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❍❡✐❣❤t ✢✉❝t✉❛t✐♦♥s t❤r♦✉❣❤ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

❱❛❞✐♠ ●♦r✐♥ ▼■❚ ✭❈❛♠❜r✐❞❣❡✮ ❛♥❞ ■■❚P ✭▼♦s❝♦✇✮ ▲❡❝t✉r❡ ✶

❋❡❜r✉❛r② ✷✵✶✼

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

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐♥ 2d s②st❡♠s

❊①❛♠♣❧❡✿ t❛❦❡ ②♦✉ ❢❛✈♦r✐t❡ r❛♥❞♦♠ t✐❧✐♥❣s ♠♦❞❡❧✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐♥ 2d 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 ✐♥ 2d 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 ✐♥ 2d 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 ✐♥ 2d 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♠ ✉♣✿ ❝❧❛ss✐❝❛❧ ❈▲❚

❚❤❡ t❡①t❜♦♦❦ ♣r♦♦❢ ♦❢ t❤❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠✿ ❚❤❡ r❛♥❞♦♠ ✈❡❝t♦r ξ(N) = (ξ1, . . . , ξk) ✐s ●❛✉ss✐❛♥ ❛s N → ∞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ E exp

• ix, ξ(N)
• = E exp

 i

k

• j=1

xjξj   N→∞ − → exp

• ix, m − 1

2Cx, x

• ❯♥❞❡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) = (ξ1, . . . , ξk) ✐s ●❛✉ss✐❛♥ ❛s N → ∞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ E exp

• ix, ξ(N)
• = E exp

 i

k

• j=1

xjξj   N→∞ − → exp

• ix, m − 1

2Cx, x

• ❯♥❞❡r t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥s t❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦
• 1

i

• ∂

∂xa

k

a=1 ln

• E exp
• ix, ξ(N)
• x1=···=xk=0 → m✱
• ✲
• ∂

∂xa ∂ ∂xb

k

a,b=1 ln

• E exp
• ix, ξ(N)
• x1=···=xk=0 → C✱
• q
• a=1

∂ ∂xja

• ln
• E exp
• ix, ξ(N)
• x1=···=xk=0 → 0,

|{xja}| > 2. ❲❤❛t ❝❛♥ ❜❡ ❛ ✉s❡❢✉❧ ❛♥❛❧♦❣✉❡ ❢♦r ✷❞ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s❄

❲❛r♠ ✉♣✿ ❝❧❛ss✐❝❛❧ ❈▲❚

❚❤❡ t❡①t❜♦♦❦ ♣r♦♦❢ ♦❢ t❤❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠✿ ❚❤❡ r❛♥❞♦♠ ✈❡❝t♦r ξ(N) = (ξ1, . . . , ξk) ✐s ●❛✉ss✐❛♥ ❛s N → ∞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ E exp

• ix, ξ(N)
• = E exp

 i

k

• j=1

xjξj   N→∞ − → exp

• ix, m − 1

2Cx, x

• ❯♥❞❡r t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥s t❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦
• 1

i

• ∂

∂xa

k

a=1 ln

• E exp
• ix, ξ(N)
• x1=···=xk=0 → m✱
• ✲
• ∂

∂xa ∂ ∂xb

k

a,b=1 ln

• E exp
• ix, ξ(N)
• x1=···=xk=0 → C✱
• q
• a=1

∂ ∂xja

• ln
• E exp
• ix, ξ(N)
• x1=···=xk=0 → 0,

|{xja}| > 2. ❲❤❛t ❝❛♥ ❜❡ ❛ ✉s❡❢✉❧ ❛♥❛❧♦❣✉❡ ❢♦r ✷❞ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s❄

❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

❙t❛rt ✇✐t❤ ❛ s❡❝t✐♦♥✿ N ♣❛rt✐❝❧❡s ℓ1 > ℓ2 > · · · > ℓN. ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P(ℓ) ❲❛♥t✿ H(u) =

N

• i=1

1(u ≤ ℓi), ❙♠♦♦t❤❡❞ ✈❡rs✐♦♥✿

N

• i=1

f

• ℓi).

❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✿ ❍♦✇ ✐s ✐t ✉s❡❢✉❧❄

❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

❙t❛rt ✇✐t❤ ❛ s❡❝t✐♦♥✿ N ♣❛rt✐❝❧❡s ℓ1 > ℓ2 > · · · > ℓN. ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P(ℓ) ❲❛♥t✿ H(u) =

N

• i=1

1(u ≤ ℓi), ❙♠♦♦t❤❡❞ ✈❡rs✐♦♥✿

N

• i=1

f

• ℓi).

❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✿ GP =

• ℓ

P(ℓ)sℓ(x1, . . . , xN) sℓ(1N) , sℓ(x1, . . . , xN) = det

• xℓj

i

N

i,j=1

• i<j(xi − xj)

❍♦✇ ✐s ✐t ✉s❡❢✉❧❄

▼❛✐♥ ❛s②♠♣t♦t✐❝ st❛t❡♠❡♥t

❚❤❡♦r❡♠✳ ❬❇✉❢❡t♦✈✲●♦r✐♥✱ ✶✸✲✶✼❪✳ ❚❛❦❡ ❛ r❛♥❞♦♠ N✕t✉♣❧❡ ℓ1 > · · · > ℓN ✇✐t❤ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ G(x1, . . . , xN) =

• ℓ

P(ℓ)sℓ(x1, . . . , xN) sℓ(1, . . . , 1) . ■❢ ❛♥❞ ♦♥❧② ✐❢ ❛s N → ∞

• 1

N (∂i)a ln(G)

• x1=···=xN=1 → ca
• (∂i)a (∂j)b ln(G)
• x1=···=xN=1 → da,b
• [k

a=1 ∂ia] ln(G)

• x1=···=xN=1 → 0✱ |{ia}| > 2✳

❚❤❡♥ ▲▲◆✿ 1 N

N

• i=1

f ℓi N

• → mf

❛♥❞ ❈▲❚✿

N

• i=1
• f

ℓi N

• − Ef

ℓi N

• → N(0, σ2

f )

❤♦❧❞ ❢♦r ♣♦❧②♥♦♠✐❛❧ f (x)✱ ❥♦✐♥t❧② ❢♦r ♠✉❧t✐♣❧❡ fi(x)✱ i = 1, . . . , m✳✳ ❊①♣❧✐❝✐t ✜♥✐t❡ ❡①♣r❡ss✐♦♥s ❢♦r mf ✱ σ2

f t❤r♦✉❣❤ ca✱ da,b✳

▼❛✐♥ ❛s②♠♣t♦t✐❝ st❛t❡♠❡♥t

G =

• ℓ

P(ℓ)sℓ(x1, . . . , xN) sℓ(1, . . . , 1)

• 1

N (∂i)a ln(G)

• x1=···=xN=1 → ca
• (∂i)a (∂j)b ln(G)
• ···=1 → da,b
• [k

a=1 ∂ia] ln(G)

• =1 → 0✱ |{ia}| > 2

✐❢ ❛♥❞ ♦♥❧② ✐❢ pk =

N

• i=1

ℓi N k

• 1

N pk → p(k)

• Epkpm − EpkEpm → cov(k, m)
• pk − Epk → ●❛✉ss✐❛♥s

p(k) = [z−1] 1 (k + 1)(1 + z)

• 1 + z

z + (1 + z)

∞

• a=1

caza−1 (a − 1)! k+1 cov(k, m) = [z−1w −1]   ∞

• a=0

za w 1+a 2 +

∞

• a,b=1

da,b (a − 1)!(b − 1)!za−1w b−1   ×

• 1 + z

z + (1 + z)

∞

• a=1

caza−1 (a − 1)! k 1 + w w + (1 + w)

∞

• a=1

caw a−1 (a − 1)! m

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s

❚❤❡♦r❡♠✳❬❇✉❢❡t♦✈✲●✳✱ ✶✸✲✶✼❪✳ ❆s②♠♣t♦t✐❝s ♦❢ ❧♦❣❛r✐t❤♠ ♦❢ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r r❛♥❞♦♠ ℓ1 > · · · > ℓN − → ▲▲◆ ❛♥❞ ❈▲❚✳ ❚❤❡♦r❡♠✳❬●✳✲P❛♥♦✈❛✲✶✷❪ ❆s②♠♣t♦t✐❝s ♦❢ ❙❝❤✉r ❢✉♥❝t✐♦♥s ♥❡❛r 1✳ ❈♦♠❜✐♥❛t✐♦♥ ♣r♦✈❡s ▲▲◆ ❛♥❞ ❈▲❚ ❢♦r ❛ ❤✉❣❡ ✈❛r✐❡t② ♦❢ s②st❡♠s✿

