st t t t Ø t q s ss P r s P 2stŁ s q t q s t rt r s t s t ss s Ø r s t r t P r røs r Łs t r t t Ø t q s r Ø r t t r t q t rs tø P r s r t 15 rs 2017 t r2 sø PrØs t r2 r ss r rs tø r r tr t Łs r ss r rs tø P r s r t P r s 7 rt r P st t tr Ør s rs tø P r s P r s 11 r t r t r r ss r rs tø s 1 t r tr Ør s rs tø P r s rt r r ss r rs tø rr t røs t s t

st t t t Ø t q s ss P r s rs tø P r s r t r rt st P r s t r P s ss P r s 1

Øs Ø Øs Ø s tt t Łs Øt Ø Ł s t rs s Ø r t rs r r rt s q ss s t 1 t s t s r r s q s t 2 q s r r t s s Łr Ł rs s r t t rt r t t Ø t r rs Ø t s t t q t str Ø s t str t st s Ør t t r t rt rtø Øt t r Ł s t rs r s Ø r s s s t r s s s tr rłs r trø q st Ø t r tr rt tr q s t Łt s s2stł s rt r s s Øt 1 s2stł s t rt r s 2 t t r t s s t t r Øs r r r tr r P r røs t t 1 s t s tø r Ø str t s r s Ø t r s st Ø ss r t tr r s t r t s 2 t r trłs Ø Ør s r Ø s s røs t ts s t Ø trøs t t 1 s2stł s t rt r s 2s t Ø Ø r r q r t tr t r s str t s s Łr s tr r t Øt s ts r t r s s st Ø s r t r t t t s 2s t Ø s t Øt s r s s Ør t r r ØtØ sø r tø t s t t Ø tr Ø t st Ø s s q r rt t s 2s s Ø r s 1trŒ s ts s Ør t rs Ø t r s s2stł s t rt r s ss s Ø r s s t r t s

t rt 3 t r s r r q t s2st s t r s r t t r t str t t s t s s st 2 r t rt s r r r t rs t r s 3 t s sts t 1 t 3 t t s 2 3 t rst s r t s r t t rt rs t s rt r t r t r t t s s r t t 2 str t t r t s r t2 str t t r s s t t t tt ts s tr s r r t rs 3 t r r s s t 2 s t t s tr t t r st s t t t t 3 t r s rt s2st s 1t s t t rt s2st s t s t 2 t r t t r tr r2 s r r r s t t s r t2 str t t t r r s r s ts r r 2 t t t t rt s2st s t r r 2 t s 2s s s s r str t s st t r t ts t r s s r t t s 2s s r r r s r s s t t s s r t s s st t 2s rt t r t 2s s 1tr r s 2 r s rs 3 t r r t rs t rt s2st s r s t r t

r ts t s 1 r r r t t r tr t Łs t t q r sø r Ł t Łs t r t r r Ø t r t t s s t s q tr r s s t t t rø s t s r r t t s r r s r r Ø t t r s r s s tø s t s Ø s t s ss s q r t t t Łs s s r s s t r ss r s tr r t r t rt s s 2 s t Ø ts t s ss 1 r r t t r t t rs tø P r s r t r r rs tø r P r s tø t rt r tø r t t r r t Ø t q s r r r Ø s t r r s r r s t r s r r Ø t s Ør ts r s s s r t tø r t t r r t Ø t q s t ss s Ør ts r s s s r t r st t t t Ø t q s ss P r s q t t rs ØtØ Ø t rs s t r ts s rt r r s s t s r t t q r t rs tø r t t r r t Ø t q s ss r st t P tr 3 t r t t q r t rs s ss r t r s r 2 t Ø s r tł t t q r t rs Ø t r r r st t P r s t st r r tr r t s r Øt r s r rts r s tr 1 t Łs t r r ss t t s s tr s rs s t t r r s Ør r t tø Œtr r r2 t Łs t s s r s r2 s t r r r ss s s t r ts st t t q tr

r t t r t s r r rs tt t r ss t q t ØtØ s r r s t r t s r s r r r r s tr s rs s rs tø s r s Ør ts t Ø r s s r s s s str t t r t q r t r t Ø t q s P r s tr

s t Łr s tr t t røs t t s røs t ts tr t PrØs t t s tr 1 t Łs 2 t Łs s Ø Ør s Øt r t s s tr 2s t Ø t rt r ss s Ø r s s t rs ss s Ø r s s t 2 q ss s Ø r s 2s t Ø t rt r t r t s t rs Łt t r t s t 2 q t r t t s 2s s s s t t s tr r s t q t s r t2 r st t s s s st t 2 rt 2 t r t s t rt t s 2 3 t t r s ss t s t r s ts ss t s r s ts t t rt t s 2s s s t s s rt t s t t rt t s t P rs P s

s t Łr s P rs s 1 rs s s st s r r s tr r r 1 t 3 t t r 2 r r 2 3 t t r s Pr r t rt 3 t r t 2 t r t s2st ss t s t r s ts ss t s r s ts 1 r 2 s r t r t t r t rt s2st t t t r t t r 2 t r t t rt s2st s t s t r r 2 s r t 2 t r t s2st Pr r Pr r 1 r2 r s ts r r t rs rt s s t s r rt s t r t rs s ts r t s rt t r2 r 3 t 1 s r

tr tr t t røs t t s røs t ts tr t Ø q q t q Ø t t r rt r røs tø r Øt t ψ st r Ø r Øq t r r i t ψ = H(1) ψ, ø H (1) st t Ø t r H (1) = ~2 2m + V, ss t s r H = 2 (R d ) 2 (Z d ) Ø s t Ø r Øt q rt t V st t t Ø r t str t r 2s q s2stł t ss t Ør t r t t r t V (x) s s t s Øq t s t r ψ(t) = e ith (1) ψ 0 ψ(0) = ψ 0. s r Ø r r rt t s Øt ts ψ(t) st t 2s r s r r ØtØs s tr s Ør t r H (1) s t 2s st t ~2 2m r t t st t s s r ~2 2m = 1 P r s s2stł s r t t r Øs t t s Ør q s V (x x 0 ) = V (x) r t t x Z d R d t rt x 0 Z d 1Ø st Øt t s q s tr Ør t r r r H (1) st s t t t st t Ø s s s rt Ø r t s t t s t røs Øs r r t s ts rt rs s r s rt s q t q s P r s Ø r r P rs tr t Ł

tr tr t t røs t t s røs t ts q rt s t q r 1 2s q Ł r s s Ł rs Øs r r st r s t r r tłr Ø t r t t 1t r V s r 1 t rt r st Ø r t r Ør t r r r Ø t r H (1) (ω) = + V (x, ω), ss t s r Ψ = Ψ(x) 2 (Z d ) t ø {V (x, ω)} st t Ø t r rs Ø t s t t q t str Ø s r t s r sø (Ω, B, P) Ø Ł rt t q r t r s Ør t rs H (1) (ω) st s t rs q s tr t r røs s tr r t t Ø r ss 1 t s t s r r s tr r r ØtØ s rt t trłs tør ss t s s r t Ø t t r s2stł st s t 2 q q 1 r t q rt r st trø s r r Ø t t s r q s H s Ø s H = H pp H c = H pp H ac H sc ø H pp H c H ac t H sc s t r s t t s s s s s r r s ts s t r t t t s t t t s Łr t t s r s tr ss Ø H (1) (ω) t s s tr s s r str t s H (1) (ω) s Ør ts s s s s r σ (H (1) (ω)) = pp, c, ac, sc røs t t sø s r s tr 1 r r s t ss s t Ø rł r Łr Ø 2 q ss Ø 1 Ør t rs H (1) (ω) s t B R r Ø Z d tr 0 t r 2 R t Λ Z d r Ø rs ψ H pp q ψ H c q lim sup X e ith ψ(x) 2 = 0, R t 0 x/ B R lim T 1 T Z T 0! X e ith ψ(x) 2 dt = 0. x Λ s s t q Øt t ψ st sø s t 1 Øt ts s Øt s rr s t s ø rt s Ø s t t s s

tr t r rs røs t ts t Ø t q t r r 1 s r s t rs t ØtØ t s r P st r s P s r s 1 t r Ør t r t r s t rs r s Ør t rs t2 H (1) (ω) ØtØ t t r Ø r 3 t r s 1 r s str t s s tø r Ø rs s 1 s rs tr s røs t ts t s t s t t s s rs t2 r P s P r t t s Øt s s t sø s s r Øt 1 s t 2 t s t s rrø t s t s Øt t s s d 2 P r d = 2 rłs s 1 Ør s 2s q s tt r s t Łt s s d = 1 s s r s 2s t Ø rr t Œtr s rt q r s d = 1 r st s t r Ø ts s s t Ø t q s s s2stł s Øs r Øs q t q s s q q tr rø sø r r t r s s s r t sø s r 2s t Ø t Øt ss t s s s Ł rs ss s Ø r s r Øs r r t Øt rt t s s t t s P r s t rt t rt t r t tr rt Øt ss t r s t t s s s tr s t t t s t rs r Øs r r ss s Ø r s P s t r r s t s t 2s t Ø t r s t r Œ s r s t rs t t 2s t Ø s t ØtØ rø sø s s t 2 q q t ØtØ t t Ø trø r 3 t 3 s s s r t Øt s ts r t r s t s s t s Øt s ts q s q t t r t s t 2 q s Ø ss t s str t s ss 3 rø Łr Ø Ør t s t t s tø r Ø s q 2s t Ø r t tr t r r s tr t t s Ø t r s t s t t s s rs t2 r r s 1 t 2s s Ø t r st r q s Ø r Ø r ss s t s r r s t s t 2 q s t 2 q r s Øt s r s t s r 2s t Ø ØtØ Ø rt Ø trø r r t t Ł r P s t r t t tr t q s st Ø s 2s t Ø tr t s t 2 q rt s s s Ør rs s r rt t s t 2 q Ø

tr tr t t røs t t s røs t ts t ØtØ r Ø r r t t s tt t Łs s s r r 2s t Ø r Øt r s t rs t 2 q t rt r 2s t Ø st r Ø Ø t s r s Ø s rs { } 0 r t rø rr = b α c + 1 1 < α < 2 r r str r Ør t r H (1) (ω) s s C (1) (u) tø t s 1 r s s 2s t Ø s st rs Øt r Ø r ss Ø Ør t 1 t t t s s Ø s s Ø Ø ts tr s Ør t r r str t H (1) (ω) r C (1) (u) tø Øt t st sø s r t tø Ø Øtr q røs t q r t Øt r r t tr s Ø Ø ts tr s s rø s t s rr s t 1 Ø s t + 1 t røs t t tr r 2s t Ø t q t s t tø Ø Øtr q røs t Øt t Ø r ss s Ø Ø ts tr s s rt r Ø r ss s s s s s s ts t r røs røs C (1) (u) r r rt rt Ø r E r tør s t q tt Ø r s t trłs r s tr Ør t r H (1) (ω) t s s r t C (1) (u) 2 t Łs t s r s Ø s { } Øt s t s st st r r tø s røs s s t s st Ø s r r s 2 t Łs s t s r Øs r r s s Ø r s 1trŒ s st r Øt r s st Ø s Ø t s s s ss s Ø r s q s tør ss tt st Ø st sø rt s r st Ø s s q r t Ø r ss s røs t s s r s Ø r s rs s tr t tr rt s r Øt s s2 t t q s s t3 s tø Øt ts tø rø s s t t s Ø t r s rrø Øs ØtØ 2sØ r r s t q t Ø trø s t rs s s ss s r s Ø r ss t s t s s s Łt t 2t q s r r r q t Øt t s t r s t t s rrø Øs rrø t s 1 t t Ø r ss t s r Øt r s t s r r 2s t Ø st Ø r t Ø Ø rø Ø t r r t t Ø rø s P s rø t 2sØ s st t st q s s tr s s rs r r s Ør t r r r t t s t rrø Øs s rø sø s Ł rs s r 1 t t rt r r rt N 2 Øt t Ø t q t s Ør t rs st

