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❊♠♣✐r✐❝❛❧ s♣❡❝tr❛❧ ❞✐str✐❜✉t✐♦♥ ✭❊❙❉✮ ❚❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✴ ❞❡♥s✐t② ♦❢ st❛t❡s ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ C ✱ n µ A = 1 � δ λ i ( A ) , n i =1 ✐✳❡✳ ❢♦r ❛♥② s❡t I ⊂ C n µ A ( I ) = 1 � 1 ( λ i ( A ) ∈ I ) n i =1 ✐s t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ ❡✐❣❡♥✈❛❧✉❡s ✐♥ I ♦r ❡q✉✐✈❛❧❡♥t❧②✱ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ t②♣✐❝❛❧ ❡✐❣❡♥✈❛❧✉❡ ✐s ✐♥ I ✳ n fdµ A = 1 � � f ( λ i ( A )) . n i =1

❑✐r❝❤♦❢❢ ▼❛tr✐①✲❚r❡❡ ❚❤❡♦r❡♠ ■❢ G ✐s ❛♥ ✉♥❞✐r❡❝t❡❞ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ t❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ s♣❛♥♥✐♥❣ tr❡❡s ♦❢ G ✐s ❡q✉❛❧ t♦ t ( G ) = 1 � | λ i | , n λ i � =0 ✇❤❡r❡ λ i = λ i ( L ) ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ � 0 − 1 log | λ | dµ L ( λ ) − 1 n log t ( G ) = n log n. −∞

❈❧♦s❡❞ ♣❛t❤s ❋♦r t ✐♥t❡❣❡r✱ ❧❡t S t = |{ ❝❧♦s❡❞ ♣❛t❤s ♦❢ ❧❡♥❣t❤ t ✐♥ G }| ❲❡ ❤❛✈❡ n � λ i ( X ) t = n S t = Tr { X t } = � λ t dµ X . i =1 ■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r z ∈ C ✱ Im ( z ) > 0 ✱ λ t 1 S t � � 1 � � z t +1 = z t +1 dµ X = z − λdµ X n t � 0 t � 0 ✐s t❤❡ ❈❛✉❝❤②✲❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ ♦❢ µ X ✳

❘❡t✉r♥ t✐♠❡s ■❢ Z t ✐s t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✇✐t❤ tr❛♥s✐t✐♦♥ ♠❛tr✐① P ✱ n 1 P ( Z t = v | Z 0 = v ) = 1 � � n Tr { P t } = x t dµ P . n v =1 ❙✐♠✐❧❛r❧②✱ ❢♦r t > 0 r❡❛❧✱ ✐❢ Z t ✐s t❤❡ ▼❛r❦♦✈ ♣r♦❝❡ss ✇✐t❤ ❣❡♥❡r❛t♦r L ✱ n 1 � � e tL dµ L . P ( Z t = v | Z 0 = v ) = n v =1

P❛rt ■■ ✿ ❙♣❡❝tr✉♠ ❛♥❞ ●r❛♣❤ Pr♦♣❡rt✐❡s ❊✐❣❡♥✈❡❝t♦rs

▲♦❝❛❧✐③❛t✐♦♥✴❉❡❧♦❝❛❧✐③❛t✐♦♥ ♦❢ ❡✐❣❡♥✈❡❝t♦rs ❚❛❦❡ X ♦r L ✐♥ t❤❡ r❡✈❡rs✐❜❧❡ ❝❛s❡✳ ▲❡t ψ 1 , . . . ψ n ❜❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ❚❤❡♥✱ ❢♦r ❛♥② k � 1 ✱ | ψ k (1) | 2 , . . . , | ψ k ( n ) | 2 � � ✐s ❛ ♣r♦❜❛❜✐❧✐t② ✈❡❝t♦r ♦♥ V ✳ ❊✐❣❡♥✈❡❝t♦rs ❛r❡ ♦❢ ♣r✐♠❡ ✐♠♣♦rt❛♥❝❡ ❢♦r t❤❡ st✉❞② ♦❢ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s ✐♥ ❞✐s♦r❞❡r ♠❡❞✐❛ ❆♥❞❡rs♦♥ ✭✶✾✺✻✮ ✱ q✉❛♥t✉♠ ♣❡r❝♦❧❛t✐♦♥ ❉❡ ●❡♥♥❡s✱ ▲❛❢♦r❡✱ ▼✐❧❧♦t ✭✶✾✺✼✮ ✳ ❊✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ ●♦♦❣❧❡ ♠❛tr✐① ❛r❡ ❛❧s♦ st✉❞✐❡❞✱ ❡✳❣✳ ❋r❛❤♠✱ ●❡♦r❣❡♦t✱ ❙❤❡♣❡❧②❛♥s❦② ✭✷✵✶✶✮ ✳

❉❡❧♦❝❛❧✐③❛t✐♦♥ ♦❢ ❡✐❣❡♥✈❡❝t♦rs ❍♦✇ ❢❛r ✐s ❛ t②♣✐❝❛❧ ❡✐❣❡♥✈❡❝t♦r ❢r♦♠ ❛ ✉♥✐❢♦r♠ ✈❡❝t♦r ♦♥ S n − 1 = { x ∈ R n : � x � = 1 } ❄ ❋♦r ❡①❛♠♣❧❡✱ ❞♦ ✇❡ ❤❛✈❡ q✉❛♥t✉♠ ❡r❣♦❞✐❝✐t②✱ ✐✳❡✳ n n 1 f ( v ) | ψ k ( v ) | 2 ≃ 1 � � � 1 ( λ k ∈ I ) f ( v ) . � k 1 ( λ k ∈ I ) n k v =1 v =1

P❛rt ■■ ✿ ❙♣❡❝tr✉♠ ❛♥❞ ●r❛♣❤ Pr♦♣❡rt✐❡s ▲♦❝❛❧ ◆♦t✐♦♥ ♦❢ ❙♣❡❝tr✉♠

❙♣❡❝tr❛❧ ♠❡❛s✉r❡ ❛t ❛ ✈❡❝t♦r ▲❡t A ❜❡ ❛ s②♠♠❡tr✐❝ ♠❛tr✐①✱ ✭❡✳❣✳ X ♦r L ✐♥ t❤❡ r❡✈❡rs✐❜❧❡ ❝❛s❡✮✳ ▲❡t ψ 1 , . . . ψ n ❜❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ❚❛❦❡ 1 � x � n ✱ ❞❡✜♥❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✱ n µ x � | ψ k ( x ) | 2 δ λ k . A = k =1 ❲❡ ❤❛✈❡ � λ t dµ x A = ( A t ) xx , ❛♥❞ n n n n n 1 A = 1 | ψ k ( x ) | 2 δ λ k = 1 | ψ k ( x ) | 2 = µ A . � µ x � � � � δ λ k n n n x =1 x =1 k =1 k =1 x =1

