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

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

❋❡❜r✉❛r② ✷✵✶✼

❖✈❡r✈✐❡✇

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

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

❚r❛♣❡③♦✐❞s

N C N ❧♦③❡♥❣❡s ♦♥ t❤❡ r✐❣❤t✿ ❛r❜✐tr❛r② ❞❡t❡r♠✐♥✐st✐❝ ♦r r❛♥❞♦♠✳ ❈♦♥❞✐t✐♦♥❛❧❧② ♦♥ t❤❡✐r ♣♦s✐t✐♦♥s✱ t❤❡ t✐❧✐♥❣ ✐s ✉♥✐❢♦r♠❧② r❛♥❞♦♠✳ ❍❡①❛❣♦♥ ✐s ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡✳

❚r❛♣❡③♦✐❞s

N C N ❧♦③❡♥❣❡s ♦♥ t❤❡ r✐❣❤t✿ ❛r❜✐tr❛r② ❞❡t❡r♠✐♥✐st✐❝ ♦r r❛♥❞♦♠✳ ❈♦♥❞✐t✐♦♥❛❧❧② ♦♥ t❤❡✐r ♣♦s✐t✐♦♥s✱ t❤❡ t✐❧✐♥❣ ✐s ✉♥✐❢♦r♠❧② r❛♥❞♦♠✳ ❍❡①❛❣♦♥ ✐s ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡✳

❙●❋ ❝♦♠♣✉t❛t✐♦♥

x5

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L N

❋✐①❡❞ (t1 > t2 > . . . tL)✱ {xj

i }✱ 1 ≤ i ≤ j ≤ L ✖

❤♦r✐③♦♥t❛❧ ❧♦③❡♥❣❡s✳ ❚♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s ❂ st(1L)✳ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ N✕♣❛rt✐❝❧❡ ✈❡rt✐❝❛❧✳

• λ

P

• (xN

1 , xN 2 , . . . , xN N ) = λ

sλ(u1, . . . , uN) sλ(1, . . . , 1) = st(u1, . . . , uN, 1L−N) st(1L) ❈♦♠❜✐♥❛t♦r✐❛❧ ❢♦r♠✉❧❛ ❢♦r ❙❝❤✉r ❢✉♥❝t✐♦♥s✳

❙●❋ ❝♦♠♣✉t❛t✐♦♥

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L N

❋✐①❡❞ (t1 > t2 > . . . tL)✱ {xj

i }✱ 1 ≤ i ≤ j ≤ L ✖

❤♦r✐③♦♥t❛❧ ❧♦③❡♥❣❡s✳ ❙●❋ ♦❢ t❤❡ N✕♣❛rt✐❝❧❡ ✈❡rt✐❝❛❧✳ st(u1, . . . , uN, 1L−N) st(1L)

• ❆♣♣r♦❛❝❤ ✶✳ ❯s❡ ❛s②♠♣t♦t✐❝ ♦❢ ❙❝❤✉r ❢✉♥❝t✐♦♥s ❬●✳✲P❛♥♦✈❛✲✶✷❪
• ❆♣♣r♦❛❝❤ ✷✳ ❯s❡ t❤❛t ♦✉r t❤❡♦r❡♠ ✐s ✐❢ ❛♥❞ ♦♥❧② ✐❢✳

❙●❋ ♥♦✲❝♦♠♣✉t❛t✐♦♥

x5

1

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L N

st(u1, . . . , uN, 1L−N) st(1L) ▲♦❣✕❞❡r✐✈❛t✐✈❡s ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ N✳ 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

❙●❋ ♥♦✲❝♦♠♣✉t❛t✐♦♥

x5

1

x5

3

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4

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2

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1

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1

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x1

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L N

st(u1, . . . , uN, 1L−N) st(1L) ▲♦❣✕❞❡r✐✈❛t✐✈❡s ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ N✳ ❲❤❛t ❞❡♣❡♥❞s❄ 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