tr t r t t rø t str t r Ø Ør t N rt r t r t st s t H (N) (ω) = + NX V (x j, ω) + U, j=1 Łr s r Ψ = Ψ(x 1,..., x ) 2 ((Z d )N) ø st Ør t r s r (Z d ) N = Z Nd V : Z d Ω R st Ø t r r t s r sø (Ω, P) q t r t r t t r t V (x, ω) = P N i=1 V (x i, ω) t U : (Z d ) N R st t t t r t tr s rt s q t q s t t ss Ør t r t t r t U(x) s r rs røs t ts s r s t rs s Ør t rs H (N) (ω) t ØtØ t s rt r s 2 t Øt t 2s t Ø 1 s2stł s t rt r s t tr rt r 3 t r3 q t rø sø tt 1t s r Øt s ts r t r s s s 1 s ØtØ s sø q Ø t r V st rs Ø t s t t q t str Ø s t q t r t U st rtø s s t rs Øt ss t s t s tr t 2 q s s rø r Øs r r t r t t t s Ø tø Øt s ts 3 t r3 t s sø q str t s r s Ø t r s t s tø r Ø Ør t t t q s Ø t r s q s 2 t t t s s t s tr t rt r s s rø r Øs r r t r s str t s Ør s tr tr q tr t s s ss s Ø r s Łt s rø Ø ts tr 1 r Ł rt t q r t s Øt s s2stł s t rt r s t t t s s t rłs s s tr st r tø r s røs t ts r tø t Ø r rt r s q t s t 2 s r tø rt r s tr s t s s s2stł rt r st r s r q r tø st s t sø s tr str tø Ø Ør r t t q s s tr st Ø t r r tø r t rłs røs t t ß P st r t r q r sq sßr t s tr σ(h (1) (ω)) Ør σ(h (1) (ω)) = Σ, ø Σ st s s s r Ø R s 1 st s s s s s Σ pp

tr tr t t røs t t s røs t ts Σ ac Σ sc t s q P r tø 1 σ pp (H (1) (ω)) = Σ pp σ ac (H (1) (ω)) = Σ ac σ sc (H (1) (ω)) = Σ sc. ø σ pp (H (1) (ω)) σ ac (H (1) (ω)) σ sc (H (1) (ω)) Øs t r s t t s tr r t t s t t t s Łr t t H (1) (ω) søq r Ør r s tr E 0 = if σ(h (1) (ω)) st Ø t r s Ø r s s r r s t s rs tr 1 t Łs PrØs t t s tr 1 t Łs s tt Øt s Łr s r tø (Ω, B, P) Ø r Ω = R Zd, B = B(R) Zd, ø B(R) st tr rø R t t t 1t r V : Z d Ω R Ø r V (x, ω) = ω x r ω = (ω i ) i Z d s tør ss rs 1 r r ØtØs s tr s s t s N rt r {H (N) (ω), ω Ω} t N 2 t r q t Œtr r tr r t r 2s t Ø Ø ss t s Ør r ss s t s H () (ω) rr s t 1 s s s2stł s rt s r t t = 1,..., N s tr t s t t s s t s r t t = 1,..., N t ν 1 s Łr r 1 s r Z ν Ø r x = max{ x 1,..., x ν }, r x = ( x 1,..., x ν Z ν s t r 1 s Ø t t s r r 1 Ø r x 1 = x 1 + + x ν. r t s2stł rt s q t q s st t s s r Z d ø d 1 st t r 1Ø st r røs tø r t r x = (x 1,..., x ) (Z d ) = Z d, ø x j = (x (1) j,..., x (d) j ) Z d j = 1,..., t st s r Z d Ø r ( Ψ)(x) = X (Ψ(y) Ψ(x)) = X Ψ(y) 2dΨ(x), y Z d y x 1 =1 y Z d y x 1 =1

PrØs t t s tr 1 t Łs r t t Ψ 2 (Z d ) x Z d t r q st r Ø t st s t s Ør r s Ør t rs r r Ø t r H () (ω) r = 1,..., N r s t Ø H () (ω) = + X V (x j, ω) + U j=1 ss t s r s rt H () = 2 (Z d ) t s t t s t t Z d = (Z d ) x = max 1 j x j, x 1 = x 1 1 + + x 1, r x = (x 1,..., x ) (Z d ) P r rø t t r t tr t r Łtr h > 0 q r t s r r r t r t P s rø sø t t s t H () h X (ω) = + V (x j, ω) + hu, j=1 ø Ø r t r t st s s q h > 0 st t t 2 t Łs s Ø Ør s t t t r t st r Ø t r U(x) = X 1 i<j U: (Z d ) R Φ( x i x j ), x = (x 1,..., x ) Z d, ø Φ: N R + st t s t s rt t r 0 N : supp Φ [0, r 0 ]. Øt rø t t r t Ø ss t s t r t U Œtr s t r tt r s tr t 2 t Łs s Ø Ør s t

tr tr t t røs t t s røs t ts t r t U: (Z d ) R s t s t r t t = 1,..., N U(x) = X Φ( x i x j ), x = (x 1,..., x ) Z d, 1 i<j ø x = (x 1,..., x ) (Z d ) t t Φ st s Ø ss r t s t s r 0 N : supp Φ [0, r 0 ]. P r Ø r r s 2 t Łs s s r t t Ø t r V : Z d Ω R, s rt r t t s Øs r F V str t s r R s r s Ø t r s V (x, ω) t E R t s t µ s r ss Ø s r R s Ø t ss s(µ, ε) = sup a R F V (E) := P {V (0, ω) E} µ([a, b]) = F V (b) F V (a). µ([a, a + ε]) = sup(f V (a + ε) F V (a)). a R ((1 + x ) 1+κ ) := {f : R R sup f(x)(1 + x ) 1+κ < }, x R r rt κ (0, 1) s rt s r µ st Ø r supp µ = {x R µ ((x ε, x + ε)) > 0, ε > 0}. P str t µ s r s Ø t r s V (x, ω) Ør 0 supp µ [0, + ) s st Ør s r s(µ, ε) := sup P {V (0, ω) [a, a + ε]} C a R l 2A ø C (0, ) t A > 0 st s s t r. P t t Ø t r V : Z d Ω R st t r Ø 1 st M (0, ) t q supp µ [ M, M] s s r r tø µ ssł s Ø r Ø P {V (0, ω) A} = µ(a) = R A ρ(λ)dλ t ρ ((1 + x ) 1+κ ) s røs t ts s ts t ØtØ t r ts rt s

PrØs t t s tr 1 t Łs Øt r t s s tr t s t s s r r rs r tø st s tr Øt r q s s tr E () 0 s t s H () (ω) st r sq sßr t st t r t røs t t st r r s t rs s s 2 t Łs s Øq t s s trłs Ø Ør s (I1) t (P1) s r t r t U t t t Ø t r V tr q r sq sßr t t t s s q t tøs E () 0 s t t q t s Ø rł t = 1,..., N s s 2 t Łs s (I1) t (P1) r tø 1 [0, 4d] σ(h () (ω)) [0, + ). P r søq t E () 0 := if σ(h () (ω)) = 0 s s Ø s r r ss t Ø s s r Ør t s r P t t s s s Ør t rs r r rt r s 2 (R d ) s tr s røs t r t tr s ts r s s r r t r s s r r t t s r t q rø tø t r [0, 4d] st s tr s 2 (Z d ) t Øt t Ø q t r t st rtø t tr r s t s s Z d s r sq s t r t st t q t t 2 t s r t s t t s s t s s r Z d s Ø Ł ts r t s t s r t s s t Ø ts s t Øt t Ø s t 2 r t s rt s trø r tr s t q s t s rt t s s t r s s t s s ø t r t st s t tr s tø r st s t 2 r t r t r st s r t t Ø t r r t r P r s Øt s s r røs t t r tr s Ł rs r t t E R t lim d card : E (H () (ω) () C (0)) E 1 st t st Ø Ø s tø Øt ts tø rø H () (ω) s s 2 t Łs s Øq t s rłs røs t t ß t s tø Øt ts tø rø t rt r H () (ω) 1 st t t rt r H (1) (ω) t s røs t t q st Øt s r s r t rłs t Ø r s r Ł r s t t Ø t r r q s s s q st tø Øt ts tø rø,

tr tr t t røs t t s røs t ts N t rt t s t3 s s s tr r r 1 Ø rł s r s r lim E&E (1) 0 l l N (E) l(e E (1) 0 ) = d 2. t t Ø q r t s s t s H () (ω) = 1,..., N E () 0 = 0 t Ø r q t s s t s H () (ω) tt t rt t s t3 s 0 s røs t ts s ts r t rø s ss s Ø r s s t Ø trøs tr 2s t Ø t rt r ss s Ø r s s s rs rt r 2s t Ø s s r s t rt r t t Ø Ø r s 2 t s s r Øs r r r st t s r s Ø s { } s s t t s r r rt s = 1,..., N t s st Ø s r s r t tr r tø s r tøs s røs s s t Øt s rt r t t t t r t s r s r t s s s rt r s r Z d t s Ø Ł ts r t s 1 s s ts s Z d s t s r Ø t Ø ts s s rt r st r tt r s q tr t t s sø r s ø r t rt r t t s s st s t rø s tr s r t s t s tr P s rø sø t Ø t t q (x) = C(1) (x 1) C (1) (x ) st J sø r tr (y) = C(1) (y 1) C (1) (y ) s 1 st s s s J {1,, } t q [ C (1) (x j) [ [ C (1) (x j) ( C (1) (y )) =, j J j / J r ( (x), C() (y)) st t sø r s x y > 7N t s s s st J sø r tr tt r r ØtØ r t r Ø t t t 1 r rt s r s Ø t r s q r t sø r tø st =1