P❛rt ■■ ✿ ❙♣❡❝tr✉♠ ❛♥❞ ●r❛♣❤ Pr♦♣❡rt✐❡s ❊①tr❡♠❛❧ ❊✐❣❡♥✈❛❧✉❡s ❛♥❞ ❊①♣❛♥❞❡rs

❈❤❡❡❣❡r✬s ❈♦♥st❛♥t ❆ss✉♠❡ t❤❛t G ✐s ✉♥❞✐r❡❝t❡❞ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ ❋♦r X, Y ⊂ V ✱ ❞❡✜♥❡ � vol( X ) = deg( x ) . x ∈ X � E ( X, Y ) = 1 ( uv ∈ E ) . x ∈ X,y ∈ Y X Y ■s♦♣❡r✐♠❡tr✐❝ ✴ ❊①♣❛♥s✐♦♥ ❝♦♥st❛♥t ✿ E ( X, X c ) h ( G ) = min min (vol( X ) , vol( X c )) . X ⊂ V

❈❤❡❡❣❡r✬s ■♥❡q✉❛❧✐t② ▲❡t 1 = λ 1 > λ 2 � · · · � λ n � − 1 ❜❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ P ✳ 1 − λ 2 ✐s ❝❛❧❧❡❞ t❤❡ s♣❡❝tr❛❧ ❣❛♣ ♦❢ P ✳ ❚❤❡♦r❡♠ h ( G ) 2 � 1 − λ 2 � 2 h ( G ) . 2

❈❤❡❡❣❡r✬s ■♥❡q✉❛❧✐t② ❙✐♥❝❡ P = D − 1 X = D − 1 / 2 ( D − 1 / 2 XD − 1 / 2 ) D 1 / 2 , t❤❡ λ i ✬s ❛r❡ ❛❧s♦ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ S ✇✐t❤ S = D − 1 / 2 XD − 1 / 2 ✳ ❙✐♥❝❡ P 1 = 1 ✱ χ = D 1 / 2 1 ✐s t❤❡ ❡✐❣❡♥✈❡❝t♦r ♦❢ S ❛ss♦❝✐❛t❡❞ t♦ λ 1 = 1 ✳ ❍❡♥❝❡✱ ❢r♦♠ ❈♦✉r❛♥t✲❋✐s❤❡r ❢♦r♠✉❧❛✱ � Sx, x � λ 2 = max � x � 2 . x : � x,χ � =0 ❖r ❡q✉✐✈❛❧❡♥t❧②✱ � ( I − S ) x, x � 1 − λ 2 = min . � x � 2 x : � x,χ � =0

❈❤❡❡❣❡r✬s ■♥❡q✉❛❧✐t② ❲❡ s❡t π ( x ) = deg( x ) = ( D 1 )( x ) ✳ ❙✐♥❝❡ π = D 1 / 2 χ ✇✐t❤ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡ f = D − 1 / 2 x ✱ ❛❢t❡r s♦♠❡ ❛❧❣❡❜r❛✱ u ∼ v ( f ( u ) − f ( v )) 2 � 1 − λ 2 = min . � v deg( v ) f ( v ) 2 f : � f,π � =0 ▲❡t X s✉❝❤ t❤❛t E ( X, X c ) h ( G ) = min (vol( X ) , vol( X c )) . ❲❡ t❛❦❡ f ( v ) = 1 ( v ∈ X ) − 1 ( v / ∈ X ) vol( X c ) . vol( X )

❈❤❡❡❣❡r✬s ■♥❡q✉❛❧✐t② ❲❡ ❤❛✈❡ deg( x ) deg( x ) � � � f, π � = vol( X ) − vol( X c ) = 0 , x ∈ X x ∈ X c ❛♥❞ u ∼ v ( f ( u ) − f ( v )) 2 � 1 − λ 2 � � v deg( v ) f ( v ) 2 2 E ( X, X c )(1 / vol( X ) − 1 / vol( X c )) 2 = 1 / vol( X ) + 1 / vol( X c ) E ( X, X c ) 2 � min(vol( X ) , vol( X c )) � 2 h ( G ) .

❆❧♦♥✲❇♦♣♣❛♥❛ ❜♦✉♥❞ ■❢ G ✐s ❛ d ✲r❡❣✉❧❛r ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ ♦♥ n ✈❡rt✐❝❡s✱ t❤❡♥ λ 1 ( X ) = d ❛♥❞ √ λ 2 ( X ) � 2 d − 1 + o (1) . ❈♦♥s❡q✉❡♥t❧②✱ √ d − 1 1 − λ 2 ( P ) � 1 − 2 + o (1) . d

❘❛♠❛♥✉❥❛♥ ❣r❛♣❤s ▲❡t G ❜❡ ❛ d ✲r❡❣✉❧❛r ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ G ✐s ❘❛♠❛♥✉❥❛♥ ✐❢ √ λ 2 ( X ) � 2 d − 1 . ❚❤❡② ❛r❡ t❤❡ ❜❡st ♣♦ss✐❜❧❡ ❡①♣❛♥❞❡rs✳ ❚❤❡r❡ ✐s ❛ ❣❡♥❡r❛❧✐③❡❞ ❞❡✜♥✐t✐♦♥ ❢♦r ♥♦♥✲r❡❣✉❧❛r ❣r❛♣❤s ✭❧✐❢ts ♦❢ ❣r❛♣❤s✮✳

❊①✐st❡♥❝❡ ♦❢ ❘❛♠❛♥✉❥❛♥ ❣r❛♣❤s ❙❡q✉❡♥❝❡ ♦❢ ❘❛♠❛♥✉❥❛♥ ❣r❛♣❤s G 1 , G 2 , · · · ✱ ✇✐t❤ | V ( G n ) | ❣r♦✇✐♥❣ t♦ ✐♥✜♥✐t②✱ ❛r❡ ❦♥♦✇♥ t♦ ❡①✐st ✇❤❡♥ ✲ d = q + 1 ✇✐t❤ q = p k ❛♥❞ p ♣r✐♠❡ ♥✉♠❜❡r ▲✉❜♦t③❦②✱ P❤✐❧❧✐♣s ✫ ❙❛r♥❛❦ ✭✶✾✽✽✮✱ ▼♦r❣❡♥st❡r♥ ✭✶✾✾✹✮ ✳ ✲ ❛♥② d � 3 ✱ ▼❛r❝✉s✱ ❙♣✐❡❧♠❛♥✱ ❙r✐✈❛st❛✈❛ ✭✷✵✶✸✮ ✳

P❛rt ■■ ✿ ❙♣❡❝tr✉♠ ❛♥❞ ●r❛♣❤ Pr♦♣❡rt✐❡s ❊①tr❡♠❛❧ ❊✐❣❡♥✈❛❧✉❡s ❛♥❞ ❈♦♠❜✐♥❛t♦r✐❛❧ ❖♣t✐♠✐③❛t✐♦♥

