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❍❡✐❣❤t ✢✉❝t✉❛t✐♦♥s t❤r♦✉❣❤ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❱❛❞✐♠ ●♦r✐♥ ▼■❚ ✭❈❛♠❜r✐❞❣❡✮ ❛♥❞ ■■❚P ✭▼♦s❝♦✇✮ ▲❡❝t✉r❡ ✶ ❋❡❜r✉❛r② ✷✵✶✼

❊①❛♠♣❧❡✿ t❛❦❡ ②♦✉ ❢❛✈♦r✐t❡ r❛♥❞♦♠ t✐❧✐♥❣s ♠♦❞❡❧✳ ●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐♥ 2 d s②st❡♠s ●♦❛❧ ❢♦r t♦❞❛②✿ ❆s②♠♣t♦t✐❝ t❤❡♦r❡♠s ❢♦r s♠♦♦t❤❡❞ ♠❛❝r♦s❝♦♣✐❝ ✢✉❝t✉❛t✐♦♥s ♦❢ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ st♦❝❤❛st✐❝ s②st❡♠s✳

●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐♥ 2 d s②st❡♠s ❊①❛♠♣❧❡✿ t❛❦❡ ②♦✉ ❢❛✈♦r✐t❡ r❛♥❞♦♠ t✐❧✐♥❣s ♠♦❞❡❧✳

❚❤❡♦r❡♠✳ ❚❤❡ ❝❡♥t❡r❡❞ ✢✉❝t✉❛t✐♦♥s ✐♥s✐❞❡ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡ ❝♦♥✈❡r❣❡ t♦ t❤❡ ♣✉❧❧❜❛❝❦ ❜② ❛♥ ❡①♣❧✐❝✐t ♠❛♣ ♦❢ t❤❡ ●❛✉ss✐❛♥ ❋r❡❡ ❋✐❡❧❞ ✇✐t❤ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡✳ ✭❖♥ s♠♦♦t❤ t❡st ❢✉♥❝t✐♦♥s✮ ❬P❡tr♦✈✱ ❉✉✐ts✱ ❈❤✐tt❛✕❏♦❤❛♥ss♦♥✕❨♦✉♥❣✱ ❇✉❢❡t♦✈✲●♦r✐♥❀ r❡❧❛t❡❞✿ ❑❡♥②♦♥✱ ❇♦r♦❞✐♥✕❋❡rr❛r✐❪ ●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐♥ 2 d s②st❡♠s ❊①❛♠♣❧❡✿ ❚❛❦❡ ❛ ❧❛r❣❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦♠♣✉t❡ ❝❡♥t❡r❡❞ ❤❡✐❣❤t ❢✉♥❝t✐♦♥✳

●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐♥ 2 d s②st❡♠s ❊①❛♠♣❧❡✿ ❚❛❦❡ ❛ ❧❛r❣❡ ❞♦♠❛✐♥ ❛♥❞ ❝♦♠♣✉t❡ ❝❡♥t❡r❡❞ ❤❡✐❣❤t ❢✉♥❝t✐♦♥✳ ❚❤❡♦r❡♠✳ ❚❤❡ ❝❡♥t❡r❡❞ ✢✉❝t✉❛t✐♦♥s ✐♥s✐❞❡ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡ ❝♦♥✈❡r❣❡ t♦ t❤❡ ♣✉❧❧❜❛❝❦ ❜② ❛♥ ❡①♣❧✐❝✐t ♠❛♣ ♦❢ t❤❡ ●❛✉ss✐❛♥ ❋r❡❡ ❋✐❡❧❞ ✇✐t❤ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡✳ ✭❖♥ s♠♦♦t❤ t❡st ❢✉♥❝t✐♦♥s✮ ❬P❡tr♦✈✱ ❉✉✐ts✱ ❈❤✐tt❛✕❏♦❤❛♥ss♦♥✕❨♦✉♥❣✱ ❇✉❢❡t♦✈✲●♦r✐♥❀ r❡❧❛t❡❞✿ ❑❡♥②♦♥✱ ❇♦r♦❞✐♥✕❋❡rr❛r✐❪