1 2 3 4 5 6 7

time T

❇♦r♦❞✐♥✕❋❡rr❛r✐ ▼✉❧t✐✕❚❆❙❊P

❉❡❝♦♠♣♦s✐t✐♦♥s ✐♥t♦ ✐rr❡❞✉❝✐❜❧❡s ♦❢ t❡♥s♦r ♣r♦❞✉❝ts ♦❢ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ U(N)✱ N → ∞✳ P❧❛♥❝❤❡r❡❧ ♣❛rt✐t✐♦♥s

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿❡①❛♠♣❧❡s

❆③t❡❝ ❞✐❛♠♦♥❞ ◆❡❡❞ t♦ ❝❤❡❝❦ ❢♦r t❤❡ kt❤ ❧✐♥❡ ✐♥ s✐③❡ n ❞✐❛♠♦♥❞ ✐♥ t❤❡ ❧✐♠✐t k, n → ∞✱ k/n → const✳

• 1

k (∂i)a ln(G)

• x1=···=1 → ca
• (∂i)a (∂j)b ln(G)
• ···=1 → da,b
• [k

a=1 ∂ia] ln(G)

• x1=···=1 → 0✱

|{ia}| > 2

❙✳●✳❋✳✿ ❙♦ t❤❡ ❝❤❡❝❦ ✐s ❡❛s②✦ ▲❡t✬s s❡❡ ❤♦✇ ✐t ✇♦r❦s ✐♥ ❞❡t❛✐❧s ❢♦r ▲▲◆✳

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿❡①❛♠♣❧❡s

❆③t❡❝ ❞✐❛♠♦♥❞ ◆❡❡❞ t♦ ❝❤❡❝❦ ❢♦r t❤❡ kt❤ ❧✐♥❡ ✐♥ s✐③❡ n ❞✐❛♠♦♥❞ ✐♥ t❤❡ ❧✐♠✐t k, n → ∞✱ k/n → const✳

• 1

k (∂i)a ln(G)

• x1=···=1 → ca
• (∂i)a (∂j)b ln(G)
• ···=1 → da,b
• [k

a=1 ∂ia] ln(G)

• x1=···=1 → 0✱

|{ia}| > 2

❙✳●✳❋✳✿ G(x1, . . . , xk) =

k

• i=1

1 + xi 2 n−k . ❙♦ t❤❡ ❝❤❡❝❦ ✐s ❡❛s②✦ ▲❡t✬s s❡❡ ❤♦✇ ✐t ✇♦r❦s ✐♥ ❞❡t❛✐❧s ❢♦r ▲▲◆✳

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿❡①❛♠♣❧❡s

❆♣♣❧✐❡s t♦ ❆③t❡❝ r❡❝t❛♥❣❧❡s ✇✐t❤ ❛r❜✐tr❛r② ❜♦✉♥❞❛r② ❧❡❛❞✐♥❣ t♦ ●❋❋✳ ❋r♦♠ ❇✉❢❡t♦✈✕❑♥✐③❡❧✕✶✻

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿❡①❛♠♣❧❡s

N = 6 ❚❤❡ ❢r❛♠❡✇♦r❦ ❛♣♣❧✐❡s t♦ tr❛♣❡③♦✐❞✲❧✐❦❡ ♣♦❧②❣♦♥s ✇✐t❤ ❛r❜✐tr❛r② ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ r✐❣❤t ♣r♦❞✉❝✐♥❣ ▲▲◆✰❈▲❚ ❛♥❞ ❧❡❛❞✐♥❣ t♦ ●❋❋✳ ❬❝❢✳ ❡❛r❧✐❡r r❡s✉❧ts ♦❢ P❡tr♦✈❪

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿❡①❛♠♣❧❡s

❚❤❡ ❢r❛♠❡✇♦r❦ ❛♣♣❧✐❡s ❢♦r N ♥♦♥✲❝♦❧❧✐❞✐♥❣ ✇❛❧❦❡rs ❛t t✐♠❡ T ❛s N, T → ∞✱ T/N → τ ❧❡❛❞✐♥❣ t♦ ▲▲◆✰❈▲❚ ✇✐t❤ ❡①♣❧✐❝✐t ❛♥s✇❡rs✳