❈♦♠❜✐♥❛t♦r✐❛❧ ❖♣t✐♠✐③❛t✐♦♥ ❆ss✉♠❡ t❤❛t G ✐s ✉♥❞✐r❡❝t❡❞ ❛♥❞ ❝♦♥♥❡❝t❡❞ ■❢ ∆ ✐s t❤❡ ♠❛①✐♠❛❧ ❞❡❣r❡❡ ❛♥❞ deg t❤❡ ❛✈❡r❛❣❡ ❞❡❣r❡❡ √ max( ∆ , deg) � λ 1 ( X ) � ∆ . ■❢ χ ✐s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ❛♥❞ λ n ( X ) � . . . � λ 1 ( X ) ✱ 1 − λ 1 ( X ) λ n ( X ) � χ � 1 + λ 1 ( X ) . ✳ ✳ ✳

P❛rt ■■ ✿ ❙♣❡❝tr✉♠ ❛♥❞ ●r❛♣❤ Pr♦♣❡rt✐❡s ❙♣❡❝tr❛❧ ●❛♣s ❛♥❞ ❈♦♥✈❡r❣❡♥❝❡ t♦ ❊q✉✐❧✐❜r✐✉♠

❙♣❡❝tr❛❧ ❣❛♣ ❚❤❡ s♣❡❝tr✉♠ ♦❢ P ❛♥❞ L ❝♦♥t❛✐♥ ♠❛♥② ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥✴♣r♦❝❡ss✳ ◆♦t❛❜❧② t❤r♦✉❣❤ t❤❡ s♣❡❝tr❛❧ ❣❛♣ − max λ � =0 Re λ ( L ) 1 − max λ � =1 Re λ ( P ) ❊✈❡♥ ♠♦r❡ ❝♦♥♥❡❝t✐♦♥s ❢♦r r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❝❤❛✐♥✴♣r♦❝❡ss✳ ❋♦r s✐♠♣❧✐❝✐t② ✇❡ ♦♥❧② ❝♦♥s✐❞❡r L ✳

❙♣❡❝tr❛❧ ❣❛♣ ❆ss✉♠❡ t❤❛t X ✐s r❡✈❡rs✐❜❧❡✳ ▲❡t Z t ❜❡ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✇✐t❤ ❣❡♥❡r❛t♦r L ✱ P x t = e tL e x ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦❢ Z t ❣✐✈❡♥ Z 0 = x ✳ ▲❡t λ 1 = 0 > λ 2 � · · · � λ n t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ L ❛♥❞ ψ 1 = 1 / √ n, . . . , ψ n ❛♥ ♦rt❤♦❣♦♥❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ❋r♦♠ t❤❡ s♣❡❝tr❛❧ t❤❡♦r❡♠ n � e tL e tλ i ψ i ψ ∗ = i i =1 n 1 P x � e tλ i ψ i ψ i ( x ) = n + t i =2

❙♣❡❝tr❛❧ ❣❛♣ ❘❡❝❛❧❧ t❤❛t Π = 1 /n ✐s t❤❡ ✐♥✈❛r✐❛♥t ❞✐str✐❜✉t✐♦♥✳ ❲❡ ❣❡t n t − Π � 2 = e 2 tλ i | ψ i ( x ) | 2 � e − 2 | λ 2 | t � P x � i =2 ❘❡❝❛❧❧ | x i | � √ n � x � . � � x � � i ❙♦✱ t − Π � TV � √ ne −| λ 2 | t . | ψ 2 ( x ) | e −| λ 2 | t � 2 � P x ✇❤❡r❡ t❤❡ t♦t❛❧ ✈❛r✐❛t✐♦♥ ♥♦r♠ ✐s � µ − ν � TV = 1 � | µ ( x ) − ν ( x ) | . 2 x

❙♣❡❝tr❛❧ ❣❛♣ ❚❤❡ ♠✐①✐♥❣ t✐♠❡ ♦❢ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✐s ✉s✉❛❧❧② ❞❡✜♥❡❞ ❛s t − Π � TV � 1 � P x τ = inf t> 0 max 2 . x max x | ψ 2 ( x ) | � τ � log n 2 | λ 2 | . | λ 2 | ✭◆♦t❡ t❤❛t max x | ψ 2 ( x ) | � 1 / √ n ✮✳ ❚❤❡r❡ ❛r❡ s✐♠✐❧❛r ❞❡✈❡❧♦♣♠❡♥ts ❢♦r r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❝❤❛✐♥s ❛♥❞ ❛❧s♦ ♣❛rt✐❛❧❧② ✐♥ t❤❡ ♥♦♥✲r❡✈❡rs✐❜❧❡ ❝❛s❡✳ ✭▲❡✈✐♥✴P❡r❡s✴❲✐❧♠❡r ✷✵✵✾✮✳

P❛rt ■■■ ✿ ❙♣❡❝tr✉♠ ♦❢ ❧❛r❣❡ r❛♥❞♦♠ ❣r❛♣❤s ❙✐♠♣❧❡ ♠♦❞❡❧s ♦❢ r❛♥❞♦♠ ❣r❛♣❤s

❆✈❡r❛❣❡ ❉❡❣r❡❡ ❚❤❡ ♥✉♠❜❡r ♦❢ ❞✐r❡❝t❡❞ ❡❞❣❡s ✐s � � | E | = deg + ( v ) = deg − ( v ) = − Tr { L } . v ∈ V v ∈ V ❍❡♥❝❡ deg( G ) = | E | n ✐s t❤❡ ❛✈❡r❛❣❡ ❞❡❣r❡❡ ♦❢ ❛ ✈❡rt❡①✳

❚❤r❡❡ ❘♦✉❣❤ ❈❧❛ss❡s ♦❢ ●r❛♣❤s ▲❡t G = ( V, E ) ✇✐t❤ n = | V | ≫ 1 ✳ ❲❡ s❛② t❤❛t G ✐s ❉❡♥s❡ ✐❢ deg( G ) = Θ( n ) . ❙♣❛rs❡ ✐❢ 1 ≪ deg( G ) = o ( n ) . ❉✐❧✉t❡❞ ✐❢ deg( G ) = O (1) .

