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❙♣❡❝tr❛❧ ❚❤❡♦r② ❢♦r ❈♦♠♣❧❡① ◆❡t✇♦r❦s

❈❤❛r❧❡s ❇♦r❞❡♥❛✈❡

❈◆❘❙ ✫ ❯♥✐✈❡rs✐t② ♦❢ ❚♦✉❧♦✉s❡

❖✈❡r✈✐❡✇ ❈♦♥s✐❞❡r ❛ ❧❛r❣❡ ♥❡t✇♦r❦ ✇✐t❤ n ≫ 1 ✈❡rt✐❝❡s✳ P❛rt ■ ❚❤❡r❡ ❛r❡ ♥❛t✉r❛❧ ♠❛tr✐❝❡s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ♥❡t✇♦r❦✳ P❛rt ■■ ✭✐✮ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♥❡t✇♦r❦ ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❢r♦♠ t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡s❡ ♠❛tr✐❝❡s✳ ✭✐✐✮ ❙♦♠❡ ❛❧❣♦r✐t❤♠s ♦r ♣r♦❝❡ss❡s r✉♥♥✐♥❣ ♦♥ t❤❡ ♥❡t✇♦r❦ ❝❛♥ ❜❡ st✉❞✐❡❞ t❤❛♥❦s t♦ t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡s❡ ♠❛tr✐❝❡s✳ P❛rt ■■■ ■♥ t❤❡ r❡❣✐♠❡ n ≫ 1✱ ✐t ✐s ♦❢t❡♥ ♣♦ss✐❜❧❡ t♦ st✉❞② t❤❡ ❛s②♠♣t♦t✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s♣❡❝tr✉♠✳

P❛rt ■ ✿ ◆❡t✇♦r❦ ▼❛tr✐❝❡s ▼❛tr✐❝❡s ♦♥ t❤❡ ❝♦♥❞✉❝t❛♥❝❡ ♠♦❞❡❧

❈♦♥❞✉❝t❛♥❝❡ ▼♦❞❡❧ ▲❡t V = {1, . . . , n} ❛♥❞ ❝♦♥s✐❞❡r ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ ❝♦♥❞✉❝t❛♥❝❡s { Xuv : u, v ∈ V } Xuv ♠❛② ❢♦r ❡①❛♠♣❧❡ r❡♣r❡s❡♥ts ❛♥ ❛✣♥✐t② ♦❢ u ❢♦r v✳ ❚❤❡ ❛ss♦❝✐❛t❡❞ ❞✐r❡❝t❡❞ ❣r❛♣❤ G = (V, E) ✐s ❞❡✜♥❡❞ ❜② uv ∈ E ✐❢ Xuv > 0✳

❆❞❥❛❝❡♥❝② ❛♥❞ ❞❡❣r❡❡ ♠❛tr✐❝❡s ❉❡✜♥❡ t❤❡ n × n ♠❛tr✐❝❡s X = (Xij)1i,jn ❛♥❞ D = diag

• ℓ

X1ℓ, . . . ,

• ℓ

Xnℓ

• .

X =     3 1 3 2 1 1 2 1 1 1     , D =     7 4 3 2     .

1 3 2 4

▼❛r❦♦✈ tr❛♥s✐t✐♦♥ ♠❛tr✐① ❆ss♦❝✐❛t❡ ❛ ▼❛r❦♦✈ ❝❤❛✐♥ ♦♥ V ✇✐t❤ tr❛♥s✐t✐♦♥ ♠❛tr✐① P Pij = Xij n

ℓ=1 Xiℓ

. ❲❡ ❤❛✈❡ P = D−1X. ■❢ Xuv ∈ {0, 1}✱ P ✐s t❤❡ tr❛♥s✐t✐♦♥ ♠❛tr✐① ♦❢ t❤❡ r❛♥❞♦♠ ✇❛❧❦ ♦♥ G✳

• ♦♦❣❧❡ ♠❛tr✐① ✿ ❢♦r α ∈ (0, 1]✱ αP + (1 − α)11∗✳

❍②♣❡rt❡①t ▲✐♥❦s ❊✐❣❡♥✈❛❧✉❡s ♦❢ tr❛♥s✐t✐♦♥ ♠❛tr✐① P ❢♦r ❤②♣❡rt❡①t ❧✐♥❦s ♦❢ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② ✐♥ ✷✵✵✻✳