●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐♥ 2 d s②st❡♠s ❚❤❡♦r❡♠✳ ❚❤❡ ❝❡♥t❡r❡❞ ✢✉❝t✉❛t✐♦♥s ❝♦♥✈❡r❣❡ t♦ t❤❡ ●❛✉ss✐❛♥ ❋r❡❡ ❋✐❡❧❞ ❲❤❛t ♠❛❦❡s ✐t ❤❛r❞❡r ❄ ◆♦❜♦❞② ♠❛♥❛❣❡❞ t♦ ❛♣♣❧② ❛ ✇❡❧❧✲❦♥♦✇♥ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ ❞✐s❝r❡t❡ ❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥s → ❝♦♥t✐♥♦✉s ❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥s✳ • ◆♦♥✲tr✐✈✐❛❧ ❧✐♠✐t s❤❛♣❡ ❛♥❞ ❝♦♠♣❧❡① str✉❝t✉r❡✳ • ❋r♦③❡♥ r❡❣✐♦♥s ✇✐t❤ ♥♦ ✢✉❝t✉❛t✐♦♥s✳ • ◆♦ str✉❝t✉r❡ ♦❢ ❉●❋❋ ✈✐s✐❜❧❡ ❜❡❢♦r❡ t❛❦✐♥❣ ❧✐♠✐ts✳ ❬ ♥✉♠❡r♦✉s r❡s✉❧ts ❢♦r r❡❧❛①❡❞ ❝❛s❡s✱ ✐♥❝❧✉❞✐♥❣ q✉✐t❡ ❣❡♥❡r❛❧❪

●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐♥ 2 d s②st❡♠s ❚❤❡♦r❡♠✳ ❚❤❡ ❝❡♥t❡r❡❞ ✢✉❝t✉❛t✐♦♥s ❝♦♥✈❡r❣❡ t♦ t❤❡ ●❛✉ss✐❛♥ ❋r❡❡ ❋✐❡❧❞ ❲❤❛t ♠❛❦❡s ✐t ❤❛r❞❡r ❄ ◆♦❜♦❞② ♠❛♥❛❣❡❞ t♦ ❛♣♣❧② ❛ ✇❡❧❧✲❦♥♦✇♥ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ ❞✐s❝r❡t❡ ❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥s → ❝♦♥t✐♥♦✉s ❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥s✳ • ◆♦♥✲tr✐✈✐❛❧ ❧✐♠✐t s❤❛♣❡ ❛♥❞ ❝♦♠♣❧❡① str✉❝t✉r❡✳ • ❋r♦③❡♥ r❡❣✐♦♥s ✇✐t❤ ♥♦ ✢✉❝t✉❛t✐♦♥s✳ • ◆♦ str✉❝t✉r❡ ♦❢ ❉●❋❋ ✈✐s✐❜❧❡ ❜❡❢♦r❡ t❛❦✐♥❣ ❧✐♠✐ts✳ ▲❡❞ t♦ s❡✈❡r❡ r❡str✐❝t✐♦♥s ♦♥ t❤❡ s②st❡♠s ✇❤❡r❡ s✉❝❤ ❧✐♠✐t t❤❡♦r❡♠s ❛r❡ ❛❝❤✐❡✈❡❞✳ Pr♦❣r❡ss✿ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s✳

❯♥❞❡r t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥s t❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ✱ ✲ ✱ ❲❤❛t ❝❛♥ ❜❡ ❛ ✉s❡❢✉❧ ❛♥❛❧♦❣✉❡ ❢♦r ✷❞ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s❄ ❲❛r♠ ✉♣✿ ❝❧❛ss✐❝❛❧ ❈▲❚ ❚❤❡ t❡①t❜♦♦❦ ♣r♦♦❢ ♦❢ t❤❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠✿ ❚❤❡ r❛♥❞♦♠ ✈❡❝t♦r ξ ( N ) = ( ξ 1 , . . . , ξ k ) ✐s ●❛✉ss✐❛♥ ❛s N → ∞ ✐❢ ❛♥❞ ♦♥❧② ✐❢   k � i � x , m � − 1 �  N →∞ � i � x , ξ ( N ) � � � E exp = E exp  i x j ξ j − → exp 2 � Cx , x � j =1