❚❤r❡❡ ❘♦✉❣❤ ❈❧❛ss❡s ♦❢ ●r❛♣❤s ❚❤❡ s♣❡❝tr✉♠ ♦❢ ❧❛r❣❡ ❞❡♥s❡ ❛♥❞ s♣❛rs❡ r❛♥❞♦♠ ❣r❛♣❤s ❝❛♥ ❜❡ st✉❞✐❡❞ ✇✐t❤ t♦♦❧s ❢r♦♠ r❛♥❞♦♠ ♠❛tr✐① t❤❡♦r②✳ ❚❤❡ s♣❡❝tr✉♠ ♦❢ ❧❛r❣❡ ❞✐❧✉t❡❞ r❛♥❞♦♠ ❣r❛♣❤s ❝❛♥ ❜❡ st✉❞✐❡❞ ✭❤♦♣❡❢✉❧❧②✮ ✇✐t❤ t♦♦❧s ❢r♦♠ r❛♥❞♦♠ ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs✳

❊r❞➤s✲❘é♥②✐ r❛♥❞♦♠ ❣r❛♣❤ ❚❛❦✐♥❣ p ∈ [0 , 1] ✳ ❆ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ X ❤❛s ❛ Ber( p ) ❧❛✇ ✐❢ P ( X = 1) = p = 1 − P ( X = 0) . ✲ ❘❡✈❡rs✐❜❧❡ ♠♦❞❡❧ ✿ ( X ij ) 1 � i<j � n ❛r❡ ✐♥❞❡♣❡♥❞❡♥t Ber( p ) ✈❛r✐❛❜❧❡s ❛♥❞ X ij = X ji ✳ ✲ ◆♦♥✲r❡✈❡rs✐❜❧❡ ♠♦❞❡❧ ✿ ( X ij ) 1 � i � = j � n ❛r❡ ✐♥❞❡♣❡♥❞❡♥t Ber( p ) ✈❛r✐❛❜❧❡s✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❝♦✉❧❞ ❝♦♥s✐❞❡r ❛ ❣❡♥❡r❛❧ ❞❡♣❡♥❞❡♥❝❡ ♦❢ ( X ij , X ji ) ✳

❊r❞➤s✲❘é♥②✐ r❛♥❞♦♠ ❣r❛♣❤ ❲❡ ❤❛✈❡ � E deg( v ) = E X uv = ( n − 1) p. u � = v p ∈ (0 , 1) : dense np → ∞ , p = o (1) : sparse p = c/n : diluted .

■♥❤♦♠♦❣❡♥❡♦✉s r❛♥❞♦♠ ❣r❛♣❤s ▲❡t W : [0 , 1] 2 → [0 , 1] ❜❡ ❛ ❝♦♥st❛♥t ❜② ❜❧♦❝❦ ❢✉♥❝t✐♦♥ ✭✐♥❞❡♣❡♥❞❡♥t ♦❢ n ✮✳ ❋♦r ❡①❛♠♣❧❡✱ � W 11 � W 12 W ([0 , 1] 2 ) = . W 21 W 22 ❋♦r t❤❡ ♥♦♥✲r❡✈❡rs✐❜❧❡ ❝❛s❡✱ ✇❡ s❡t ( X ij ) 1 � i,j � n ❛r❡ ✐♥❞❡♣❡♥❞❡♥t Ber( p ij ) ✈❛r✐❛❜❧❡s ✇✐t❤ � i � n, j p ij = p W , n ❛♥❞ p = p ( n ) ❣✐✈❡s t❤❡ ♦r❞❡r ♦❢ t❤❡ ❛✈❡r❛❣❡ ❞❡❣r❡❡✳ ■♥ ❛ s❡♥s❡✱ ❢♦r p = 1 ✱ t❤❡s❡ ❣r❛♣❤s ❛r❡ ❞❡♥s❡ ❛♠♦♥❣ ❞❡♥s❡ ❣r❛♣❤s✳

❯♥✐❢♦r♠ d ✲r❡❣✉❧❛r ❣r❛♣❤s ❚❛❦❡ dn ❡✈❡♥✱ ✇❡ ♠❛② ❞❡✜♥❡ G ❛s ❛ r❛♥❞♦♠ d ✲r❡❣✉❧❛r s❛♠♣❧❡❞ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠ ✭♣r♦✈✐❞❡❞ t❤❡ s❡t ✐s ♥♦t✲❡♠♣t②✮✳ ■❢ d = o ( √ n ) ✱ G ❝❛♥ ❜❡ st✉❞✐❡❞ t❤❛♥❦s t♦ ❛ s✐♠♣❧❡ ♣r♦❜❛❜✐❧✐st✐❝ ♠♦❞❡❧✱ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♠♦❞❡❧✱ ❇♦❧❧♦❜ás ✭✶✾✽✶✮ ✳ ❚❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♠♦❞❡❧ ❝❛♥ ❛❧s♦ ❜❡ ✉s❡❞ t♦ st✉❞② ✉♥✐❢♦r♠ r❛♥❞♦♠ ❣r❛♣❤s ✇✐t❤ ❣✐✈❡♥ ❞❡❣r❡❡ s❡q✉❡♥❝❡s✳

❲❤② st✉❞②✐♥❣ t♦② ♠♦❞❡❧s ❄ ❱❡r② s✐♠♣❧❡ ♠♦❞❡❧s ♦❢ r❛♥❞♦♠ ♥❡t✇♦r❦s ✿ ♥♦ ✉♥❞❡r❧②✐♥❣ ❣❡♦♠❡tr②✳ ❚❤❡✐r st✉❞② ♠❛② ❤❡❧♣ t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ✐s ❞✉❡ t♦ ♥♦✐s❡ ✐♥ ❛ r❡❛❧ ✇♦r❧❞ ♥❡t✇♦r❦✳ ❚❤❡② ❛r❡ ♥❛t✉r❛❧ ❝❛♥❞✐❞❛t❡s ❢♦r ❜❡st ❡①♣❛♥❞❡rs✱ ❜❡st ♠✐①✐♥❣ t✐♠❡s✱ ❣r❛♣❤s ✇✐t❤ ❞❡❧♦❝❛❧✐③❡❞ ❡✐❣❡♥✈❡❝t♦rs✱ ❡t❝✳

P❛rt ■■■ ✿ ❙♣❡❝tr✉♠ ♦❢ ❧❛r❣❡ r❛♥❞♦♠ ❣r❛♣❤s ❉❡♥s❡ ✉♥❞✐r❡❝t❡❞ r❛♥❞♦♠ ❣r❛♣❤s

❉❡♥s❡ ❊r❞➤s✲❘é♥②✐ ❲❡ ✜① p ∈ (0 , 1) ✳ ❘❡✈❡rs✐❜❧❡ ♠♦❞❡❧ ✿ ( X ij ) 1 � i<j � n ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s Ber( p ) ❛♥❞ X ij = X ji ✳ ❊✐❣❡♥✈❛❧✉❡s ♦❢ X ❢♦r n = 100 ✱ p = 1 / 2 ✳

❉❡♥s❡ ❊r❞➤s✲❘é♥②✐ ❍✐st♦❣r❛♠ ♦❢ ❡✐❣❡♥✈❛❧✉❡s λ 2 ( X ) � . . . � λ n ( X ) ❢♦r n = 100 ❛♥❞ p = 1 / 2 ✳