❋r❛❤♠✱ ●❡♦r❣❡♦t✱ ❙❤❡♣❡❧②❛♥s❦② ✭✷✵✶✶✮✳

▲❛♣❧❛❝✐❛♥ ♠❛tr✐① ❋♦r i = j✱ Lij = Xij ❛♥❞ Lii = −

• ℓ=i

Xiℓ. ❚❤❡ ▲❛♣❧❛❝✐❛♥ L ✐s t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ t❤❡ ▼❛r❦♦✈ ♣r♦❝❡ss ✇❤❡r❡ t❤❡ ❥✉♠♣ ✐♥t❡♥s✐t② ❢r♦♠ i t♦ j ✐s Xij✳ ❲❡ ❤❛✈❡ L = X − D. ◆♦r♠❛❧✐③❡❞ ▲❛♣❧❛❝✐❛♥ ✿ D−1/2LD−1/2 = D1/2(P − I)D−1/2.

◆♦♥✲❇❛❝❦tr❛❝❦✐♥❣ ♠❛tr✐① ❚❤❡r❡ ❛r❡ ♠❛♥② ♦t❤❡r r❡❧❛t❡❞ ♠❛tr✐❝❡s✱ ♠♦r❡ ♦r ❧❡ss ✇❡❧❧ ✉♥❞❡rst♦♦❞✳ ■❢ e = uv, f = xy ❛r❡ ✐♥ E✱ Bef = 1(v = x)1(u = y), ❞❡✜♥❡s ❛ |E| × |E| ♠❛tr✐① ♦♥ t❤❡ ♦r✐❡♥t❡❞ ❡❞❣❡s✳

e f e f u v = x y

◆♦♥✲❇❛❝❦tr❛❝❦✐♥❣ ♠❛tr✐① ❯s❡❞ ♥♦t❛❜❧② ❢♦r ❝♦♠♠✉♥✐t② ❞❡t❡❝t✐♦♥✳

❑r③❛❦❛❧❛✴▼♦♦r❡✴▼♦ss❡❧✴◆❡❡♠❛♥✴❙❧②✴❩❞❡❜♦r♦✈á✴❩❤❛♥❣ ✭✷✵✶✸✮

P❛rt ■ ✿ ◆❡t✇♦r❦ ▼❛tr✐❝❡s ❇❛s✐❝ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙♣❡❝tr✉♠

❙♣❡❝tr✉♠ ❋♦r A ∈ Mn(R)✱ ✇❡ ❞❡♥♦t❡ ✐ts ❡✐❣❡♥✈❛❧✉❡s ❜② |λ1(A)| . . . |λn(A)|. ■❢ A ✐s s②♠♠❡tr✐❝✱ t❤❡♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ r❡❛❧ ❛♥❞ t❤❡r❡ ✐s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ■❢ Xuv = Xvu t❤❡♥ X ❛♥❞ L ❛r❡ s②♠♠❡tr✐❝ ♠❛tr✐❝❡s ❛♥❞ P = D−1/2(D−1/2XD−1/2)D1/2 ❤❛s ❛❧s♦ r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s✳

P❡rr♦♥✲❋r♦❜❡♥✐✉s ❚❤❡♦r❡♠ ❆ss✉♠❡ t❤❛t t❤❡ ❣r❛♣❤ G ♦❢ X ✐s str♦♥❣❧② ❝♦♥♥❡❝t❡❞✳ ❚❤❡♥ X ❛♥❞ P ❛r❡ s❛✐❞ t♦ ❜❡ ✐rr❡❞✉❝✐❜❧❡✳ ❚❤❡♥ λ1 ✐s ♣♦s✐t✐✈❡ ❛♥❞ ✐s ❛ s✐♠♣❧❡ ❡✐❣❡♥✈❛❧✉❡✳ ■ts ❧❡❢t ❛♥❞ r✐❣❤t ❡✐❣❡♥✈❡❝t♦r ❤❛✈❡ ♣♦s✐t✐✈❡ ❝♦♦r❞✐♥❛t❡s✳ ❋♦r t❤❡ tr❛♥s✐t✐♦♥ ♠❛tr✐① P✱ λ1(P) = 1 ❛♥❞ πP = π ✇✐t❤

• v π(v) = 1 ✐s t❤❡ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦❢ t❤❡ ▼❛r❦♦✈