❙●❋ ♥♦✲❝♦♠♣✉t❛t✐♦♥

x5

1

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L N

st(u1, . . . , uN, 1L−N) st(1L) ▲♦❣✕❞❡r✐✈❛t✐✈❡s ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ N✳ ❲❤❛t ❞❡♣❡♥❞s❄ ❲❤❛t ✐❢ N = L❄ 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

❙●❋ ♥♦✲❝♦♠♣✉t❛t✐♦♥

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L N

st(u1, . . . , uN, 1L−N) st(1L) ▲♦❣✕❞❡r✐✈❛t✐✈❡s ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ N✳ ❈♦♥❝❧✉s✐♦♥✿ ▲❡✈❡❧ N ❣✐✈❡s ❝♦♠♣❧❡t❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❡♥t✐r❡ ♣✐❝t✉r❡✳ 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

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋

◆♦❜♦❞② ❦♥♦✇s ❙✳●✳❋✳ ♦r ❡✈❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s✦ ❙♣❡❝✐❛❧ t❤❛♥❦s t♦ ▲❡♦♥✐❞ P❡tr♦✈ ❢♦r s❛♠♣❧✐♥❣✦ ❈♦♥s✐❞❡r t❤❡ ❤❡①❛❣♦♥ ✇✐t❤ ❛ ❤♦❧❡ ♦❢ ✜①❡❞ ❤❡✐❣❤t✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋

◆♦❜♦❞② ❦♥♦✇s ❙✳●✳❋✳ ♦r ❡✈❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s✦ ❍♦✇❡✈❡r✱ ❚❤❡♦r❡♠✳ ❚❤❡ ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ❝♦♥✈❡r❣❡ t♦ ❛ ●❛✉ss✐❛♥ ❋✐❡❧❞ ❛s t❤❡ ♠❡s❤ s✐③❡ ❣♦❡s t♦ 0✳ ❬❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❇✉❢❡t♦✈✕●♦r✐♥✲✶✼ ❛♥❞ ❇♦r♦❞✐♥✕●♦r✐♥✕●✉✐♦♥♥❡t✲✶✻❪

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋

◆♦❜♦❞② ❦♥♦✇s ❙✳●✳❋✳ ♦r ❡✈❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s✦ ❍♦✇❡✈❡r✱ ❚❤❡♦r❡♠✳ ❚❤❡ ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ❝♦♥✈❡r❣❡ t♦ ❛ ●❛✉ss✐❛♥ ❋✐❡❧❞ ❛s t❤❡ ♠❡s❤ s✐③❡ ❣♦❡s t♦ 0✳ ❍♦✇ ❞♦❡s ✐t ✇♦r❦❄ ❬❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❇✉❢❡t♦✈✕●♦r✐♥✲✶✼ ❛♥❞ ❇♦r♦❞✐♥✕●♦r✐♥✕●✉✐♦♥♥❡t✲✶✻❪

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋

◆♦❜♦❞② ❦♥♦✇s ❙✳●✳❋✳ ♦r ❡✈❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s✦ ❍♦✇❡✈❡r✱ ❖❜s❡r✈❛t✐♦♥✳ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❤♦r✐③♦♥t❛❧ ❧♦③❡♥❣❡s ❛❧♦♥❣ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s ♦❢ t❤❡ ❤♦❧❡ ✭❛♥❞ ♦♥❧② ❛❧♦♥❣ ✐t✦✮ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞✳ (a)n = a(a+1) · · · (a+n−1)

• i<j

(ℓi−ℓj)2

N

• i=1
• (A+B+C+1−t−ℓi)t−B (ℓi)t−C (H−ℓi)D (H−ℓi)D

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋

1 Z

• 1≤i<j≤N

Γ(ℓj − ℓi + 1)Γ(ℓj − ℓi + θ) Γ(ℓj − ℓi)Γ(ℓj − ℓi + 1 − θ)