❉❡♥s❡ ❊r❞➤s✲❘é♥②✐ ❍✐st♦❣r❛♠ ♦❢ ❡✐❣❡♥✈❛❧✉❡s λ 2 ( X ) � . . . � λ n ( X ) ❢♦r n = 1000 ❛♥❞ p = 1 / 2 ✳

❲✐❣♥❡r✬s s❡♠✐✲❝✐r❝❧❡ ❧❛✇ ❲✐❣♥❡r ♠❛tr✐① ✿ Y = ( Y ij ) 1 � i,j � n ✇✐t❤ ( Y ij ) i>j ✐✐❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ( Y ii ) i � 1 ✐✐❞ ❛♥❞ Y ji = Y ij ✱ ❚❤❡♦r❡♠ ■❢ E Y 11 = E Y 12 = 0 ✱ E Y 2 12 = 1 t❤❡♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❢♦r ❛♥② ✐♥t❡r✈❛❧ I ⊂ R ✱ n √ n ( I ) = 1 � λ i ( Y ) � � µ Y √ n ∈ I → µ sc ( I ) . 1 n i =1 √ 4 − x 2 dx ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛✳s✳ µ Y/ √ n ✇❤❡r❡ µ sc ( dx ) = 1 | x | � 2 ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ µ sc ✳

❙❡❝♦♥❞ ♠♦♠❡♥t ❆ss✉♠❡ Y ii = 0 ✳ � Y � 2 � 1 x 2 dµ Y = n Tr √ n √ n 1 � Y 2 = ij n 2 i,j 2 � Y 2 = ij n 2 1 � i<j � n ❚❛❦✐♥❣ ❡①♣❡❝t❛t✐♦♥✱ ❢r♦♠ E Y 2 12 = 1 ✱ � √ n = n ( n − 1) x 2 dµ Y = 1 + o (1) . E n 2 ❍❡♥❝❡✱ ❢r♦♠ t❤❡ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ � x 2 dµ Y √ n → 1 + o (1) .

▼❡t❤♦❞ ♦❢ ♠♦♠❡♥ts ❲❡ ❤❛✈❡ � � x 2 k +1 dµ sc = 0 x 2 k dµ sc = c k , ❛♥❞ ✇❤❡r❡ c k ✐s t❤❡ k ✲t❤ ❈❛t❛❧❛♥ ♥✉♠❜❡r 1 � 2 k � c k = . k + 1 k ❇② ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥t✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t � Y � k √ n = 1 � x k dµ Y √ n n Tr k 1 � � = Y i ℓ i ℓ +1 n 1+ k/ 2 i 1 , ··· ,i k ℓ =1 � x k dµ sc + o (1) . =

❙❡♠✐✲❝✐r❝❧❡ ❧❛✇ ❢♦r ❛❞❥❛❝❡♥❝② ♠❛tr✐① ❚❤❡♦r❡♠ ❋✐① p ∈ (0 , 1) ❛♥❞ ❧❡t σ 2 = p (1 − p ) ✳ ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❢♦r ❛♥② ✐♥t❡r✈❛❧ I ⊂ R ✱ n σ √ n ( I ) = 1 � λ i ( X ) � � σ √ n ∈ I µ → µ sc ( I ) . 1 X n i =1

❋r♦♠ ❲✐❣♥❡r ♠❛tr✐① t♦ ❆❞❥❛❝❡♥❝② ♠❛tr✐① ❲❡ ❤❛✈❡ σ 2 = Var( X 12 ) = E ( X 12 − E X 12 ) 2 = E ( X 2 12 ) − ( E X 12 ) 2 = p (1 − p ) . ❆❧s♦✱ ✇✐t❤ J = 11 ∗ ✱ E X = p ( J − I ) . ◆♦✇✱ σ √ n = ( X − E X ) X + pJ pI σ √ n σ √ n − σ √ n.

❋r♦♠ ❲✐❣♥❡r ♠❛tr✐① t♦ ❆❞❥❛❝❡♥❝② ♠❛tr✐① σ √ n = ( X − E X ) X + pJ pI σ √ n σ √ n − σ √ n. ( X − E X ) ✲ ✐s ❛ ❲✐❣♥❡r ♠❛tr✐①✱ ✐ts ❊❙❉ ❝♦♥✈❡r❣❡ t♦ t❤❡ σ √ n s❡♠✐✲❝✐r❝✉❧❛r ❧❛✇✳ σ √ n ❤❛s r❛♥❦ ♦♥❡ ✭❜✉t ♥♦r♠ ♦❢ ♦r❞❡r √ n ✦✮✳ pJ ✲ σ √ n ❤❛s ♥♦r♠ ♦❢ ♦r❞❡r 1 / √ n ✭❜✉t ❢✉❧❧ r❛♥❦ ✦✮✳ pI ✲

■♥t❡r❧❛❝✐♥❣ ■♥❡q✉❛❧✐t✐❡s ❚❛❦❡ A ∈ M n ( R ) s②♠♠❡tr✐❝ ❛♥❞ A ′ ∈ M n − 1 ( R ) ❛ ♠✐♥♦r✱ � A ′ � v A = v ∗ u ❚❤❡♥✱ ✐❢ λ i +1 � λ i ✱ λ i +1 ( A ) � λ i ( A ′ ) � λ i ( A ) . ❚❛❦❡ A, B ∈ M n ( R ) s②♠♠❡tr✐❝ ❛♥❞ rank( A − B ) = k t❤❡♥ λ i + k ( A ) � λ i ( B ) � λ i − k ( A ) .

■♥t❡r❧❛❝✐♥❣ ■♥❡q✉❛❧✐t✐❡s ❉❡✜♥❡✱ t❤❡ ❑♦❧♠♦❣♦r♦✈✲❙♠✐r♥♦✈ ❞✐st❛♥❝❡ ❛s d KS ( µ, ν ) = sup | µ (( −∞ , t )) − ν (( −∞ , t )) | . t ■❢ d KS ( µ n , µ ) → 0 t❤❡♥ µ n ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ µ ✳ ▲❡♠♠❛ ■❢ A, B ❛r❡ ❍❡r♠✐t✐❛♥ n × n ♠❛tr✐❝❡s✱ t❤❡♥ d KS ( µ A , µ B ) � rank( A − B ) . n

❋r♦♠ ❲✐❣♥❡r ♠❛tr✐① t♦ ❆❞❥❛❝❡♥❝② ♠❛tr✐① σ √ n = ( X − E X ) X + pJ pI σ √ n σ √ n − σ √ n. ( X − E X ) ✲ ✐s ❛ ❲✐❣♥❡r ♠❛tr✐①✱ ✐ts ❊❙❉ ❝♦♥✈❡r❣❡ t♦ t❤❡ σ √ n s❡♠✐✲❝✐r❝✉❧❛r ❧❛✇✳ σ √ n ❤❛s r❛♥❦ ♦♥❡ ✭❜✉t ♥♦r♠ ♦❢ ♦r❞❡r √ n ✦✮✳ pJ ✲ σ √ n ❤❛s ♥♦r♠ ♦❢ ♦r❞❡r 1 / √ n ✭❜✉t ❢✉❧❧ r❛♥❦ ✦✮✳ pI ✲