PrØs t t s tr 1 t Łs r r ØtØ r t ss t r r t s r s s s s t st ts s t r s s st s s t r r t t s r sø r tø 1 s s d = 1 t = 3 t s x = (a, a, b) Z 3 t y = (a, b, b) Z 3 rs r t t 0 s s t C (3) (x) = ([ + a, a + ] [ + a, a + ] [ + b, b + ]) Z3 C (3) (y) = ([ + a, a + ] [ + b, b + ] [ + b, b + ]) Z3 q st a b q t Œtr r tr r t r s a 6= b s t s sø r s r s r t s t rs t t rs r t t t tr t ss s r rq t s r r rt s t s r r t r t t = 1,..., N r s s sø r s t s r s Ø s +1 t s t q s r t s t t s s s s t s t s t s r Ø t Ø r r tø t rs t Ø Ł ts r t s s r t s t tt r Łr r r ØtØ st s r str t q sø r tø t r str r 2s t Ø q t t s s st s s s t r Øt r s t rs s Ø q s t s s t q r t r s t t r rt s Z d 1 st tø s st s s t r t t s r t s t t s t rs t t r r tr t t st s ss r s rø r Ø t s s s t s r t s t t s t rs t t r r r t st s r t s s sø r s s t t st r Ør t s s rt s s 1 r s r D Z d D = {x (Z d ) : x = (x,..., x), x Z d }, t s tr s t t t Łtr t q r r s t s tr s Ø s t t t t r t s t 1 s s t Ø Øs r D t s tr s t r s Łtr s r tr r t r q s r t Ø Øs s rt t t r t s s s t t t t r t s t st s s t s t 1 t s r t s t t s s t s t s r tt t s r r r t s r s Ø s s s rt r st q st t st x y > 7N s Ø t P r tr r 2s s

tr tr t t røs t t s røs t ts s rt t t r t s st Ø ss r Ø r s t tt t q t r t s Ø s s r t Ø Ø søq t s Ø s ss s 1 t s 2 t r rt s str t t s t t Ø st Ø r s r t s s r t r rt r s r t s s r s s s t s ø t s r r rt s s Ø t røs tø r s 2 t s s rø r Øs r r t s Œtr r t t t sø s s s ss s Ø r s t ß Œtr Ø P r 1 s Ø t s t q rt r Łtr p(n, g) + q Øs r r g + P r Ø rr r 2s t Ø ss s Ø r s s s rt st Ø s s t st Ø r Ø t r r ss s {V (x, ω)} x Z d rs Ø t s t t q t str Ø s t 1 N t t Ø m > 0 t Ø r E R (u) Z d st t (E, m) s r (E, m) s røs t (H () (u) E) 1 Ør max v Z d v u = (H() (u) E) 1 (u, v) e m(1+ 1/8 ) N +1, s s tr r t q st (E, m) s r (E, m) s s Ø rł s t q Øt t tø s st Ø s 2s t Ø s r t t s s Ø s Ø rł t = 1,..., N s s 2 t Łs s (I1) t (P1) r t t p > 6Nd 1 st m > 0 t E > E (N) 0 t q r t t r s sø r (u) t (v) P E I : (u) t 2p 4N (v) s t (E, m), r t t 0, ø I = (, E ] s t rs ss s Ø r s røs t t t s s st Ø s 2s t Ø t s Ø q s r s s q Ø r ss 1 t s t s r r s t r r 1 Øt Ø Ø r r s t 1 s2s tł s t rt r s

PrØs t t s tr 1 t Łs Ø rł s s 2 t Łs s (I1) t (P1) 1 st E > E (N) 0 t q P r tø 1 s tr H (N) (ω) s [E (N) 0, E ] st t r t t s t s r r s Ψ i (x, ω) rr s t 1 rs r r s E i (ω) [E (N) 0, E ] Ø r ss t 1 t t t 1 st m > 0 t C i (ω) > 0 t s q Ψ i (x, ω) C i (ω)e m x. s t 2 q ss s Ø r s t t s rø sø t s t 2 q t r r rt t tt rs st s q s tr s r s s t 2 q rø Ø t Øt s t ØtØ t t Ø Ø Ø r r t t s t 1t s Ør t rs Ør t s s R d t B 1 s s t s s r s r Ø s f : R R t s q f 1 Ø rł s s 2 t Łs s (I1) t (P1) 1 st E > E (N) 0 t s > 0 t q r t t rø r Ø K Z Nd t t t 0 < s < s E X s sup 2 f(h (N) (ω))p I (H (N) 2 (ω))1 K f B 1 HS <, ø ( X Ψ)(x) := x Ψ(x) P I (H (N) (ω)) st r t r s tr H (N) (ω) s r t r I := [E (N) 0, E ] st ss Øt r s røs t ts ss s 1 t t s Ø t r s rrø Øs Ør t t Ø r st s r t sø r tø t s s r s r t s st s s t tr t rt t st st Ø r Ø t r r ss s {V (x, ω)} x Z d q st s t t q s s r s Ø t r s V (x, ω) s t rrø Ø s s røs t ts s t q r t rø s s t r t s s t Ø trøs tr 2s t Ø t rt r t r t s Øt rø t t r t st 1 tr r r Łtr r tt t tr r r t r t U s rø r t

tr tr t t røs t t s røs t ts s t s rt r H () (ω) = 1,..., N s t X H () (ω) = + V (x j, ω) + hu, h R. j=1 r st Ør t r r rt r sø t t st Ø rr st Ø r t 0 q rø Ør s r r s r s s rt st Ø s s r Øt r 2s t Ø t s q s rø t t r t s s rt s røs t ts s t rt r s 1 rø t t r t r q rt Ø Łr t r r ØtØ s r tø t t Ø r tt r r ØtØ r r Łtr h r tt t tr r r t r t rsq h = 0 s2stł t rt r st s sø s s t r t st t (E, m, h) s r r r rt t H () (ω) str tø s rø r s s r s rs Øt s h rt r st Ø s2stł s s t r t Øt t s2stł t t r t r r t rt r t t s t s t tøs røs t Ø t r s2stł t t r t Ø r 1 t Ø t r s t st Ø Ø r r q s s s 2 t Łs s (I2) (P2) P r tø 1 σ(h (N) h (ω)) [ N(4d + M) h U, N(4d + M) + h U]. s r st s t s I = [ 1 N(4d + M) h U, N(4d + M) + h U + 1] t Ø t r rt > 0 t s ts x, y Z Nd F x,y (E) = (H C (N) E) 1 (x, y), F x (E) = max F x,y (E). (x) y C (N) røs t t st s t Ø rł t = 1,..., N t θ (0, 1/3) s s 2 t Łs s (I2) t (P2) 1 st h > 0 t q r t t h h t r t t r s sø r s (x) (y) P E I mi{f x (E) F y (E)} 2 p(1+θ) 5 C( I, N, d) p(+θ) 5 +(2N +1)d, r t t 0 ø I = [ 1 N(4d + M) h U, N(4d + M) + h U + 1] t p > 6Nd/(1 3θ)

PrØs t t s tr 1 t Łs s t rs Łt t r t Ø rł s s 2 t Łs s (I2) (P2) 1 st h > 0 t q r t t h ( h, h ) t H (N) h t r t t h t s t rs Łt P r tø 1 s tr H () r t t s t s r r s Ψ i (x, ω) r t 1 rs r r s E i (ω) I Ø r ss t r t r q Ψ j r t t x Z Nd t s st t s a, c, C i (ω) > 0 Ψ i (x, ω) C i (ω)e a(l x )1+c. s t 2 q t r t Ø t B 1 s s t s r Ø s s r s f : R R s t s s t f 1 Ø rł s s 2 t Łs s (I2) t (P2) 1 st h > 0 t q r t t h ( h, h ) t t t t rø f : R R t t rø r Ø K Z Nd t t t s > 0 E X s sup 2 f(h (N) (ω))p I (H (N) 2 h (ω))1 K f B 1 HS <. ø ( X Ψ)(x) := x Ψ(x) P I (H (N) (ω)) st r t s tr H (N) (ω) s t r I

t r t s 2s s s t s t r r s t t t s r s t t s 2s s t t r 2 t t r t r s 2 r 2 r ss t s r t t r t s s rt r t str t t r t t t st r t s s ss t s (I1) (P1) r r2 s r s ss t s (I2) (P2) r t s t t s tr r s t q t s 2 t s s t Λ Z d ts t r r2 Λ = v Z d : dist ( v, Z d \ Λ = 1, ts 1t r r2 + Λ = v Z d \ Λ : dist (v, Λ) = 1, ts r2 Λ = (v, v 0 ) Z d Z d v v 0 1 = 1 t r v Λ v 0 / Λ r v / Λ v 0 Λ. st t t s s ts Λ, Λ 0 Z d s 2 dist(λ, Λ 0 ) = mi x Λ,x 0 Λ 0 x x0. st 2 r t t r str t Λ t t s s t Λ Z d t s r2 t s ts tr 1 ts Λ (x, y) = (x, y) r x, y Λ Λ (x, y) = 0 t r s r t t st t t s 2s s t r 2 t t t s r rst t

t r t s 2s s s r str t N Λ s q r t r s 2 ϕ, ( ) N Λ φ = 1 X (ϕ(x) ϕ(y))(φ(x) φ(y)) 2 ϕ, φ 2 (Λ), x,y Λ x y 1 =1 t s t q t2 N Λ Λ r r r t r2 t s t tt s r r t t 2 rs r t t s tr H () Λ 2 σ(h() Λ ) ts r s t 2 1, G Λ (E) := H () Λ E E R \ σ H Λ. ts tr 1 ts G Λ (x, y; E) r s 2 t r t s t r t r H () Λ t Λ Zd t r2 r t r Γ Λ 2 1 (x, y) Λ Γ Λ (x, y) = 0 t r s. t H () r tt s s H () = H () Λ H() Λ c + Γ Λ, r t 2 (Z d ) t 2 (Λ) 2 (Λ c ) tr 1 ts H () Λ r 2 H () Λ c (H () Λ H() Λ c )(x, y) = H () Λ (x, y) x, y Λ, H () Λ (x, y) x, y Λ c, c 0 t r s. s r t s s ts Λ 1 Λ 2 Z d t r t r2 Λ2 Λ 1 = (x, y) x y 1 = 1 x Λ 1 y Λ 2 \ Λ 1 r x Λ 2 \ Λ 1 y Λ 1. t t rt t r t t H () Λ 2 Γ Λ 2 Λ 1 (x, y) = = H () Λ 1 H () Λ 2 \Λ 1 + Γ Λ 2 Λ 1, 0 x = y x Λ 1, 0 x = y x Λ 2 \ Λ 1, 1 (x, y) Λ2 Λ 1, 0 t r s.