N

• i=1

w(ℓi; N), ❚❤❡♦r❡♠✳ ❬❇♦r♦❞✐♥✕●♦r✐♥✕●✉✐♦♥♥❡t✲✶✻❪ ❆ss✉♠❡ t❤❛t w(·; N) ❛♥❞ V (·) ❛r❡ ❛♥❛❧②t✐❝ ✭x ln(x) ❜❡❤❛✈✐♦r ♦❢ V ❛t ❡♥❞✕♣♦✐♥ts ✐s ♦❦✮✱ ❛❧❧ ❞❛t❛ ❞❡♣❡♥❞s ♦♥ N r❡❣✉❧❛r❧②✱ ❛♥❞ µ(x)dx ✐s s✉❝❤ t❤❛t t❤❡r❡ ✐s ♦♥❡ ❜❛♥❞ ✐♥ ❡❛❝❤ r❡❣✐♦♥✳ ❚❤❡♥ ✉♥❞❡r t❡❝❤♥✐❝❛❧ ❛ss✉♠♣t✐♦♥s✱ ❢♦r ❛♥❛❧②t✐❝ f1(x), . . . , fm(x) lim

N→∞ N

• i=1
• fj

ℓi N

• − Efj

ℓi N

• ,

j = 1, . . . , m. ❛r❡ ❥♦✐♥t❧② ●❛✉ss✐❛♥ ✇✐t❤ ❡①♣❧✐❝✐t ❝♦✈❛r✐❛♥❝❡✳ Pr♦♦❢ ✐s ❜❛s❡❞ ♦♥ ◆❡❦r❛s♦✈ ❡q✉❛t✐♦♥s ✖ ❛♣♣r♦♣r✐❛t❡ ❞✐s❝r❡t❡ ✈❡rs✐♦♥s ♦❢ ❧♦♦♣ ♦r ❙❝❤✇✐♥❣❡r✕❉②s♦♥ ❡q✉❛t✐♦♥s✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋

❚❤❡♦r❡♠✳ ❚❤❡ ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ❝♦♥✈❡r❣❡ t♦ ❛ ●❛✉ss✐❛♥ ❋✐❡❧❞ ❛s t❤❡ ♠❡s❤ s✐③❡ ❣♦❡s t♦ 0✳ ❙tr❛t❡❣② ♦❢ t❤❡ ♣r♦♦❢✿

• ❬❇♦✲●✉✲●♦✲✶✺❪✿ ❈▲❚ ❛❧♦♥❣ t❤❡ ♠✐❞❞❧❡ ✈❡rt✐❝❛❧ s❡❝t✐♦♥
• ❬❇✉✲●♦✲✶✸✲✶✼✱ ♦♥❧② ✐❢ ♣❛rt❪✿ ❙✳●✳❋✳ ❛❧♦♥❣ t❤❡ ♠✐❞❞❧❡ s❡❝t✐♦♥
• ❙✐♠♣❧❡ ❝♦♠❜✐♥❛t♦r✐❝s ❡①t❡♥❞s ❙✳●✳❋✳ t♦ ❛❧❧ ✈❡rt✐❝❛❧ s❡❝t✐♦♥s
• ❬❇✉✲●♦✲✶✸✲✶✼✱ ✐❢ ♣❛rt❪ ❣✐✈❡s ❈▲❚ ❡✈❡r②✇❤❡r❡✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋

❲❤❛t ❛r❡ t❤❡ ◆❡❦r❛s♦✈ ❡q✉❛t✐♦♥s ✇❤✐❝❤ ❤❡❧♣ ❤❡r❡❄ 1 Z

• 1≤i<j≤N

Γ(ℓj − ℓi + 1)Γ(ℓj − ℓi + θ) Γ(ℓj − ℓi)Γ(ℓj − ℓi + 1 − θ)

N

• i=1

w(ℓi; N), ❚❤❡♦r❡♠✳❬❇♦r♦❞✐♥✕●♦r✐♥✕●✉✐♦♥♥❡t✕✶✺❪ ❆ss✉♠❡ w(x; N) w(x − 1; N) = φ+

N(x)

φ−

N(x),

❢♦r ❛♥❛❧②t✐❝ φ±

N.