■♥❤♦♠♦❣❡♥❡♦✉s r❛♥❞♦♠ ❣r❛♣❤ ❆ s✐♠✐❧❛r st✉❞② ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r t❤❡ ❊❙❉ ♦❢ ❞❡♥s❡ ✉♥❞✐r❡❝t❡❞ ✐♥❤♦♠♦❣❡♥❡♦✉s ❣r❛♣❤s✱ ✇❤❡r❡ d X ij = Ber( W ( i/n, j/n )) , ❛♥❞ W : [0 , 1] 2 → [0 , 1] ✐s ❛ ❝♦♥st❛♥t ❜② ❜❧♦❝❦ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❧✐♠✐t ✐s t❤❡ s❡♠✐✲❝✐r❝❧❡ ✐❢ � 1 σ 2 ( x ) = W ( x, y )(1 − W ( x, y )) dy 0 ✐s ❛ ❝♦♥st❛♥t ❢✉♥❝✐t♦♥ ♦❢ x ✳

❊①tr❡♠❛❧ ❡✐❣❡♥✈❛❧✉❡s ✐♥ ❞❡♥s❡ ❊r❞➤s✲❘é♥②✐ ❲❡ ✜① p ∈ (0 , 1) ✳ ( X ij ) 1 � i<j � n ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s Ber( p ) ❛♥❞ X ij = X ji ✳ ❊✐❣❡♥✈❛❧✉❡s ♦❢ X ❢♦r n = 100 ✱ p = 1 / 2 ✳

❋➯r❡❞✐✲❑♦♠❧ós ❚❤❡♦r❡♠ ❲✐❣♥❡r ♠❛tr✐① Y = ( Y ij ) 1 � i,j � n ✇✐t❤ ( Y ij ) i>j ✐✐❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ( Y ii ) i � 1 ✐✐❞ ❛♥❞ Y ji = Y ij ✱ ❚❤❡♦r❡♠ ■❢ E Y 11 = E Y 12 = 0 ✱ E Y 2 12 = 1 ✱ E Y 2 11 < ∞ ✱ E Y 4 11 < ∞ ✱ t❤❡♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ � Y � Y � � √ n √ n λ 1 = 2 + o (1) = − λ n . ❘❡❝❛❧❧ t❤❛t supp( µ sc ) = [ − 2 , 2] ✳

❊①tr❡♠❛❧ ❊✐❣❡♥✈❛❧✉❡s ♦❢ ❉❡♥s❡ ❊r❞➤s✲❘é♥②✐ ❚❤❡♦r❡♠ ❋✐① p ∈ (0 , 1) ❛♥❞ ❧❡t σ 2 = p (1 − p ) ✳ ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ λ 1 ( X ) = pn + O ( √ n ) , λ 2 ( X ) = 2 σ √ n + o ( √ n ) = − λ n ( X ) , ❛♥❞✱ ✐❢ ψ 1 ✐s t❤❡ P❡rr♦♥ ❡✐❣❡♥✈❡❝t♦r✱ � ψ 1 − 1 / √ n � = O (1 / √ n ) .

❇❛✉❡r✲❋✐❦❡ ❚❤❡♦r❡♠ ❚❤❡♦r❡♠ ■❢ A, B ✐s n × n ❍❡r♠✐t✐❛♥✱ | λ i ( A + B ) − λ i ( A ) | � � B � . ❋♦r ❣❡♥❡r❛❧ B ✱ ❢♦r s♦♠❡ ♣❡r♠✉t❛t✐♦♥ σ ✱ � � � � � B � . � λ i ( A + B ) − λ σ ( i ) ( A )

❊①tr❡♠❛❧ ❊✐❣❡♥✈❛❧✉❡s ♦❢ ❉❡♥s❡ ❊r❞➤s✲❘é♥②✐ X = p ( J − I ) + ( X − E X ) . ❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ J − I ❛r❡ ✿ n − 1 ❛♥❞ − 1 ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② n − 1 ✳ ❋r♦♠ ❋➯r❡❞✐✲❑♦♠❧ós ❚❤❡♦r❡♠✱ � X − E X � = 2 σ √ n + o ( √ n ) ❍❡♥❝❡✱ O ( √ n ) | λ 1 ( X ) − p ( n − 1) | = (2 + o (1)) σ √ n. | λ 2 ( X ) + p | � ❍♦✇❡✈❡r✱ ❢r♦♠ t❤❡ s❡♠✐✲❝✐r❝❧❡ ❧❛✇✱ ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t λ 2 ( X ) � (2 + o (1)) σ √ n ✳

❘❛♥❦ ♦♥❡ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❊✐❣❡♥✈❡❝t♦r ❆ss✉♠❡ t❤❛t B = A + uv ∗ . ■❢ λ ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ B ❛♥❞ ♥♦t ♦❢ A t❤❡♥ 1 + v ∗ ( A − λ ) − 1 u = 0 ❛♥❞ ψ = ( A − λ ) − 1 u. ✐s ❛♥ ❡✐❣❡♥✈❡❝t♦r✳

❘❛♥❦ ♦♥❡ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❊✐❣❡♥✈❡❝t♦r ❆♣♣❧② t❤✐s X = ( X − E X − pI ) + p 11 ∗ , ❛♥❞ λ = λ 1 ( X ) = pn + O ( √ n ) . ❋r♦♠ � X − E X − pI � = O ( √ n ) ✱ ✇❡ ❣❡t � ψ 1 − 1 / √ n � = O (1 / √ n ) .

▼❛r❦♦✈ ❚r❛♥s✐t✐♦♥ ▼❛tr✐① ❆ s✐♠✐❧❛r ❛♥❛❧②s✐s ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r P = D − 1 X. � 1 � 2 σ λ 2 ( P ) = p √ n + o √ n = − λ n ( P ) . ❊✐❣❡♥✈❛❧✉❡s ♦❢ P ❢♦r n = 100 ✱ p = 1 / 2 ✳

▼❛r❦♦✈ ❚r❛♥s✐t✐♦♥ ▼❛tr✐① ❍✐st♦❣r❛♠ ♦❢ λ n ( P ) � . . . � λ 2 ( P ) ❢♦r n = 1000 ✱ p = 1 / 2 ✳

▲❛♣❧❛❝✐❛♥ L = X − D. ❊✐❣❡♥✈❛❧✉❡s ♦❢ L ❢♦r n = 100 ✱ p = 1 / 2 ✳

▲❛♣❧❛❝✐❛♥ L = X − D. ❍✐st♦❣r❛♠ ♦❢ λ n ( X ) � . . . � λ 2 ( L ) ❢♦r n = 1000 ✱ p = 1 / 2 ✳