s t t s tr r s t q t s s r 3 r t tr r s t q t s r r t s t s 2 rt t r t t st t t s 2s s t 2 t s r s t 2 r Λ 1 Λ 2 x Λ 1 y Λ 2 \ Λ 1 E / (σ(h () Λ 1 ) σ(h () Λ 2 )) t G () Λ 2 (x, y; E) = X (v,v 0 ) Λ 1 v Λ 1,v 0 Λ 2 G () Λ 1 (x, v; E)G () Λ 2 (v 0, y; E). t Λ Z d E / σ(h () Λ ) Ψ s s t t q t H() Ψ = EΨ t r 2 x Λ Ψ(x) = X (y,z) Λ y Λ,z + Λ G () Λ (x, y; E)Ψ(z). Pr t E / (σ(h () Λ 1 ) σ(h () Λ 2 ) σ(h () Λ 2 \Λ 1 )) ss rt Pr s t t 2 2 t (H () Λ 2 E) 1 = (H () Λ 1 H () Λ 2 \Λ 1 E) 1 (H () Λ 1 H () Λ 2 \Λ 1 E) 1 Γ Λ 2 Λ 1 (H () Λ 2 E) 1. s s t t (H () Λ 1 H () Λ 2 \Λ 1 E) 1 = (H () Λ 1 E) 1 (H () Λ 2 \Λ 1 E) 1 x Λ 1 y Λ 2 \ Λ 1 (H () Λ 1 H () Λ 2 \Λ 1 E) 1 (x, y) = 0 s t s r t t (H () Λ 2 E) 1 (x, y) = X (H () Λ 1 H () Λ 2 E) 1 (x, v)γ Λ 2 Λ 1 (v, v 0 )(H () Λ 2 E) 1 (v 0, y) v,v 0 Λ 2 = X G () Λ 1 (x, v; E)G () Λ 2 (v 0, y; E). (v,v 0 ) Λ 1 v Λ 1,v 0 Λ 2 s r s r E / σ(h () Λ 2 \Λ 1 ) s r t t t tt r q t s t t Λ 2 \ Λ 1 t s s 1 st r 2t ts σ(h () Λ 1 ) σ(h () Λ 2 ) s t r s tr r E / σ(h () Λ 1 ) σ(h () Λ 2 ) t Ψ s t t q t H () Ψ = EΨ t 2 0 = (H () E)Ψ = (H () Λ H() Λ c + Γ Λ E)Ψ,

t r t s 2s s s s t t (H () Λ H() Λ c E)Ψ = Γ Λ Ψ r E / σ(h () Λ ) r 2 x Λ Ψ(x) = [(H () Λ E) 1 Γ Λ Ψ](x) = X (y,z) Λ G () Λ (x, y; E)Ψ(z). r t t r str t r t s r tt r t s r u = (u 1,..., u ) Z d s q t s { i : i = 1,..., } r (u) = Y i=1 C (1) i (u i ), C (1) i (u i ) = x Z d : x u i i. 2 (u) t t rt (u) = x Z d : x u. r 2 (u) := card C() (u) = (2 + 1)d t s t s r st t (u) (3)d r 2 u = (u 1,..., u ) Z d ts r t Πu 2 Πu := {u 1,..., u } Z d, r u = (u 1,..., u ). r r 2 r 2 t2 s s t J {1,..., } t J r t 2 Π J u := {u j, j J } Z d. t t t r J = {1,..., } Π J = Π s t 2 r rt r t (u) = C 1 (u 1 ) C (u ) Z d s r ts J r t r 2 J {1,..., } Π J (u) = [ j J C j (u j ) Z d. r J = {1,..., } r t Π (u) st Π J (u) r t2 t s r t2 s r r t r t r st t s r rt t t t r t2 r s s 2 r t2 s t 2 tr t r r t t rt t s 2s s t s r t s t t

r t2 t t (x) = Q i=1,..., C(1) i (x i ) (y) = Q i=1,..., C(1) (y i ) t r t s (x) s J r s r r (y) t r 1 sts t2 s s t J {1,, } s t t [ Π j (x) [ Π j (x) Π (y) =. j J j / J 0 i r ( (x), (y)) s r s r t r t s J r s r r t t r t = max{ i, 0 i : i = 1,..., } r (C() (x), (y)) s s r 2 t s r s r x y > 7N st t s tr ts r rs s r s t > 1 x Z d r 1 sts t rt s 2 (x( ) ) t x ( ) {x 1,..., x } r = 1,, = 1,..., κ() κ() s t t r t y s t s s y x > 7N y / κ() [ =1 2 (x( ) ), t t s (x) C() (y) r s r t (y) Zd rt 2 (x) t y x > max 1 i,j y i y j + 5N s J s r r (y) r s J {1,..., } Pr r s t > 0 6= J {1,..., } y Z d {y j } j J s st r t [ C (1) (y j) t s t t t2 s t s s ts r t s x, y Z d r s s j J 1t t s t t r y t 1 st rs Γ 1,..., Γ M t r 2 t M

t r t s 2s s s s t y i rr s s t 1 t 2 st r Γ j j = j(i) {1,..., M} t r 1 sts j {1,..., M} s t t Γ j Π (y) C() (x) r s r (x) = t s s r t r = 1,..., M Γ Π (x) 6= s r = 1,..., M i = 1,..., s t t Γ C (1) (x i) 6= r 2 j = 1,..., t r 1 sts = 1,..., M s t t y j Γ r r r s 2 2 t s s t r 1 sts i = 1,..., s t t Γ C (1) (x i) 6= 1t t z Γ C (1) (x i) s t t z x i t t y j x i y j z + z x i 2 + = 2 s y j, z Γ t t t t y j z 2 st 2 s y j s t r t st r Γ r j = 1,..., y j st t t s C (1) 2 (x i) r t s t s (y 1,..., y ) t κ() = r 2 t st κ() ss t s y = (y 1,..., y ) st t t rt s r t s s 3 2 t d s s 3 2 t ss rt t s t R(y) = max 1 i,j y i y j + 5N s r (x) t y x > R(y) t r 1 sts i 0 {1,..., } s t t y i0 x i0 > R(y) s r t 1 t t Λ x := S i J C(1) (x i) t Si C(1) (x i) t x i0 ts t r s 2 2 dist(λ x, Π (y)) = mi u v, u,v s t x i0 y i0 x i0 u + u v + v y i0, dist(λ x, Π (y)) = mi u v u,v x i0 y i0 diam(λ x ) max v,y i0 v y i0. t t diam(λ x ) 2 max v,y i0 v y i0 max v y j + max y j y i0, v y i0

r st t s r s j = 1, s t t v C (1) (y j) 2 t dist(λ x, Π (y)) > R(y) diam(λ x) (2 + diam(πy)) > 0, t s s t t (x) s J s r r C() (y) t J t 1 s s t r t t Λ x r st t s r st t s t 2 tr 2 r 2 r r t t st t t s 2s s r s s t s s r 2 2 t r s rt r s s t t r t t t t rt s2st s 2 s 2 t t t t s s 1t t rr t r t t s 2 s 2 rs r t t t rt r st t s r s t 2 t s str t s t s t2 t J t2 s t t r t2 J = p J s t t {1,..., p} t R J = R p t s t s s (e 1,..., e p ) s r R J + = {q = (q 1,..., q p ) R J : q j 0, j = 1,..., p}. t Φ : R R s 2 t r r R J + v RJ Φ(v + r) Φ(v). r r r e = e 1 + + e p R J r v R J t > 0 Φ(v + te) Φ(v) t. t s s t 2 r t r r r t s r s t t r s r r t r t t s t r t2 str t µ r t t r t 2 s r r t s r s t r µ J = µ µ R J t s t J 1 s t t r t2 J = p µ r t2 s r R µ J = µ µ r t s r R J t t Φ : R J R s 2 t t r 2 t r I R µ J {q : Φ(q) I} p s(f V, I ).

t r t s 2s s s Pr t I = (a, b) b a = ε > 0 t A = {q : Φ(q) a}, A ε 0 = A A ε j = A ε j 1 + [0, ε]e j := q + te j : q A ε j 1, t [0, ε]. r 2 A ε j s r s s q j t t2 2 t s s Φ s {q : Φ(q) < b} A ε p. Φ(q) < b t r r := q ε e Φ(r) Φ(r + εe) ε = Φ(q) ε b ε a. s r {Φ a} = A q = r + ε e A ε p. t t {q : Φ(q) I} = {q : Φ(q) (a, b)} = {q : Φ(q) < b} \ {q : Φ(q) a} A ε p \ A. r r µ p {q : Φ(q) I} µ p ( A ε p \ A = µ p p[ px j=1 j=1 (! A ε j \ A ε j 1 µ ( p A ε j \ A ε j 1 r q 0 6=1 = (q 2,, q p ) R p 1 s t I 1 (q 0 6=1) = {q 1 R : (q 1, q 0 6=1) A ε 1 \ A}. 2 t (q 1, q6=1 0 ) Aε 1 \ A t r 1 st (a 1,..., a p ) A ε 1 \ A s t t q6=1 0 = (a 2,..., a 2 ) q 1 = a 1 + t r s t ]0, ε] r r I 1 (q6=1 0 ) s t r t ε µ p s r t s r Z Z µ p (A ε 1 \ A) = dµ p 1 (q6=1) 0 dµ(q 1 ) s(f V, ε). I 1

r st t s r 2 t r j = 2,, p µ p (A ε j \ A ε j 1) s(f V, ε). r r µ p {q : Φ(q) I} px µ p (A ε j \ A ε j 1) p s(f V, ε). j=1 t t 1 β = 1/2 E R s r r t (u) = Q i=1 C(1) i (u i ) s t = mi i=1,..., { i } (u) s E r s t E t r s t s E r s t E h dist E, σ ( H () C (u) i < e β. () t t E R (v) Zd s 3 2 s E t 2 r s t E t s t t 2 E s 3 2/3 rt r C (v) s ts E r t r s r r t s (u) = Q Q i=1 C(1) (u 0 0 i ) i r 2 ε > 0 E R h i P dist σ(h () (u) ), E < ε r 2 ε > 0 h i P dist σ(h () ), σ(h() (u) (u 0 ) ) < ε i=1 C(1) i (u i ) (u 0 ) = (u) mi i=1,..., Π i (u) s(f V, 2ε), (u 0 ) (u) max i=1,..., max u,u 0 { Π i (u), Π i (u 0 ) } s(f V, 2ε). Pr t t ss r t2 ss t t 1 = mi i=1,..., { i } s t J = Π 1 (u) p = J ss t t J = {y 1,..., y p } t {λ () (u) : = 1,..., C() (u) } t s H () (ω) r 2 (u) s s t Λ Z d t 2 P Λ E Λ t r t2 t 1 t t t r s t t r r s {ω x : x Λ} r t H () (u)

t r t s 2s s s r t s H () (u) (ω) = + U(x) + X i=1 ω xi X px = + U(x) + δ xi,y j ω yj i=1 j=1 X X + δ xi,yω y. i=1 y Π (u)\j P ω : dist(e, σ(h () (u) (ω))) ε = = (u) X =1 (u) X =1 (u) X =1 s r t t Φ(ω) := λ () t P E λ () (u) (ω) ε P ΠC () (u)\)j P J { E λ () (ω) ε} (u) E ΠC () (u)\j[µ J ((ω x ) x J : E λ () (ω) ε)]. (u) H () (u) (ω + t (1,..., 1)) = + U(x) + X + t (ω) s t t Φ s 2 X i=1 px j=1 δ xi,y j i=1 H () (ω) + t Id, (u) px δ xi,y j ω yj s x 1 J s t t P i=1 P p j=1 δ x i,y j 1 r r 2 t 1 r r j=1 λ () (ω + t (1,..., 1) λ() (ω) + t. (u) (u) t 2 t t s t µ J (ω = (ω x ) x J : λ () (ω) E ε) J s(µ, ε). (u)