❚❤❡♥ φ−

N(ξ) · E

N

• i=1
• 1 −

θ ξ − ℓi

• + φ+

N(ξ) · E

N

• i=1
• 1 +

θ ξ − ℓi − 1

• .

✐s ❛♥❛❧②t✐❝ ✐♥ t❤❡ D ⊂ C✱ ✇❤❡r❡ φ±

N ❛r❡✳

◆✐❝❡❧② ❝♦♠❜✐♥❡s ✇✐t❤ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r ❧♦❣✕❣❛s❡s t♦ ♣r♦❞✉❝❡ ❈▲❚✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋

1 Z

• 1≤i<j≤N

Γ(ℓj − ℓi + 1)Γ(ℓj − ℓi + θ) Γ(ℓj − ℓi)Γ(ℓj − ℓi + 1 − θ)

N

• i=1

w(ℓi; N), ❚❤❡♦r❡♠✳❬❇♦r♦❞✐♥✕●♦r✐♥✕●✉✐♦♥♥❡t✕✶✺❪ ❆ss✉♠❡ w(x; N) w(x − 1; N) = φ+

N(x)

φ−

N(x),

❢♦r ❛♥❛❧②t✐❝ φ±

N.

❚❤❡♥ φ−

N(ξ) · E

N

• i=1
• 1 −

θ ξ − ℓi

• + φ+

N(ξ) · E

N

• i=1
• 1 +

θ ξ − ℓi − 1

• .

✐s ❛♥❛❧②t✐❝ ✐♥ t❤❡ D ⊂ C✱ ✇❤❡r❡ φ±

N ❛r❡✳

• ❚❤✐s ✐s ❛ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ ✭◆❡❦r❛s♦✈✕P❡st✉♥✮✱ ✭◆❡❦r❛s♦✈✕❙❤❛t❛s❤✈✐❧✐✮✱ ✭◆❡❦r❛s♦✈✮
• ❑♥♦✇✐♥❣ t❤❡ st❛t❡♠❡♥t✱ t❤❡ ♣r♦♦❢ ✐s ❡❧❡♠❡♥t❛r②✳
• ❉✐s❝r❡t❡ ❛♥❛❧♦❣✉❡ ♦❢ ❧♦♦♣ ✴ ❙❝❤✇✐♥❣❡r✕❉②s♦♥ ❡q✉❛t✐♦♥s✳

◆✐❝❡❧② ❝♦♠❜✐♥❡s ✇✐t❤ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r ❧♦❣✕❣❛s❡s t♦ ♣r♦❞✉❝❡ ❈▲❚✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋

1 Z

• 1≤i<j≤N

Γ(ℓj − ℓi + 1)Γ(ℓj − ℓi + θ) Γ(ℓj − ℓi)Γ(ℓj − ℓi + 1 − θ)

N

• i=1

w(ℓi; N), ❚❤❡♦r❡♠✳❬❇♦r♦❞✐♥✕●♦r✐♥✕●✉✐♦♥♥❡t✕✶✺❪ ❆ss✉♠❡ w(x; N) w(x − 1; N) = φ+

N(x)

φ−

N(x),

❢♦r ❛♥❛❧②t✐❝ φ±

N.

❚❤❡♥ φ−

N(ξ) · E

N

• i=1
• 1 −

θ ξ − ℓi

• + φ+

N(ξ) · E

N

• i=1
• 1 +

θ ξ − ℓi − 1

• .