▲❛♣❧❛❝✐❛♥ ❚❤❡♦r❡♠ ❋✐① p ∈ (0 , 1) ❛♥❞ ❧❡t σ 2 = p (1 − p ) ✳ ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❢♦r ❛♥② ✐♥t❡r✈❛❧ I ⊂ R ✱ µ L + npI σ √ n ( I ) → µ ( I ) . ✇❤❡r❡ µ = µ sc ⊞ N (0 , 1) ✳ ▼♦r❡♦✈❡r✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ � λ 2 ( L ) = − np + σ (1 + o (1)) 2 n log n. ✭❉✐♥❣✴❏✐❛♥❣ ✷✵✶✵✮✱ ✭❏✐❛♥❣ ✷✵✶✷✮

❍❡✉r✐st✐❝s L + npI X − E X − D − E D + pJ σ √ n σ √ n σ √ n σ √ n. = X − E X ✲ ✿ ❲✐❣♥❡r ♠❛tr✐① ✿ ✐ts ❊❙❉ ❝♦♥✈❡r❣❡s t♦ t❤❡ σ √ n s❡♠✐✲❝✐r❝✉❧❛r ❧❛✇✳ D − E D ✲ ✿ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ✇✐t❤ ❛♣♣r♦①✐♠❛t❡❧② ✐✐❞ ●❛✉ss✐❛♥ σ √ n N (0 , 1) ❝♦❡✣❝✐❡♥ts✱ D ii = � j X ij ✳ σ √ n ✿ ♦♥❡ ❡✐❣❡♥✈❛❧✉❡ p √ n pJ ✲ → ∞ ✱ ❛❧❧ ♦t❤❡rs 0 ✳ σ

❋r❡❡ ❝♦♥✈♦❧✉t✐♦♥ ▲❡t A n ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❞❡t❡r♠✐♥✐st✐❝ ❍❡r♠✐t✐❛♥ n × n ♠❛tr✐❝❡s s✉❝❤ t❤❛t ❢♦r ❛♥② ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ f ✱ � � fµ A n → fdµ. ❚❤❡♥✱ ✐❢ Y ✐s ❛ ❲✐❣♥❡r ♠❛tr✐①✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ � � fµ Y √ n + A n → fdν. ❛♥❞ ν := µ sc ⊞ µ. ■♥ ❤✐❣❤ ❞✐♠❡♥s✐♦♥✱ t❤❡ s♣❡❝tr❛ ❛❞❞ ✉♣ ✦ ✦

▼❛①✐♠✉♠ ♦❢ ●❛✉ss✐❛♥ ✈❛r✐❛❜❧❡s L + npI X − E X − D − E D + pJ σ √ n = σ √ n σ √ n σ √ n. ■❢ ( Z i ) i � 1 ❛r❡ ✐✐❞ N (0 , 1) ✈❛r✐❛❜❧❡s t❤❡♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ � 1 � i � n Z i = (1 + o (1)) max 2 log n. ■♥ ♣❛rt✐❝✉❧❛r✱ � D − E D � � � � σ √ n � = (1 + o (1)) 2 log n, � � � ❛♥❞ ✳ ✳ ✳ � λ 2 ( L ) = − np + σ (1 + o (1)) 2 n log n.

P❛rt ■■■ ✿ ❙♣❡❝tr✉♠ ♦❢ ❧❛r❣❡ r❛♥❞♦♠ ❣r❛♣❤s ❙♣❛rs❡ ✉♥❞✐r❡❝t❡❞ r❛♥❞♦♠ ❣r❛♣❤s

❙♣❛rs❡ ❊r❞➤s✲❘é♥②✐ ❚❤❡ ❛❜♦✈❡ r❡s✉❧ts r❡♠❛✐♥ ✈❛❧✐❞ ❛s ❧♦♥❣ ❛s np → ∞ , ❢♦r t❤❡ ❊❙❉ ♦❢ X ♦r np log n → ∞ . ❢♦r ❛❧❧ ♦t❤❡r st❛t❡♠❡♥ts✳ p (1 − p ) ∼ √ p ✇❤❡♥ p = o (1) ✳ � ◆♦t❡ t❤❛t σ =

❙♣❛rs❡ ❊r❞➤s✲❘é♥②✐ ❆ ❦❡② t❡❝❤♥✐❝❛❧ st❛t❡♠❡♥t ✐s ❑❤♦r✉♥③❤② ✭✷✵✵✶✮✱ ❱✉ ✭✷✵✵✼✮ ✿ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ � � X − E X � � σ √ n � = 2 + o (1) � � � ✇❤❡♥ np log n → ∞ . � � � � log n � X − E X ❍♦✇❡✈❡r✱ ✇❤❡♥ p = o ✱ � ≫ 1 ✳ � σ √ n � n

P❛rt ■■■ ✿ ❙♣❡❝tr✉♠ ♦❢ ❧❛r❣❡ r❛♥❞♦♠ ❣r❛♣❤s ❙♣❛rs❡✴❞❡♥s❡ ❞✐r❡❝t❡❞ r❛♥❞♦♠ ❣r❛♣❤s

❉✐r❡❝t❡❞ ❊r❞➤s✲❘é♥②✐ ■rr❡✈❡rs✐❜❧❡ ♠♦❞❡❧ ✿ ( X ij ) 1 � i � = j � n ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s Ber( p ) ✳ ❊✐❣❡♥✈❛❧✉❡s ♦❢ X ❢♦r n = 100 ✱ p = 1 / 2 ✳

❈✐r❝✉❧❛r ▲❛✇ ❚❤❡♦r❡♠ ❆ss✉♠❡ t❤❛t ( Y ij ) i,j � 1 ❛r❡ ✐✐❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s E Y 12 = 0 ✱ ❛♥❞ E | Y 12 | 2 = 1 ✱ ❝♦♥s✐❞❡r t❤❡ ♠❛tr✐① Y = ( Y ij ) 1 � i,j � n . ❚❤❡♥✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❢♦r ❛♥② ❇♦r❡❧ s❡t I ⊂ C ✱ µ Y/ √ n ( I ) → µ c ( I ) , ✇✐t❤ µ c ( dxdy ) = 1 π 1 | z | � 1 dz. ✭▼❡❤t❛ ✶✾✻✼✮✱ ✭●✐r❦♦ ✶✾✽✹✮✱ ✭❇❛✐ ✶✾✾✼✮✱ ✭P❛♥✴❩❤♦✉ ✷✵✶✵✮✱ ✭●öt③❡✴❚✐❦❤♦♠✐r♦✈ ✷✵✶✵✮ . . . ✱ ✭❚❛♦✴❱✉ ✷✵✶✵✮✳