r st t s 2 P dist(e, σ(h () (u) (ω))) ε (u) Π 1 (u) s(f V, 2ε), r s t t ss r t2 ss t t (u) s r s r r (u 0 ) J {1,..., } : Π J (u) Π J c (u) Π (u 0 ) =. t r s 1 t r s (u) (u 0 ) t {λ : = 1,..., (u) }, {λ 0 : 0 = 1,, (u 0 ) }, t s H () H() (u) (u 0 ) r s t 2 r 2 i J r s s t J = Π i (u) s ss s t t J = {y 1,..., y p } r r P dist(σ(h () ), σ(h() (u) (u 0 ) )) ε = P ΠC () (u) Π (u 0 )\J P J {dist(σ(h () ), σ(h() )) ε} (u) (u 0 ) (u) X i=1 (u 0 ) X j=1 E ΠC () (u) $ (u )\J[µ J ((ω 0 x ) x J : λ (i) (ω) λ(j) (ω) ε)]. (u) (u 0 ) s s t t t t Φ(ω) := λ (i) t r r t t s s t t (u) (ω) s 2 µ J {ω = (ω x ) x J : λ (i) (ω) λ(j) (u) (u 0 ) (ω) ε} J s(f V, 2ε), 2 P dist(σ(h () ), σ(h() (u) (u 0 ) )) ε (u 0 ) (u) max i=1,..., max u,u 0 { Π i (u), Π i (u 0 ) } s(f V, 2ε). r r2 t α = 3/2 p > 6Nd ss t t t r t t s t s s ss t (P1) t A > 4 N αp + 6αNd t r 2 E R P (x) s t E 4N p,

t r t s 2s s s P E R : t r (x) r C() (y) s E 4N p. Pr P (x) s t E = P y (x) : 1/α, (x) ( + 1) P (2 + 1) d ( + 1) P (y) s E (y) s E dist[e, σ(h () )] < e β (y) (3) d+1 (y) C (1) (y 1 ) s(µ, 2e β ) Cst 3d+1 A/α < 4Np, r s r q t ss t (P1) r t t2 t r str t µ q t r s r rst t t t r 2 r s 1 (u 1 ) 2 (u 2 ) t 1/α 1, 2 s P E R 1 (u 1 ) 2 (u 2 ) r E = P E R : dist[e, σ(h () C j (u j ) )] < e β j, j = 1, 2 P dist[σ(h () C 1 (u 1 )), σ(h() C 2 (u 2 ))] < 2e β/α (3) (2+1)d Cst A/α, 2 r ss t (P1) t t r ss t rs u 1, u 2 t r s t s r s t r ss s 1, 2 t t P E R : t r (x) r C() (y) s E = P{ E R u 1 (x) u2 (y) 1, 2 t 1/α 1, 2 : 1 (u 1 ) 2 (u 2 ) r E } (3) 2d 2 (3) (2+1)d Cst A/α 4N p, s A > 4 N pα + 9Nd

s s st t r r2 ss t t t r t t s t s s ss t (P2) t r 2 E R P (x) s t E e τ 1 1/2, r s τ 1 (0, 1) P E R : t r (x) r C() (y) s E e τ 2 1/2, r s τ 2 (0, 1) Pr r s t s s s t t r r2 t t s r s 1 t 2 s s t r t t2 t r t2 str t st t r t t2 s r r2 s s st t s s st t r s r s rr r r r t s t ss s t r t r r s t r t r t r2 s s t r t Λ Z d A r t r 2 (Λ) t tr 1 ts A(u, v) A s r r Pr t s s t a 1 = sup u Λ a 2 = sup v Λ X A(u, v) < v Λ X A(u, v) <. u Λ A a 1/2 1 a 1/2 2. B(u, v) = A(u, v) 1/2, C(u, v) = A(u, v) 1/2 s A(u, v) t s A(u, v) = 0 A(u, v) = 0 s A(u, v) = A(u,v) A(u,v) A(u, v) 6= 0 A(u, v) = B(u, v)c(u, v).

t r t s 2s s s Af 2 = X u X B(u, v) 2 a 1 < v Λ X C(u, v) 2 a 2 <. u Λ X A(u, v) hδ v, fi v 2 = X X 2 B(u, v)c(u, v) hδ v, fi u v X " X B(u, v) X # 2 C(u, v) hδ v, fi 2 u v v X a 1 C(u, v) 2 hδ v, fi 2 u,v X a 1 a 2 hδ v, fi 2 v = a 1 a 2 f 2 s Af a 1/2 1 a 1/2 2 f r f 2 (Λ) r s r tt r r r t r H Λ = Λ + W (x) t 2 (Λ) Λ Z ν s t s s t ν 1 t r tr r2 t t W : Λ R s t t E R s s t t dist(e, σ(h)) η t η (0, 1] x, y Λ (H E) 1 (x, y) 2η 1 e η 12ν x y. Pr t µ > 0 t s t r 1 x 0 Λ t r t r t t F = F x0 2 (Λ) 2 F u(x) = F x0 u(x) = e µ x 0 x 1 u(x). r 2 r t r A ( F 1 x 0 AF x0 (x, y) = e µ x 0 x 1 A(x, y)e µ x 0 y 1. s s t s s s t rt r t rs t t r 2 2 t r t t s

s s st t tt F = F x (H E) 1 (x, y) = e µ x y 1 F 1 (H E) 1 F (x, y) = e µ x y 1 (F 1 HF E) 1 (x, y) e µ x y 1 (F 1 HF E) 1. r r s t r s t q t Pr s t (F 1 HF E) 1 = (H E) 1 (F 1 HF E) 1 (F 1 HF H)(H E) 1 2 (F 1 HF E) 1 (I + (F 1 HF H)(H E) 1 ) = (H E) 1. (F 1 HF H)(H E) 1 < 1 t t r t r (I+(F 1 HF H)(H E) 1 ) s rt t (F 1 HF E) 1 = (H E) 1 (I + (F 1 HF H)(H E) 1 ) 1. s t r t r F 1 HF H q t s X (Fx 1 HF x H)(u, v) X e µ x u 1 e µ x v 1 1 v Λ v: u v 1 =1 2νµe µ. r t st q t2 s t t r u v 1 1 rt r r a 1 µ > 0 x u 1 x v 1 u v 1 1. e µa 1 e µ 1 = e µ 1 Z 1 0 µe µt dt µe µ s st t t r u s t 2 1 t r s u v s F 1 HF H 2νµe µ. t µ = η 12ν η 1 (F 1 HF H)(H E) 1 (F 1 HF H)(H E) 1 2νµe µ 1 η = 2ν η 12ν e η 1 12ν η 1 2.

t r t s 2s s s s t t e η 12ν e 3 s η 1 r r t r t r I + (F 1 HF H)(H E) 1 s rt s t s r s t t (I + (F 1 HF H)(H E) 1 ) 1 2. q t s t (F 1 HF E) 1 = (H E) 1 (I + (F 1 HF H)(H E) 1 ) 1 2 2 η (H E) 1 (x, y) e µ x y 1 (F 1 HF E) 1 2 η e η 12ν x y 1. 2 rt 2 t r t s t 2 rt 2 t r t rt (u) Zd s 2 t r t rt 2 t r t P t r s diam Πu max i6=j u i u j (2 + r 0 ), P rt 2 t r t 2 t r t s r s t r r t t st t t s 2s s t 1ts t t rs r 2 t t P s t s st t t (u) s P t t r 1 sts s s t J {1,..., } t 1 card J 1 s t t dist Π J (u), Π J cc() (u) > r 0. Pr t s t t s t t Z d, R d t t t t ss R d s ts r tt s t R := 2 + r 0 ss t t diam Πu = max i,j u i u j > R t s R/2 (u i) 1 i r t s t t r r s t r s t t t ts t r 2 (2(R/2)) = R diam Πu R tr ts t 2 t s s r r t r 1 sts

2 rt 2 t r t s 1 s s t J {1,..., } s t t u j1 u j2 > 2(R/2) r j 1 J j 2 J c t s s t t dist Π J (u), Π J cc() (u) = mi dist C (1) j 1 J,j 2 J c (u j 1 ), C (1) (u j 2 ) mi j 1 J,j 2 J c u j 1 u j2 2 > r 0. t (u) s P 2 t r t t s (u) = C(0 ) (u0 ) C (00 ) (u00 ), dist ΠC (0 ) (u0 ), ΠC (00 ) (u00 ) > r 0, r u 0 = u J = (u j : j J ) u 00 = u J c = (u j : j J c ) 0 = card J 00 = card J c r t s t (u) s t 2 ss t t t s t s t t r s s t t P (u) t t r rt t r t s s s U(u) = U(u 0 ) + U(u 00 ) s s t t t r rt s s r t 0 rt s s2st s r 2 0 < t r t tr r rt s s t 2 > 2r 0 s r t rt 2 t r t (u) C() (v) t x y > 7 s Pr r s R > 0 Π (u) ΠC() (v)) =. R < x y = max 1 j x j y j, t t r 1 sts 1 j 0 s t t x j0 y j0 > R t s r 2 t r t s r t t rs x, y r t x j0 x i diam Πx (2 + r 0 ), y j0 y j diam Πy (2 + r 0 ). 2 tr q t2 r 2 1 i, j R > 7 > 6 + 2 r 0 x i y j x j0 y j0 x j0 x i y j0 y j > 6 + 2r 0 2(2 + r 0 ) = 2.

t r t s 2s s s r r r 2 1 i, j mi dist(c (1) i,j (x i), C (1) (y j)) mi x i y j 2 > 2( 1) 2. i,j s s t t dist(π (x), ΠC() t t α = 3/2 (y)) = mi i,j dist C (1) (x i), C (1) (y j) > 0. t s 2s s s s t s { } 0 s s s s t s { } 0 s s q t rs 2 t t t s 0 > 0(r 0, N, d) r s r 0 (N, d, r 0 ) > 0 2 t r rr r t = b α c + 1. s t s { } 0 s ss t s t t t t s 2s s 1 t t t t rs t 2s s t s t r q r t t 0 r t t m > 0 E R (u) Zd 1 N s (E, m) s r (E, m) E / σ(h () max v (u) G () C r γ(m,, ) = γ(m,,, N) s 2 (u)) (u)(u, v; E) e γ(m,,), γ(m,, ) = m(1 + 1/8 ) N +1 > m. t r s t s (E, m) s r (E, m) rt (u) E R t 2 M sep ( (u), E) t 1 r r s s r (E, m) s r s (u (j) ) (u) 2 M sep PI (C() (u), E) t 1 r r s s r (E, m) s r P s (u (j) ) (u) 2 M PI ( (u), E) t 1 r t ss r 2 s r (E, m) s r P s (u (j) ) (u) t u (j) u (j0) > 7N r j 6= j 0