✐s ❛♥❛❧②t✐❝ ✐♥ t❤❡ D ⊂ C✱ ✇❤❡r❡ φ±

N ❛r❡✳

◆✐❝❡❧② ❝♦♠❜✐♥❡s ✇✐t❤ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r ❧♦❣✕❣❛s❡s t♦ ♣r♦❞✉❝❡ ❈▲❚✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❞♦♥✬t ❧✐❦❡ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s

✳✳✳ ❜✉t ♣r❡❢❡r ❛s②♠♣t♦t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✐♥st❡❛❞✳ ❯♥✐t❛r② ❣r♦✉♣ U(N) ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ✭❂❤♦♠♦♠♦r♣❤✐s♠s t♦ GL(V )✮ Tλ✱ λ1 ≥ λ2 ≥ · · · ≥ λN ▲✐tt❧❡✇♦♦❞✕❘✐❝❤❛r❞s♦♥ ❝♦❡✣❝✐❡♥ts✳ ❈♦♠❜✐♥❛t♦r✐❛❧ ❢♦r♠✉❧❛s ❡①✐st✱ ❜✉t ❜❡❝♦♠❡ ✐♥tr❛❝t❛❜❧❡ ❛s ✳ ✖ ❤❛r❞ t❤❡♦r❡♠ ❬❍♦r♥❀ ❑❧②❛❝❤❦♦❀ ❑♥✉ts♦♥✕❚❛♦❪✳ ❬❇✐❛♥❡❪ ❚r❡❛t ❛s ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ❛s ✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❞♦♥✬t ❧✐❦❡ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s

✳✳✳ ❜✉t ♣r❡❢❡r ❛s②♠♣t♦t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✐♥st❡❛❞✳ ❯♥✐t❛r② ❣r♦✉♣ U(N) ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ✭❂❤♦♠♦♠♦r♣❤✐s♠s t♦ GL(V )✮ Tλ✱ λ1 ≥ λ2 ≥ · · · ≥ λN Tλ ⊗ Tµ =

• ν

cν

λ,µTν

▲✐tt❧❡✇♦♦❞✕❘✐❝❤❛r❞s♦♥ ❝♦❡✣❝✐❡♥ts✳ ❈♦♠❜✐♥❛t♦r✐❛❧ ❢♦r♠✉❧❛s ❡①✐st✱ ❜✉t ❜❡❝♦♠❡ ✐♥tr❛❝t❛❜❧❡ ❛s N → ∞✳ cν

λ,µ ?

= 0 ✖ ❤❛r❞ t❤❡♦r❡♠ ❬❍♦r♥❀ ❑❧②❛❝❤❦♦❀ ❑♥✉ts♦♥✕❚❛♦❪✳ ❬❇✐❛♥❡❪ ❚r❡❛t ❛s ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ❛s ✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❞♦♥✬t ❧✐❦❡ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s

✳✳✳ ❜✉t ♣r❡❢❡r ❛s②♠♣t♦t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✐♥st❡❛❞✳ ❯♥✐t❛r② ❣r♦✉♣ U(N) ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ✭❂❤♦♠♦♠♦r♣❤✐s♠s t♦ GL(V )✮ Tλ✱ λ1 ≥ λ2 ≥ · · · ≥ λN Tλ ⊗ Tµ =

• ν

cν

λ,µTν

▲✐tt❧❡✇♦♦❞✕❘✐❝❤❛r❞s♦♥ ❝♦❡✣❝✐❡♥ts✳ ❈♦♠❜✐♥❛t♦r✐❛❧ ❢♦r♠✉❧❛s ❡①✐st✱ ❜✉t ❜❡❝♦♠❡ ✐♥tr❛❝t❛❜❧❡ ❛s N → ∞✳ cν

λ,µ ?

= 0 ✖ ❤❛r❞ t❤❡♦r❡♠ ❬❍♦r♥❀ ❑❧②❛❝❤❦♦❀ ❑♥✉ts♦♥✕❚❛♦❪✳ ❬❇✐❛♥❡❪ ❚r❡❛t cν

λ,µ ❛s ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ❛s N → ∞✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❞♦♥✬t ❧✐❦❡ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s

✳✳✳ ❜✉t ♣r❡❢❡r ❛s②♠♣t♦t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ✐♥st❡❛❞✳ Tλ ⊗ Tµ =