❈✐r❝✉❧❛r ❧❛✇ ❢♦r ❛❞❥❛❝❡♥❝② ♠❛tr✐① ❚❤❡♦r❡♠ ❆ss✉♠❡ (log n ) 6 /n ≪ p � 1 − δ ❛♥❞ ❧❡t σ 2 = p (1 − p ) ✳ ❋♦r ❛♥② ❇♦r❡❧ I ⊂ C ✱ ✐♥ ♣r♦❜❛❜✐❧✐t②✱ µ σ √ n ( I ) → µ c ( I ) . X ✭❇♦r❞❡♥❛✈❡✱ ❈❛♣✉t♦✱ ❈❤❛❢❛ï ✷✵✶✹✮

❍❡✉r✐st✐❝s σ √ n = ( X − E X ) X + pJ pI σ √ n σ √ n − σ √ n. ( X − E X ) ✲ ✐s ❛ r❛♥❞♦♠ ✐✐❞ ♠❛tr✐①✱ ✐ts ❊❙❉ ❝♦♥✈❡r❣❡s t♦ t❤❡ σ √ n ❝✐r❝✉❧❛r ❧❛✇✳ σ √ n ❤❛s r❛♥❦ ♦♥❡ ✭❜✉t ♥♦r♠ ♦❢ ♦r❞❡r √ n ✦ ✮✳ pJ ✲ σ √ n ❤❛s ♥♦r♠ ♦❢ ♦r❞❡r 1 / √ n ✭❜✉t ❢✉❧❧ r❛♥❦ ✦ ✮✳ pI ✲

P❡rt✉r❜❛t✐♦♥ ♦❢ ♥♦♥✲❍❡r♠✐t✐❛♥ ♠❛tr✐❝❡s ❚❛❦❡     0 1 0 · · · 0 1 0 · · · 0 0 1 · · · 0 0 1 · · ·     N = ❛♥❞ C =  .  ✳ ✳   ✳ ✳  ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳ ✳ ✳ ✳     ✳ ✳ ✳ ✳    0 0 . . . ε 0 . . . ❆❧❧ ❡✐❣❡♥✈❛❧✉❡s ♦❢ N ❛r❡ 0 ✱ µ N = δ 0 . ❋♦r ε = 1 ✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ C ❛r❡ e 2 iπk/n ✱ 1 � k � n ✿ � 2 π � fdµ C → 1 f ( e 2 iθ ) dθ. 2 π 0 ❚r✉❡ ❛s s♦♦♥ ❛s ε 1 /n → 1 ✦ ✦

▼❛r❦♦✈ ❚r❛♥s✐t✐♦♥ ▼❛tr✐① ❋♦r (log n ) 6 /n ≪ p � 1 − δ ✱ t❤❡r❡ ✐s ❛ ❝✐r❝✉❧❛r ❧❛✇ ❢♦r P = D − 1 X ✇✐t❤ r❛❞✐✉s σ/ ( p √ n ) ✳ ❊✐❣❡♥✈❛❧✉❡s ♦❢ P ❢♦r n = 100 ✱ p = 1 / 2 ✳

▲❛♣❧❛❝✐❛♥ ■♥ t❤❡ s❛♠❡ r❡❣✐♠❡✱ t❤❡r❡ ✐s ❛ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ ❊❙❉ ♦❢ L = X − D s❤✐❢t❡❞ ❜② pn ❛♥❞ r❡s❝❛❧❡❞ ❜② σ √ n ✳ ❊✐❣❡♥✈❛❧✉❡s ♦❢ L ❢♦r n = 100 ✱ p = 1 / 2 ✳

■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❚❤❡♦r❡♠✳ ■❢ p ≫ (log n ) /n ✱ t❤❡♥✱ ❛✳s✳ ❢♦r n ≫ 1 ✱ t❤❡ ▼❛r❦♦✈✐❛♥ ❣❡♥❡r❛t♦r L ✐s ✐rr❡❞✉❝✐❜❧❡ ❛♥❞ � √ σ � � � σ log n log n � � Π − 1 /n � ❚❱ = O + O . n 3 / 4 p n p ❙✐♠✐❧❛r❧②✱ ❢♦r t❤❡ ▼❛r❦♦✈ tr❛♥s✐t✐♦♥ ♠❛tr✐① P ✱ � √ σ log n � σ � � � π − 1 /n � ❚❱ = O p √ n + O , pn 3 / 4 ✭❇♦r❞❡♥❛✈❡✱ ❈❛♣✉t♦✱ ❈❤❛❢❛ï ✷✵✶✹✮

P❛rt ■■■ ✿ ❙♣❡❝tr✉♠ ♦❢ ❧❛r❣❡ r❛♥❞♦♠ ❣r❛♣❤s ❉✐❧✉t❡❞ ✉♥❞✐r❡❝t❡❞ r❛♥❞♦♠ ❣r❛♣❤s

❑❡st❡♥✲▼❝❑❛② ▲❛✇ ❋✐① ✐♥t❡❣❡r d � 1 ✳ ▲❡t G = G n ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ d ✲r❡❣✉❧❛r ❣r❛♣❤s ♦♥ n ✈❡rt✐❝❡s s✉❝❤ t❤❛t ❢♦r ❛♥② k ✱ |{ ❝②❝❧❡s ♦❢ ❧❡♥❣❤t k }| = o k ( n ) . ■♥ ✇♦r❞s✱ G ✐s ❧♦❝❛❧❧② tr❡❡✲❧✐❦❡✳ � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �� �� �� �� �� �� �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �� �� �� � � � � �� �� �� �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ❋♦r ❡①❛♠♣❧❡✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❛ s❡q✉❡♥❝❡ ♦❢ ✉♥✐❢♦r♠❧② s❛♠♣❧❡❞ d ✲r❡❣✉❧❛r ❣r❛♣❤s ♦♥ n ✈❡rt✐❝❡s ✐s ❧♦❝❛❧❧② tr❡❡✲❧✐❦❡✳

❑❡st❡♥✲▼❝❑❛② ▲❛✇ ❚❤❡♦r❡♠ ❋✐① ✐♥t❡❣❡r d � 2 ✳ ▲❡t G = G n ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛❧❧② tr❡❡✲❧✐❦❡ d ✲r❡❣✉❧❛r ❣r❛♣❤s ♦♥ n ✈❡rt✐❝❡s t❤❡♥ ❢♦r ❛♥② I ⊂ R ✱ n 1 � 1 ( λ k ( X ) ∈ I ) = µ X ( I ) → µ KM ( I ) . n k =1 ✇❤❡r❡ � 4( d − 1) − x 2 µ KM ( dx ) = d √ d − 1 dx. 1 | x | � 2 d 2 − x 2 2 π √ ❲❡ ❤❛✈❡ µ KM ( I d ) → µ sc ( I ) ✇❤❡♥ d → ∞ ✳