2 rt 2 t r t s 2 M FI ( (u), E) t 1 r (E, m) s r s (u (j) ) (u) t u (j) u (j0) > 7N r j 6= j 0 2 M( (u), E) t 1 r (E, m) s r s (u (j) ) (u) t dist(u (j), (u)) > 2 u (j) u (j0) > 7N r j 6= j 0 rt r s t M PI ( (u), I) := sup E I M FI ( (u), I) := sup E I M PI ( (u), E). M FI ( (u), E). r 2 2 M PI ( (u), E) + M FI ( (u), E) M( (u), E). M( (u), E) κ()+2 t κ() = t M sep ( (u), E) M PI ( (u), E) κ() + 2 t M sep PI (C() (u), E) 2 Pr ss t t M sep ( (u), E) < 2 t r s r s r s r s (u) t M( (u), E) κ()+2 (u) st t t st κ() + 2 s (v i ) 0 i κ() + 1 r s r t s t s 2 v i v i 0 > 7N r i 6= i 0 t t r 2 t r r t st κ() s 2 (y i ) s t t 2 (x) t x / S κ() j=1 C() 2 (y j ) s s r r (v 0 ) r i 6= i 0 v i v i 0 > 7N t r st t st t r v i r 2 (y j ) 1 j κ() t tr t κ() + 1 κ() v i S j C() 2 (y j ) i = 1,, κ() + 1 s 2s s s tr s r 2 P s t t 1 N s 2 t t t s s ts A, B Z d t A B 6= r t r 1 st x A y B s t t x y = 1 t t t 2 t s (u (j) ) (u (j0) ) t u (j) u (j0) > 7N r t t 2 s r

t r t s 2s s s t t r t r t rt s t r st t t 1t r r s 2 t r t r J := + 5 t m = (12Nd) 1/2 0 E R s t t (u) s E M PI ( (u), E) + M FI ( (u), E) J t r 1 sts 2(J, N, d) > 0 s t t 0 2(J, N, d) t t (u) s (E, m) Pr 2 t r r t st J s s 3 t (u) t t rs t st > 7N t t r (E, m) r r x i (u) t dist(x i, (u)) 2 i = 1,, r J s t t x (u) \ r[ i=1 2 (x i ), t (x) s (E, m) s r t t t s 2 (x i ) 2 (x i 0) r i 6= i 0 t t dist( 2 (x i ), 2 (x i 0)) x i x i 0 4 > 7N 4 > 1. t t t x + 2 (x j ) r s j = 1,, r t x / S r i=1 C() 2 (x i ) 1t t Λ (u) x Λ y (u) \ Λ t s r t r s t t t2 t t s G () (x, y; E) = X (u) +1 G () (x, y; E) (u) +1 (z,z 0 ) Λ " X z Λ G () Λ G () Λ (x, z; E) (x, z; E)G() (u) (z0, y; E). +1 # G () (z 1, y; E) r s z 1 + Λ 1 v (u) t x (u) t dist(x, (u)) t s s

2 rt 2 t r t s (x) s (E, m) t s s X G () (x, z; E) 2 d d d 1 e γ(m,,), z (x) 2 G () (x, v; E) (u) 2d d d 1 e γ(m,,) G () (z 1, y; E), +1 r s z 1 + (x) (x) s (E, m) t s s t t x 2 (x i ) r s i = 1,, r dist( 2 (x i ), (u)) +1 t q t s G () (x, v; E) (u) +1 r s z 1 + 2 (x i ) + 1 s X z 2 (x i ) G () 2 (x i ) (x, z; E) G () (z 1, v; E) (u) +1 s r t t dist( 2 (x i ), (u)) dist(z 1, (u)). t s 2 (x i ) r E G () (u) (x, v; E) 2Nd Nd(2 ) d 1 e (2 ) 1/2 G () (u) (z 1, v; E) s t (z 1 ) s (E, m) 2 tt G () (x, v; E) (u) (2Nd Nd) 2 (2 ) d 1 d 1 +1 r s z 2 + (z 1 ) s e (2 ) 1/2 γ(m,,) G () (u) (z 2, v; E) (2 Nd Nd) 2 (2) d 1 ( + 1) 2(d 1) e (2 ) 1/2 γ(m,,) G () +1 (u)(z 2, v; E) G () (x, v; E) (m,,) G () (z 2, v; E) (u) e γ0 +1 (u) +1

t r t s 2s s s r γ 0 (m,, ) = γ(m,, ) 1 [(2 ) 1/2 + 2(d 1) l( + 1) s m = (12Nd) 1/2 x (u) dist W (x) = + l((2 Nd Nd) 2 )(2) d 1 )] γ(m,, ) 6Nd + 2 l(2 Nd Nd) 1/2 > m 6Nd + 2 l(2 Nd Nd) 1/2 0 > 0, 0 x, (u) t 2 d d d 1 e γ(m,,) x s s s e γ0 (m,,) x s s s. r s r s s 2 t t G () (u) (x, v; E) W (x) G() (u) +1 (z, v; E), r s z (u) st t G () (u) (u, v; E) r v (u) st rt r u z 1, z 2,... 2 2 t r r r t 2 ss tt t r r st s G () (u) (u, v; E) W (u) G() (z 1, v; E) W (u)w (z 1 ) G () (z 2, v; E) (u) +1 (u) +1 W (u)w (z 1 ) W (z r 1 ) G () (u) (z r, v; E). r t s t ss z 1,... z r 1 st s t s 2 t t s t r r s r t t t rst st r x = u t r s r s t s s t t t s t z 1,..., z r 1 s t2 (u) s (E, m) t r s t r s (u) s (E, m) u 2 (x i ) r s i 1t t 2 diam( 2 (x i )) t t r 2 (x i )) dist( 2 (x i ), (u)) diam( 2 (x i )) 4 > + 1,

2 rt 2 t r t s 0 > C(N, J) r s st t C(N, J) > 0 s t t s s r s t s t r 1 r 2 t r t s t s r s t 2 G () (u) (u, v; E) 2Nd Nd d 1 (e γ(m,,) ) r 1 (e γ0 (m,,) ) r 2 G () (u) (z r 1 +r 2 +1, v; E). t t t r r r 1 rr s t t s r t s i r t 2 r 1 4J 1 4J. 2 t ss t t (u) s E s G C () (u) (E) < e 1/2 +1 γ 0 (m,, ) > 0 e r 2γ 0 (m,,) 1 r r G () (u, v; E) 2 Nd Nd d 1 (u) e γ(m, r1,) e 1/2 +1 +1 r 1 e m0 m 0 = 1 ( r1 γ(m,, ) r 1 l(2 Nd Nd) d 1 1. 1/2 +1 4J r 1 1 t m 0 γ(m,, ) γ(m,, ) 4J 1 l(2 Nd Nd) d 1 ) 1 1/2 +1 γ(m,, ) γ(m,, ) 4J 1/2 1 (l(2nd Nd)) + (d 1) l( )) 3/4 γ(m,, )[1 (4J + l(2 Nd Nd) + Nd) 1/2 ] 0 2(J, N, d) r s 2(J, N, d) > 0 r γ(m,, ) = m(1 + 1/8 ) N +1 γ(m,, ) γ(m,, ) = 1 + 1/8 1 + 3/16! N +1 1 + 1/8 1 + 3/16

t r t s 2s s s r r t γ(m,, ) γ(m,, ) (1 (4J + l(2nd Nd) + Nd) 1/2 ) 1 + 1/8 1 + 3/16 (1 (4J + l(2 Nd Nd) + Nd) 1/2 ) > 1, r 0 2(J, N, d) r s r 2(J, N, d) > 0 t 2 C(J, N, d) t st t 4J + l(2 Nd 2Nd) + Nd s r t t 0 2(J, N, d) r s r 2(J, N, d) > 0 t s t t 2 t s 1 > C(J, N, d) 3/8 + C(J, N, d) 5/8 + 3/16, 1/8 > C(J, N, d) 1/2 + C(J, N, d) 5/8 + 3/16, 1 C(J, N, d) 1/2 + 1/8 C(J, N, d) 5/8 > 1 + 3/16, (1 + 1/8 )(1 C(J, N, d) 1/2 ) > 1 + 3/16, 2 t s r 1 + 1/8 1 + 3/16 (1 C(J, N, d) 1/2 ) > 1. r r t t t m 0 > γ(m, l +1, ) G C () (u) (u, v; E) e γ(m,,) s t s t r

t r t rt t s 2 3 t t r s t s t r r 3 t r t rt s2st s t r rt s 2 s t r N < N 2 2 r tr r2 t t s s t s 1 r t r st t s 2s s s r rt t r t rs s r s N ss t s t r s ts ss t s t t t r rt t r t s t r U(x) = X 1 i<j U: (Z d ) R Φ( x i x j ), x = (x 1,..., x ) Z d, r Φ: N R + s t 2 s rt t t r 0 N : supp Φ [0, r 0 ]. r 0 t r t t r t U P t s ss t t 0 supp µ [0, + ) rt r t r t2 str t t F V s r t s r r s 2 s(f V, ε) := sup(f V (a + ε) F V (a)) C a R l 2A r s C (0, ) A > 3 2 4N p + 9Nd. st t s t r t r p tr r rt2 (DS.,, N)

t r t rt t s 2 3 t t r s r s ts r 2 = 1,..., N t 2 σ(h () (ω)) t s tr H () (ω)) E () 0 t σ(h () (ω)) r t 1 N r ss t s (I1) (P1) t r t2 [0, 4d] σ(h () (ω)) [0, + ). s q t 2 E () 0 := if σ(h () (ω)) = 0 s r r t ss t s (I1) (P1) t r 1 sts E > E (N) 0 s t t t P r t2 t s tr H (N) (ω) [E (N) 0, E ] s t2 r t 2 t Ψ i (x, ω) t E i (ω) [E (N) 0, E ] s 1 t 2 2 t t2 t r 1 st r st t m + > 0 r st t C i (ω) > 0 s t t Ψ i (x, ω) C i (ω)e m + x. 1 t 2 B 1 t s t r t s f : R R s t t f r r t ss t s (I1) (P1) r 2 s > 0 t r 1 sts E > E (N) 0 s t t r 2 s t K Z Nd 2 s (0, s ) X s E sup 2 f(h (N) (ω))p I (H (N) 2 (ω))1 K <, f B 1 HS r ( X Ψ)(x) := x Ψ(x) P I (H (N) (ω)) s t s tr r t H (N) (ω) t t t r I := [E (N) 0, E ] rr2 t t t t r rt s r2 r 1 t N t st r 1 t r q r s r t 0 rt s2st s r 0 = 1,..., 1 s r 2 1 2 t t t t t t st s t 2s s t s t s rt s2st t t s s2st s t 0 00 = 0 rt s r 2 0 = 1,..., 1 t s ss t t ss r2 s r r 0 rt s2st s t 2 0 = 1,..., 1

t t rt t s 2s s s r t s t rt t t t t s 2 s s t t st r 1 t s s s t t r s q s s 0 st rt t t s t r q r s 2 s t t s 2s s t st s t t t s 0 r t t s t r ss t t t t s t t t s 2s s s t s t 0 0(N, d, r 0 ). r s r 0(N, d, r 0 ) > 0 = 1,..., N t r rt2 r s m > 0 E > 0 2 rt t r r str t 2 (DS.,, N) r 2 r s r s (u) (v) P E I : (u), 2p 4N (v) r (E, m), r m > 0 E > 0 p > 6Nd I = (, E ] r 1 t t rt t s 2s s s t s s r 2 = 1,..., N rst r t r s t r s t s s s r t s st t r t st 2 t s t3 s2 t t s t s rt t H () (ω) = + V (x 1, ω) + + V (x, ω) + U(x) rt r r r r t r 2 (Z d ) r U V s t s 2 (I1) (P1) r s t 2 r 2 C > 0 t r 1 st r tr r 2 r 0 (C) > 0 C 1, c > 0 s t t r 2 0 (u) t st E () 0 (ω) H () (ω) s t s s (u) 0 P E () 0 (ω) 2C 1/2 0 C1 d 0e c1/4 0. Pr t t r t t t U s t t s r t 1 r t t t st E () 0 (ω) H () (ω) s r 0 2 t st E e() 0 (ω) t r t r t t t r t eh () 0 (u) (ω) = + V (x 1, ω) + + V (x, ω).