1 2 3 4 5 6 7

time T s❦✐♣

• N ✐♥❞❡♣❡♥❞❡♥t s✐♠♣❧❡

r❛♥❞♦♠ ✇❛❧❦s

• ♣r♦❜❛❜✐❧✐t② ♦❢ ❥✉♠♣ p
• st❛rt❡❞ ❛t ❛r❜✐tr❛r② ❧❛tt✐❝❡

♣♦✐♥ts

• ❝♦♥❞✐t✐♦♥❡❞ ♥❡✈❡r t♦

❝♦❧❧✐❞❡

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s

❚❤❡♦r❡♠✳❬❇✉❢❡t♦✈✲●✳✱ ✶✸✲✶✼❪✳ ❉❡r✐✈❛t✐✈❡s ♦❢ ❧♦❣❛r✐t❤♠ ♦❢ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r r❛♥❞♦♠ ℓ1 > · · · > ℓN − → ▲▲◆ ❛♥❞ ❈▲❚✳ ❈♦♠♣❛r❡ ✇✐t❤ ❝♦✉♥t❡r♣❛rt ❢r♦♠ ❝❧❛ss✐❝❛❧ ✶❞ ♣r♦❜❛❜✐❧✐t② t❤❡♦r②✳ ❈✉♠✉❧❛♥ts ka(ξ) : ∂ ∂t a ln

• E(exp(itξ))
• t=0 = (i)aka
• ξn → c✱ ✐✛ k1(ξn) → c✱ k2(ξn) → 0✳
• ξn − Eξn → N(0, σ2)✱ ✐✛ k2(ξn) → σ2✱ ka(ξn) → 0✱ a > 2✳

❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦♥ t❤❡ ❧✐♥❡ ✭♦r ♦♥ ✮ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦♥ ❝❤❛r❛❝t❡rs ♦❢ ✭♦r ♦❢ ✮ ❝❤❛r❛❝t❡rs ♦❢ ✉♥✐t❛r② ❣r♦✉♣ ✳

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s

❚❤❡♦r❡♠✳❬❇✉❢❡t♦✈✲●✳✱ ✶✸✲✶✼❪✳ ❉❡r✐✈❛t✐✈❡s ♦❢ ❧♦❣❛r✐t❤♠ ♦❢ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r r❛♥❞♦♠ ℓ1 > · · · > ℓN − → ▲▲◆ ❛♥❞ ❈▲❚✳ ❈♦♠♣❛r❡ ✇✐t❤ ❝♦✉♥t❡r♣❛rt ❢r♦♠ ❝❧❛ss✐❝❛❧ ✶❞ ♣r♦❜❛❜✐❧✐t② t❤❡♦r②✳ ❈✉♠✉❧❛♥ts ka(ξ) : ∂ ∂t a ln

• E(exp(itξ))
• t=0 = (i)aka
• ξn → c✱ ✐✛ k1(ξn) → c✱ k2(ξn) → 0✳
• ξn − Eξn → N(0, σ2)✱ ✐✛ k2(ξn) → σ2✱ ka(ξn) → 0✱ a > 2✳

E(exp(itξ)) E

• sℓ(x1,...,xN)

sℓ(1,...,1)

• ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦♥ t❤❡ ❧✐♥❡

✭♦r ♦♥ Z✮ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦♥ ℓ1 > · · · > ℓN ❝❤❛r❛❝t❡rs ♦❢ R ✭♦r ♦❢ S1✮ ❝❤❛r❛❝t❡rs ♦❢ ✉♥✐t❛r② ❣r♦✉♣ U(N)✳

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿ ❣❧✐♠♣s❡ ♦❢ ♣r♦♦❢s

❚❤❡♦r❡♠✳❬❇✉❢❡t♦✈✲●✳✱ ✶✸✲✶✼❪✳ ❉❡r✐✈❛t✐✈❡s ♦❢ ❧♦❣❛r✐t❤♠ ♦❢ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r r❛♥❞♦♠ ℓ1 > · · · > ℓN − → ▲▲◆ ❛♥❞ ❈▲❚✳ ❍♦✇ ❞♦ ②♦✉ ♣r♦✈❡ s✉❝❤ t❤✐♥❣s❄ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ❝♦♠❡s ✐♥ ♣❛❝❦❛❣❡ ✇✐t❤ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✦