❲❤❛t ❝❛♥ ❜❡ ❛ ✉s❡❢✉❧ ❛♥❛❧♦❣✉❡ ❢♦r ✷❞ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s❄ ❲❛r♠ ✉♣✿ ❝❧❛ss✐❝❛❧ ❈▲❚ ❚❤❡ t❡①t❜♦♦❦ ♣r♦♦❢ ♦❢ t❤❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠✿ ❚❤❡ r❛♥❞♦♠ ✈❡❝t♦r ξ ( N ) = ( ξ 1 , . . . , ξ k ) ✐s ●❛✉ss✐❛♥ ❛s N → ∞ ✐❢ ❛♥❞ ♦♥❧② ✐❢   k � i � x , m � − 1 �  N →∞ � i � x , ξ ( N ) � � � E exp = E exp  i x j ξ j − → exp 2 � Cx , x � j =1 ❯♥❞❡r t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥s t❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ � k � 1 ∂ i � x , ξ ( N ) � � � �� • a =1 ln E exp x 1 = ··· = x k =0 → m ✱ i ∂ x a � k � ∂ ∂ i � x , ξ ( N ) � • ✲ � � �� a , b =1 ln E exp x 1 = ··· = x k =0 → C ✱ ∂ x a ∂ x b � q � ∂ i � x , ξ ( N ) � � � �� • � ln E exp x 1 = ··· = x k =0 → 0 , |{ x j a }| > 2 . ∂ x ja a =1

❲❛r♠ ✉♣✿ ❝❧❛ss✐❝❛❧ ❈▲❚ ❚❤❡ t❡①t❜♦♦❦ ♣r♦♦❢ ♦❢ t❤❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠✿ ❚❤❡ r❛♥❞♦♠ ✈❡❝t♦r ξ ( N ) = ( ξ 1 , . . . , ξ k ) ✐s ●❛✉ss✐❛♥ ❛s N → ∞ ✐❢ ❛♥❞ ♦♥❧② ✐❢   k � i � x , m � − 1 �  N →∞ � i � x , ξ ( N ) � � � E exp = E exp  i x j ξ j − → exp 2 � Cx , x � j =1 ❯♥❞❡r t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥s t❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ � k � 1 ∂ i � x , ξ ( N ) � � � �� • a =1 ln E exp x 1 = ··· = x k =0 → m ✱ i ∂ x a � k � ∂ ∂ i � x , ξ ( N ) � • ✲ � � �� a , b =1 ln E exp x 1 = ··· = x k =0 → C ✱ ∂ x a ∂ x b � q � ∂ i � x , ξ ( N ) � � � �� • � ln E exp x 1 = ··· = x k =0 → 0 , |{ x j a }| > 2 . ∂ x ja a =1 ❲❤❛t ❝❛♥ ❜❡ ❛ ✉s❡❢✉❧ ❛♥❛❧♦❣✉❡ ❢♦r ✷❞ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s❄

❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✿ ❍♦✇ ✐s ✐t ✉s❡❢✉❧❄ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❙t❛rt ✇✐t❤ ❛ s❡❝t✐♦♥✿ N ♣❛rt✐❝❧❡s ℓ 1 > ℓ 2 > · · · > ℓ N . ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P ( ℓ ) N � ❲❛♥t✿ H ( u ) = 1 ( u ≤ ℓ i ) , i =1 N � � ❙♠♦♦t❤❡❞ ✈❡rs✐♦♥✿ f ℓ i ) . i =1

❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❙t❛rt ✇✐t❤ ❛ s❡❝t✐♦♥✿ N ♣❛rt✐❝❧❡s ℓ 1 > ℓ 2 > · · · > ℓ N . ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P ( ℓ ) N � ❲❛♥t✿ H ( u ) = 1 ( u ≤ ℓ i ) , i =1 N � � ❙♠♦♦t❤❡❞ ✈❡rs✐♦♥✿ f ℓ i ) . i =1 ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✿ x ℓ j � N � det P ( ℓ ) s ℓ ( x 1 , . . . , x N ) i i , j =1 � G P = , s ℓ ( x 1 , . . . , x N ) = s ℓ (1 N ) � i < j ( x i − x j ) ℓ ❍♦✇ ✐s ✐t ✉s❡❢✉❧❄