t r t rt t s 2 3 t t r s t H i = 2 (C (1) 0 (u i )) r t r r r tt s s X eh () (ω) = 1 H1 1 Hj 1 H (1) (u) j 0 j=1 {z } j 1 t s 1 Hj+1 1 H {z } j t s r H (1) j (ω) = (1) + V (x C (1) (u j j, ω) j = 1,..., s t H j t ) 0 st E e() 0 (ω) H e () (ω) s t r (u) 0 r E (1) 0,j ee () 0 (ω) = X j=1 E (1) 0,j (ω), s t st H(1) j s E (1) 0,j t t t t t2 t r t rs H (1) j t t s t t r 2 s 0 P ee () 0 (ω) s P E (1) 0,1 (ω) s. r t t 1t r r r s r r t t s rt r r r t r H (1) 1 r t s t s 3 t t r t t s t t t t r t t t s rr t r t t t s t t r r s t t r t s t s t3 t s r t s st t r ss t s (I1) (P1) r 2 p > 0 t r 1 sts r tr r 2 r 0 (N, d, p) s t t m := 12Nd 1/2 0 E := (12Nd)(2 N+1 m) t (DS.0, N) s r 2 = 1,..., N Pr t C = (12Nd) 2 2 N+1 t 0 (x) Z d s r ω Ω s t t t rst E () 0 (ω) H () 0 (ω) r E C 1/2 0 = E s t s s E() 0 (ω) > 2C 1/2 0 dist(e, σ(h () (ω))) = E() (u) 0 (ω) E > C 1/2 0 =: η > 0. 0 r r r 2 v 0 (u) t 2 t s s st t r G C () 0 (u) (E, u, v) 2C 1 1/2 0 e C 1/2 0 12Nd 0.

t t rt t s 2s s s s r t t C 1/2 0 12Nd r 0 r s = 2N+1 m s G C () 0 (u) (E, u, v) 2C 1 1/2 0 e 2N+1 m 0 2C 1 1/2 0 e 2γ(m, 0,) 0 e γ(m, 0,) 0, γ(m,, ) = m(1 + 1/8 ) N +1 < m 2 N. s s t t 0 (u) s (E, m) s ω {ω Ω (u) s (E, m) E E }. r r P 2 E E 0 (u) s (E, m) P E () 0 (ω) 2C 1/2 0, P E () 0 (ω) 2C 1/2 0 C1 d 0e c1/4 0. 2 t q t t2 C 1 d 0e cd0/4 2p 4N r 0 s t 2 r s ss t 0 s r s t s t r t2 r t s t s r t t s r 2 s 2 t r t2 t r t t s r t r st t t r t r t rs m E r s r t 2 I t t r (, E ] rt t s t st st s t s rt t s r rt2(ds.0, 1 N) t r s t r r t t st + 1 r t s rt r t H (1) (ω) 2 r r r ss t s (I1) (P1) t r 1 sts e (d) > 0 s t t 0 e r 0 (DS., 1 N) s tr t (DS. + 1, 1 N) s tr r t r t t r 1 r2 r s t t tm(c (1) (u); E) s t t t r (E, m) s C (1) (v j ) C (1) (u) t v j v j0 > 7N r j 6= j 0

t r t rt t s 2 3 t t r s r ss t s (I1) (P1) t 0 ss t t r rt2 (DS., 1 N) s tr r rs s r s rt s r 2 1 P M(C (1) (u); E) 2 C(, d) 2 dα 2 p 4N 1. Pr t r 1 st 2 r s s r s rt s C (1) (u (j) ) C (1) (u) 1 j 2 t t s s r s ts r 2 r C (1) (u (2i 1) ) C (1) (u (2i) ) t t s H (1) C (1) H(1) r t s r t r (u (2i 1) ) C (1) (u (2i) ) s tr r t s r i = 1,, s t A i = { E I : C (1) (u (2i 1) ) C (1) (u (2i) ) r (E, m) }. s ss t (DS., 1 N) r i = 1,, 2p 4N 1 P {A i }, 2 ts A 1,, A ( \ ) P A i i=1 2p 4N 1 s t t t t r r t s 2 s C (1) (u (j) ) s 2 1 (2 )! C(1). (u) 2 2 dα C(, d). Pr r t C (1) (x) C (1) (y) r s r s t B +1 = { E I : C (1) (x) C (1) (y) r (E, m) } Σ = { E I : t r C (1) (x) r C (1) (y) s E } M x = { E I : M(C (1) (x); E) 6} M y = { E I : M(C (1) (y); E) 6} s + 5 = 6 r = 1 t ω B +1 \ Σ M x t E I C (1) (x) r C (1) (y) s E M(C (1) (x); E) 3 C (1) (x) t E 2 t (E, m) t C (1) (y) s E (E, m) s s 2 t t M(C (1) (y); E) 6

t rt t s t s ω M y B +1 Σ M x M y s r r2 2 t P {B +1 } P {Σ} + P {M x } + P {M y } 4N p +1 + C(d) 6 α p 4N 1 +6d +1 r = 2 1 4N 1 2p +1 + 1 4N 1 2p +1 = 2 2 2p 4N 1 +1. p > 4αNd 2 ss t r 0 e r s r e > 0 t rt t s t = 2,..., N ss (DS. 1, 0 N) r 0 = 1,..., 1 (DS., 0 N) r 2 0 = 1,..., 1 r (DS.+1,, N) s r t 2 r t t r t2 s rs s (x) (y) r t P s (x) (y) r t s t s (x) r (y) s P t t r s

t r t rt t s 2 3 t t r s P rs P s D Z 2 y C (2) (y) x C (2) (x) C (1) (x 2 ) C (1) (x 1 ) C (1) (y 2 ) C (1) (y 1 ) P rs P s r d = 1 = 2 x = (x 1, x 2 ) y = (y 1, y 2 ) s r r r s r rt 2 t r t s (x) (y) r t t (DS. + 1,, N) s r s r t (u) = C (0 ) (u 0 ) C (00 ) (u 00 ) P s r t x = (x 0, x 00 ) r 2 t x (u) t s 2 s (u 0, u 00 ) rr s t H () C s t r (u) +1 H () (u) Ψ(x) = ( Ψ)(x) + [U(x0 ) + V(x 0, ω) + U(x 00 ) + V(x 00, ω)] Ψ(x) r t r H () = ) (u) H(0 +1 C (0 ) C (00 ) (u 0 ) I + I H(00 ) (u 00 ). r r s t t s r t 2 G (0) (u 0, v 0 ; E) G (00) (u 00, v 00 ; E) t r t s r t t H (0 ) H(00 ) C (0 ) (u 0 ) C (00 ) r s t 2 t +1 (u 00 ) +1 {(λ i, ϕ i )} i=1,..., C ( 0 ) (u 0 ) {(µ j, φ j )} j=1,..., C ( 00 ) (u 00 )

t rt t s t t s rr s t s H (0 ) s t rs Ψ ij H () (u) C (0 ) C (00 ) (u 0 ) H(00 ) s t s r r ts (u 00 ) Ψ ij (x) = ϕ i (x 0 ) φ j (x 00 ), rr s s r 2 t s s E ij = λ i + µ j. t rs t t s r s s q t r ts t s ss r 3 tr t t t 1 N E R (u) = ) C(0 (u0 ) C (00 ) (u00 ) E s r P (u) s E 2 r s t r µ j σ(h (00 ) ) t ) C (00 ) (u 00 ) C(0 (u0 ) s (E µ j ) r λ i σ(h (0 ) ) t C(00 ) H (0 ) (u0 ) (u00 ) s (E λ i ) t (E, m) t t 1 N E R m > 0 s r P (u) = ) C(0 (u0 ) C (00 ) (u00 ) (u) s (E, m) t t (E, m) µ j σ(h (00 ) ) s t t ) C (00 ) (u 00 ) C(0 (u0 ) t s t s r (E µ j, m) s C (0 ) l (v 1 ) C (0 ) l (v 2 ) t = bl α c + 1 t r s t s (E, m) t t (E, m) (E, m) r t t (E, m) λ i σ(h (0 ) ) s t t C(00 ) C (0 ) (u0 ) t s t s r (E λ i, m) s C (00 ) l (v 1 ) C (00 ) l (v 2 ) t = bl α c + 1 t r s t s (E, m) r t t (E, m) (u00 ) (E, m) t (E, m) t r t s (E, m) r (E, m) t r s t s (E, m) t (E, m) r r t r r r t 1t t E R P (u) = ) C(0 (u0 ) C (00 ) E t (u00 ) s t

t r t rt t s 2 3 t t r s t r t r 1 st 1/α x C (0 ) (u0 ) s t t t rt r t = C (0 ) (x) C (00 ) (u00 ) (u) s E r t r 1 st 1/α x C (00 ) (u00 ) s t t t rt r t = C (0 ) (u0 ) C (00 ) (x) (u) s E Pr 2 t (u) s t E t t r t r 1 sts µ j σ(h (00 ) ) s t t ) C (00 ) (u 00 ) C(0 (u0 ) s t E µ j r t r 1 sts λ i σ(h (0 ) ) s t t C(00 ) C (0 ) (u0 ) (u00 ) s t E λ i t s rst s s C (0 ) (u0 ) s t E µ j t r 1 sts 1/α x C (0 ) (u0 ) s t t C (0 ) (x) C (0 ) (u0 ) C (0 ) (x) s E µ j dist(e µ j, σ(h (0 ) )) < C (0 ) (x) e β r r t r 1 sts η σ(h (0 ) s r = C (0 ) (x) C (00 ) σ(h () ) = σ(h (0 ) C ) + σ(h(00 ) () C (0 ) (x) C( 00 ) (u 00 ) ) s t t E µ C (0 ) j η < e β (x) (u00 ) s t (u) s P ) dist(e, σ(h () )) E µ j η < e β. s s E s r ts s t t s r s s rs t α = 3/2 r 2 p > 6Nd ss t t t r t t s t s s (P1) t A > 4 N α p + 6αNd t (u) = ) C(0 (u0 ) C (00 ) (u00 ) (v) = ) C(0 (v0 ) C (00 ) (v00 ) t s s t t C ) (u) C() (v) r P s t P E R t r (u) r C() (v) s E 4Np. s t t (u) s P C() (v) s P E R (u) s t E C() (v) s t E 4N p.