• ✐✈❡s ❝♦♠♣❧❡t❡ ✐♥❢♦r♠❛t✐♦♥ ❢♦r ♣♦❧②♥♦♠✐❛❧ t❡st ❢✉♥❝t✐♦♥s✱ ❛s s♦♦♥

❛s ❛ ❝♦♠❜✐♥❛t♦r✐❝s ♦❢ t❤❡s❡ ♦♣❡r❛t♦rs ✐s ✉♥❞❡rst♦♦❞ ✭✇❤❛t ✇❡ ❞♦✦✮

❈▲❚ ❢♦r ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿ ❣❧✐♠♣s❡ ♦❢ ♣r♦♦❢s

❚❤❡♦r❡♠✳❬❇✉❢❡t♦✈✲●✳✱ ✶✸✲✶✼❪✳ ❉❡r✐✈❛t✐✈❡s ♦❢ ❧♦❣❛r✐t❤♠ ♦❢ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r r❛♥❞♦♠ ℓ1 > · · · > ℓN − → ▲▲◆ ❛♥❞ ❈▲❚✳ ❍♦✇ ❞♦ ②♦✉ ♣r♦✈❡ s✉❝❤ t❤✐♥❣s❄ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ❝♦♠❡s ✐♥ ♣❛❝❦❛❣❡ ✇✐t❤ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs✦ Gt(x1, . . . , xN) =

• ℓ

P(ℓ)sℓ(x1, . . . , xN) sℓ(1, . . . , 1) . Dk =

• i<j

(xi − xj)−1 N

• i=1
• xi

∂ ∂xi k

i<j

(xi − xj) (Dk)mGN(x1, . . . , xN)

• x1=···=xN=1 = EP

N

• i=1

(ℓi)k m .

• ✐✈❡s ❝♦♠♣❧❡t❡ ✐♥❢♦r♠❛t✐♦♥ ❢♦r ♣♦❧②♥♦♠✐❛❧ t❡st ❢✉♥❝t✐♦♥s✱ ❛s s♦♦♥

❛s ❛ ❝♦♠❜✐♥❛t♦r✐❝s ♦❢ t❤❡s❡ ♦♣❡r❛t♦rs ✐s ✉♥❞❡rst♦♦❞ ✭✇❤❛t ✇❡ ❞♦✦✮

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿ ❣♦✐♥❣ ✷❞

❍♦✇ ❞♦ ②♦✉ ❡①t❡♥❞ ❢r♦♠ ❛ s✐♥❣❧❡ s❧✐❝❡ t♦ t❤❡ ✇❤♦❧❡ 2d ♣✐❝t✉r❡ ♥❡❝❡ss❛r② t♦ s❡❡ t❤❡

• ❛✉ss✐❛♥ ❋r❡❡ ❋✐❡❧❞❄

❈♦♥s✐❞❡r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❢♦r s❡✈❡r❛❧ s❧✐❝❡s ❛♥❞ ♣❧❛② t❤❡ s❛♠❡ ❣❛♠❡✳ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿ ❣♦✐♥❣ ✷❞

❍♦✇ ❞♦ ②♦✉ ❡①t❡♥❞ ❢r♦♠ ❛ s✐♥❣❧❡ s❧✐❝❡ t♦ t❤❡ ✇❤♦❧❡ 2d ♣✐❝t✉r❡ ♥❡❝❡ss❛r② t♦ s❡❡ t❤❡

• ❛✉ss✐❛♥ ❋r❡❡ ❋✐❡❧❞❄

❈♦♥s✐❞❡r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❢♦r s❡✈❡r❛❧ s❧✐❝❡s ❛♥❞ ♣❧❛② t❤❡ s❛♠❡ ❣❛♠❡✳ ℓ1 > ℓ2 · · · > ℓN, r1 > r2 · · · > rM, ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P(ℓ, r) G(x1, . . . , xN; y1, . . . , yM) =

• ℓ,r

P(ℓ, r)sℓ(x1, . . . , xN) sℓ(1N) ·sr(y1, . . . , yM) sr(1M) .

❙✉♠♠❛r②

G =

• ℓ

P(ℓ)sℓ(x1, . . . , xN) sℓ(1, . . . , 1)

• 1

N (∂i)a ln(G)

• x1=···=xN=1 → ca
• (∂i)a (∂j)b ln(G)
• ···=1 → da,b
• [k

a=1 ∂ia] ln(G)

• =1 → 0✱ |{ia}| > 2

■❢ ❛♥❞ ♦♥❧② ✐❢✿ pk =

N

• i=1

ℓi N k

• 1

N pk → p(k)

• Epkpm − EpkEpm → cov(k, m)
• pk − Epk → ●❛✉ss✐❛♥s

❚♦♠♦rr♦✇✿ ❲❤❛t ✐❢ ②♦✉ ❞♦ ◆❖❚ ❦♥♦✇ ❙✳●✳❋✳❄