• ν

cν

λ,µTν,

P(ν) = cν

λ,µ dim Tν

dim Tλ dim Tµ ❚❤❡♦r❡♠✳ ❬❇✉❢❡t♦✈✲●♦r✐♥✲✶✸✲✶✻✱ ❝♦r♦❧❧❛r②❪ ❆ss✉♠❡ t❤❛t 1 N

N

• i=1

δ λi + N − i N

• → ρ1,

1 N

N

• i=1

δ µi + N − i N

• → ρ2

❚❤❡♥ P✕r❛♥❞♦♠ ν s❛t✐s✜❡s t❤❡ ▲▲◆ ❛♥❞ ❈▲❚✿ 1 N

N

• i=1

νi + N − i N k →

• xk(ρ1 ⊗ ρ2)(dx),

N

• i=1

νi + N − i N k − E νi + N − i N k ❛s②♠♣t♦t✐❝❛❧❧② ●❛✉ss✐❛♥ ✇✐t❤ ❡①♣❧✐❝✐t ❝♦✈❛r✐❛♥❝❡✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❞♦♥✬t ❧✐❦❡ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s

Tλ ⊗ Tµ =

• ν

cν

λ,µTν,

P(ν) = cν

λ,µ dim Tν

dim Tλ dim Tµ 1 N

N

• i=1

νi + N − i N k →

• xk(ρ1 ⊗ ρ2)(dx),

Gρ(z) = ρ(dx) z − x , Rq(z) = Gρ(z)(−1) − 1 1 − e−z . ◗✉❛♥t✐③❡❞ ❱♦✐❝✉❧❡s❝✉ R✕tr❛♥s❢♦r♠✳ Rq

ρ1⊗ρ2(z) = Rq ρ1(z) + Rq ρ2(z)

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿ s❡♠✐❝❧❛ss✐❝❛❧ ❧✐♠✐t

❚❡♥s♦r ♣r♦❞✉❝ts ❘❛♥❞♦♠ ♠❛tr✐❝❡s Tλ ⊗ Tµ =

ν cν λ,µTν

❡✐❣❡♥✈❛❧✉❡s(Aλ + Bµ) Rq

ρ (z) = Gρ(z)(−1) − 1 1−e−z

Rρ(z) = Gρ(z)(−1) − 1

z

▲✐♥❡❛r✐③❛t✐♦♥ ✐♥ ▲▲◆ ✭q✉❛♥t✐③❡❞ ✈❡rs✐♦♥✮ ❋r❡❡ ❈♦♥✈♦❧✉t✐♦♥ ❬❇✉✲●✲✶✸❪ ❬❱♦✐❝✉❧❡s❝✉✲✾✶❪ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ✐♥ ❈▲❚ ❋♦r♠✉❧❛s ♦❢ ❬❇✉✲●✲✶✻❪ ❙❡❝♦♥❞ ♦r❞❡r ❢r❡❡♥❡ss

❬❈♦❧❧✐♥s✲▼✐♥❣♦✲❙♥✐❛❞②✲❙♣❡✐❝❤❡r✲✵✹❪

❆♥♦t❤❡r✱ ❢❛st❡r s❝❛❧✐♥❣ ♦❢ λi ✐♥ t❡♥s♦r ♣r♦❞✉❝ts ❧❡❛❞s t♦ ❢r❡❡ ♣r♦❜❛❜✐❧✐t② ❞✐r❡❝t❧② ❬❇✐❛♥❡✲✾✺❪✱ ❬❈♦❧❧✐♥s✕❙♥✐❛❞②✲✵✾❪✳ P❡r❡❧♦♠♦✈✕P♦♣♦✈ ♠❡❛s✉r❡s ❧❡❛❞ t♦ ❢r❡❡ ♣r♦❜❛❜✐❧✐t② ❞✐r❡❝t❧②✳ ❬❇✉✲●✲✶✸❪✱ ❬❈♦❧❧✐♥s✕◆♦✈❛❦✕❙♥✐❛❞②✲✶✻❪ ❚❤❡ ❧✐♠✐t s❤❛♣❡s ❢♦r r❛♥❞♦♠ t✐❧✐♥❣s ❝❛♥ ❜❡ ❛❧s♦ ❞❡s❝r✐❜❡❞ ✐♥ t❤✐s ❧❛♥❣✉❛❣❡✱ ❧✐♥❦✐♥❣ t❤❡♠ t♦ ✭q✉❛♥t✐③❡❞✮ ❢r❡❡ ♣r♦❥❡❝t✐♦♥s✳