t rt t s t Pr s P{ E R t r (u) r C() (v) s E } P{ E R 1, 2 ; 1/α 1, 2 x C (0 ) (u0 ) y C (0 ) (v0 ) 1 (x) C (00 ) (u00 ) (u) ) C(0 2 (y) C (00 ) (v00 ) (v) r E } C (0 ) + P{ E R 1, 2 1/α 1, 2 x C (00 ) (u00 ) y C (00) (v 00 ) C (0 ) (u0 ) C (00 ) 1 (x) (u) ) C(0 (v0 ) C (00 ) 2 (y) (v) r E } + P{ E R 1, 2 1/α 1, 2 x C (0 ) (u0 ) y C (00 ) (v00 ) 1 (x) C (00 ) (u00 ) (u) C(0 (v0 ) C (00 ) 2 (y) (v) r E } C (0 ) + P{ E R, 1, 2 : 1/α 1, 2 x C (00 ) (u00 ) y C (0 ) (v0 ) C (0 ) (u0 ) C (00 ) 1 (x) (u) ) C(0 2 (y) C (00 ) (v00 ) (v) r E } t s t t rst t r t r ts r t t r t r s r s r st t { E R, dist(e, σ(h C ( 0 ) 1 (x) C (00 ) (u 00 ) )) < e β 1, dist(e, σ(h C ( 0 ) 2 (y) C (00 ) (v 00 ) )) < e β 2 } {dist(σ(h C ( 0 ) 1 (x) C (00 ) (u 00 ) ), σ(h C (0 ) (3) 2+1)d Cst A/α. (y) C (00 ) 2 )) < e β/α } (v 00 ) s r t (P1) t t r ss t rs t r s t s r s t r ss s 1, 2 t (3)2d 2 (3) (2+1)d Cst A/α 1 4 4N p, s A > 4 N pα + 9Nd 2 t s t r r t2 t r s 2 s t ss rt t

t r t rt t s 2 3 t t r s P{ E R (u) s t E C() (v) s t E } P{ E R 1, 2 1/α 1, 2 x C (0 ) (u0 ) y (v), 1 (x) C (00 ) (u00 ) (u) C() 2 (y) (v) r E } + P{ E R 1, 2 1/α 1, 2 x C (00 ) 1 (u 000 ) y (v) C (0 ) C (0 ) (u0 ) C (00 ) 1 (x) C() (u) 2 (y) (v) r E }. t t rst t r t t t r t s t r s s r t s s s rt (3) 2d 2 (3) 2+1)d Cst A/α 1 2 4N p, s A > 4 N pα + 9Nd r r t s t t r t2 t r s 2 s t ss rt s ts r t 2 t r t r r ss t (P2) st t t r t s t r t s r s r 2 s s r st r s t E I (u) P ss t t (u) s (E, m) E (u) s (E, m) Pr s r P (u) = C (0 ) (u 0 ) C (00 ) (u 00 ) t {λ i, ϕ i } {µ j, φ j } t s rr s t rs H (0 ) H(00 ) C (0 ) (u 0 ) C (00 ) r s t 2 s t t rs Ψ ij rr s s E ij H () C (ω) s s (u) Ψ ij = ϕ i φ j, E ij = λ i + µ j. (u 00 ) 2 t ss E r rt2 t (u) r s λ i s C (00 ) (u00 ) s E λ i 1t 2 ss t E C (00 ) (u 00 ) s t t 2 r s r (E λ i, m) s r s 1 t r r 2 M( (u), E) < κ() + 2 s t t t s s

t rt t s t (E λ i, m) 2 max {λ i } max v 00 C (00 ) (u 00 ) s 2s s r C (0 ) max {µ j } max v 0 C (0 ) (u 0 ) G (00) (u 00, v 00 ; E λ i ) (u0 ) s s G (0) (u 0, v 0 ; E µ j ) e γ(m,, 00 ). e γ(m,, 0 ). r 2 v (u) u v = t s t r v 0 u 0 = r v 00 u 00 = s r rst t tt r s q t s t G () (u, v; E) = X i (2 + 1) ( 1)d max {λ i } t 2 t X ϕ i (u 0 )ϕ i (v 0 )φ j (u 00 )φ j (v 00 ) E λ i,j i µ j ϕ i (u 0 )ϕ i (v 0 ) G () (u 00, v 00 ; E λ i ) max v 00 (u 00 ) G () (u 00, v 00 ; E λ i ), (2 + 1) ( 1)d e γ(m,, 1) = e [γ(m,, 1) 1 l(2) ( 1)d ]. γ(m,, ) = m(1 + 1/8 ) N +1, t m = (12Nd) 1/2 0 r r r 2 N γ(m,, 1) γ(m,, ) > 1 l(2 + 1) ( 1)d. s tt C 1 = 12Nd γ(m,, 1) γ(m,, ) = m 1/8 (1 + 1/8 r 0 s t 2 r 1 ) N +1 s ϕ 1 = C 1 1/2 0 1/8 (1 + 1/8 ) N +1 > C 1 5/8, l(2 + 1) ( 1)d 1 ( 1)d(3 ) 3/8 C 1 5/8. s (u) s (E, m) 2 t s u 0 v 0 = s s r

t r t rt t s 2 3 t t r s t 2 N ss r rt2 (DS., 0 N) r 2 1 0 < r 2 P (y) s P E I, (y) s (E, m) 1 4N 4p +1. 2 Pr s r P (y) = C (0 ) (y 0 ) C (00 ) (y 00 ) t ts T y 4 r 1 sts E I s t t (y) s (E, m) t 6 T y 4 r 1 st E I µ j σ(h (00 ) ) s t t ) C (00 ) (y 00 ) C(0 (y 0 ) t s +1 t s r (E µ j, m) s r s s 3 6 RT y 4 r 1 st E I λ i σ(h (0 ) ) s t t C(00 ) C (0 ) (y 0 ) (y 00 ) t s +1 t s r (E λ i, m) s r s s 3 6 2 t T y T y 0 RT y 00 s E I µ j 0 E µ j E r 2 j E µ j I rt r s r rt2 (DS., 0 N) s r r t s s s t t s t t ) P {T y } C(0 (y 0 ) 2 C (00 ) 2 +1 (y 00 ) 2p4N 0 C(, N, d) 2p 4 P {RT y } C(, N, d) 2p 4 N ( 1) α +3( 1)d +1. N ( 1) α +3( 1)d +1, P E I : (y) s (E, m) C(, N, d) 4N ( 1) 2p α +3( 1)d +1. ss rt s 2 s r t t 2p 4 N ( 1) /α 3( 1)d > 4p 4 N r α = 3/2 r 0 s r p > 4αNd = 6Nd r t = 2,..., N r 1 sts 1 = 1(N, d) > 0 s t t 0 1 r 0 (DS., 0 N) s tr r 2 0 = 1,..., 1 t (DS. + 1,, N) s tr r 2 r s r P s (x) (y)

t rt t s t Pr t (x) (y) t s r P s s r t ts B +1 = E I : (x) (y) r (E, m), R = E I : t r (x) r (y) s E, T x = E I : (x) s (E, m), T y = E I : (y) s (E, m). ω B +1 \ R t E I (x) r (y) s E (y) s E t t st (E, m) t r s t (E, m) 2 r 2 (x) s E t t st (E, m) s s t t B +1 R T x T y. r r P {B +1 } P {R} + P{T x } + P{T y } P {R} + 1 4N 4p +1 + 1 4N 4p +1, 2 2 r s t st t P{T x } P{T y } s 2 t t P {R} 4N p +1 2 P {B +1 } 4N p +1 + 4p4N +1 < 2p4N +1. t t t t s ss t t (DS. 1, 0 N) s tr r 0 = 1,..., 1 t P M PI ( (u), I) κ() + 2 32d 2 2d +1 4N p + 4p 4N. Pr s t t M PI ( (u), I) κ()+2 t 2 M sep PI (C() (u), I) 2 t r r t st t s r (E, m) s r P s (u (j 1) ) (u (j 2) ) s (u) r ss rs t rs {u (j 1), u (j 2) } s t t (u (j 1) ), (u (j 2) ) (u)

t r t rt t s 2 3 t t r s s 2 32d 2 2d +1 s tt t P {B } 4N p B = { E I (u (j 1) ) (u (j 2) ) r (E, m) }, P M sep PI (C() (u), I) 2 32d 2 2d +1 P {B } 4p 4N + 2 r B s s r P rs s D Z 2 y C (2) (y) x C (2) (x) C (1) (x 2 ) C (1) (x 1 ) C (1) (y 2 ) C (1) (y 1 ) P rs s r d = 1 = 2 x = (x 1, x 2 ) y = (y 1, y 2 ) r s t r (DS. + 1,, N) r r s r 2 t r t s (x) (y) r s t t s s s t s r t r r2 r s t 0 ss t t r rt2 (DS.,, N) s tr r rs s r s r 2 1 P M FI ( (u), I) 2 C(, N, d, ) 2 dα 2 p 4N.

t rt t s t Pr s t r 1 st 2 r s s r 2 t r t s (u (j) ) (u) j = 1,..., 2 2 r 2 r (u (2i 1) ) (u (2i) ) t rr s r t s H () (u (2i 1) ) H () r t s r t r s tr t r r (u (2i) ) t s r i = 1,..., s r t ts A i = E I : (u (2i 1) ) (u (2i) ) r (E, m) 2 ss t (DS.,, N) r i = 1,..., 2 ts A 1,..., A \ Y P A i = P(A i ) ( 1 i 2p 4N P {A i }, i=1 2p 4N.. t t r t t t t t t r r t s 2 s (u (j) ) (u) j = 1,..., 2 s 2 1 (2 )! +1 (u) 2 C(, N,, d) 2 dα. r t = 2,..., N r 1 sts 2 = 2(N, d) > 0 s t t 0 2 r 0 (DS. 1, 0 N) r 0 = 1,..., 1 s tr (DS., N) s tr r rs s t (DS. + 1,, N) s tr r 2 r s r s (x) (y) s t t t t (DS. 1, N) s ss t Pr s r r s r s (x) (y) s t J = κ() + 5 B +1 = E I : (x) (y) r (E, m), Σ = E I : t r (x) r (y) s E, S x = E I : M( (x); E) J + 1, S y = E I : M( (y), E) J + 1.

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