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s✿ s❡♠✐❝❧❛ss✐❝❛❧ ❧✐♠✐t

❚❡♥s♦r ♣r♦❞✉❝ts ❘❛♥❞♦♠ ♠❛tr✐❝❡s Tλ ⊗ Tµ =

ν cν λ,µTν

❡✐❣❡♥✈❛❧✉❡s(Aλ + Bµ) Rq

ρ (z) = Gρ(z)(−1) − 1 1−e−z

Rρ(z) = Gρ(z)(−1) − 1

z

▲✐♥❡❛r✐③❛t✐♦♥ ✐♥ ▲▲◆ ✭q✉❛♥t✐③❡❞ ✈❡rs✐♦♥✮ ❋r❡❡ ❈♦♥✈♦❧✉t✐♦♥ ❬❇✉✲●✲✶✸❪ ❬❱♦✐❝✉❧❡s❝✉✲✾✶❪ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ✐♥ ❈▲❚ ❋♦r♠✉❧❛s ♦❢ ❬❇✉✲●✲✶✻❪ ❙❡❝♦♥❞ ♦r❞❡r ❢r❡❡♥❡ss

❬❈♦❧❧✐♥s✲▼✐♥❣♦✲❙♥✐❛❞②✲❙♣❡✐❝❤❡r✲✵✹❪

❚❤❡ ❧✐♠✐t s❤❛♣❡s ❢♦r r❛♥❞♦♠ t✐❧✐♥❣s ❝❛♥ ❜❡ ❛❧s♦ ❞❡s❝r✐❜❡❞ ✐♥ t❤✐s ❧❛♥❣✉❛❣❡✱ ❧✐♥❦✐♥❣ t❤❡♠ t♦ ✭q✉❛♥t✐③❡❞✮ ❢r❡❡ ♣r♦❥❡❝t✐♦♥s✳ α · Rq

ρ (z; α) = Rq ρ (z; 1).

• ❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❞♦♥✬t ❧✐❦❡ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s

✳✳✳ ❜✉t ♣r❡❢❡r q✉❛♥t✉♠ r❛♥❞♦♠ ✇❛❧❦s ✐♥st❡❛❞✳ ❚❛❦❡ ❛♥② ✜♥✐t❡ ✜①❡❞ s❡q✉❡♥❝❡ λ1 ≥ · · · ≥ λk ≥ 0 ❛♥❞ tr❡❛t ✐t ❛s ❧❡♥❣t❤ N s❡q✉❡♥❝❡ ❜② ❛❞❞✐♥❣ 0✬s✳ (Tλ)⊗τN2 =

• cνTν,

P(ν) = cν dim(Tν) (dim Tλ)τN2 . ❚❤❡♦r❡♠✳ ❬❈♦r♦❧❧❛r② ♦❢ ❇✉❢❡t♦✈✲●♦r✐♥✲✶✸✲✶✼❪ ❆s N → ∞✱ P✕r❛♥❞♦♠ ν s❛t✐s✜❡s t❤❡ ▲▲◆ ❛♥❞ ❈▲❚ ✇✐t❤ ✉♥✐✈❡rs❛❧ ❛♥s✇❡rs✳ ❘❡♠❛r❦✳ Tλ ❞♦❡s ♥♦t ❤❛✈❡ t♦ ❜❡ ✐rr❡❞✉❝✐❜❧❡✳ ❘❡♠❛r❦✳ N = 1✱ (T0 ⊕ T1)⊗t✱ t = 1, 2, . . . , ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ s✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦ ♦♥ Z ✇✐t❤ st❡♣s {0, 1}✳

❙✉♠♠❛r②

(Tλ)⊗τN2 =

• cνTν

Tλ⊗Tµ =

• ν

cν

λ,µTν

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