❝❤❛✐♥✳

❊①❛♠♣❧❡ π ≃ (0.18, 0.06, 0.28, 0.22, 0.05, 0.03, 0.08, 0.11)✳

▲❛♣❧❛❝✐❛♥ ♠❛tr✐① ❆ss✉♠❡ t❤❛t t❤❡ ❣r❛♣❤ G ♦❢ X ✐s str♦♥❣❧② ❝♦♥♥❡❝t❡❞✳ ❚❤❡♥ 0 ✐s ❛ s✐♠♣❧❡ ❡✐❣❡♥✈❛❧✉❡ ♦❢ L✳ ■ts ❧❡❢t ❛♥❞ r✐❣❤t ❡✐❣❡♥✈❡❝t♦r ❤❛✈❡ ♣♦s✐t✐✈❡ ❝♦♦r❞✐♥❛t❡s✳ ❚❤❡♥ ΠL = 0 ✇✐t❤

v Π(v) = 1 ✐s t❤❡ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t②

♠❡❛s✉r❡ ♦❢ t❤❡ ▼❛r❦♦✈ ♣r♦❝❡ss✳ ❲❡ ❤❛✈❡ Π ∝ πD−1✳ ❆❧❧ ♦t❤❡r ❡✐❣❡♥✈❛❧✉❡s ❤❛✈❡ ♥❡❣❛t✐✈❡ r❡❛❧ ♣❛rt ✭❢r♦♠ ●❡rs❤❣♦r✐♥ ❚❤❡♦r❡♠✱ ❛❧❧ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ✐♥ ∪iB(−Dii, Dii)✮

❊①❛♠♣❧❡ Π ≃ (0.18, 0.06, 0.28, 0.22, 0.05, 0.03, 0.08, 0.11)✳

❘❡✈❡rs✐❜❧❡ ❝❛s❡ ❲❤❡♥ Xuv = Xvu✱ t❤❡ ❣r❛♣❤ G ✐s ✉♥❞✐r❡❝t❡❞✳ ❚❤❡♥ P ❛♥❞ L ❛r❡ r❡✈❡rs✐❜❧❡ ♣r♦❝❡ss❡s ❛♥❞ ✇❡ ✜♥❞ π = 1 S (D11, · · · , Dnn) ❛♥❞ Π = 1 n, · · · , 1 n

• ,

✇✐t❤ S =

n

• u=1

Duu =

• u,v

Xuv.

■♥❝✐❞❡♥❝❡ ♠❛tr✐❝❡s ❋♦r s✐♠♣❧✐❝✐t②✱ ❢r♦♠ ♥♦✇ ♦♥ Xuv ∈ {0, 1} ❛♥❞ Xuu = 0. ❚❤❡♥ Duu =

• (uv)∈E

1 = deg+(u), ✐s t❤❡ ♦✉t❡r ❞❡❣r❡❡ ♦❢ u✳

u deg+(u) = 3

❘❡❣✉❧❛r ❣r❛♣❤s ■❢ ❢♦r s♦♠❡ d ❛♥❞ ❢♦r ❛♥② v ∈ V ✱ deg+(v) = d. t❤❡♥ G ✐s ❛♥ ♦✉t❡r✲r❡❣✉❧❛r ❣r❛♣❤✳ ❲❡ ❤❛✈❡ D = dI ❛♥❞ t❤❡ ♠❛tr✐❝❡s X✱ L = dI − X✱ P = d−1X ❤❛✈❡ t❤❡ s❛♠❡ s♣❡❝tr✉♠ ✉♣ t♦ tr❛♥s❧❛t✐♦♥✴❞✐❧❛t✐♦♥✳ ❆❧s♦ λ1(X) = d ❛♥❞ t❤❡ ✈❡❝t♦r 1 ✐s ✐ts ❡✐❣❡♥✈❡❝t♦r✳

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

❚②♣✐❝❛❧ ✈s ❊①tr❡♠❛❧ ❊✐❣❡♥✈❛❧✉❡s ❚❤❡r❡ ❛r❡ ❡ss❡♥t✐❛❧❧② t✇♦ t②♣❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❡♥❝♦❞❡❞ ♦♥ t❤❡ s♣❡❝tr✉♠✳ ✲ t❤❡ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡s ✭❛♥❞ t❤❡✐r ❡✐❣❡♥s♣❛❝❡s✮ ❣✐✈❡ s♦♠❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ❣❧♦❜❛❧ ❣r❛♣❤ ♣r♦♣❡rt✐❡s ✭❡①♣❛♥s✐♦♥✱ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r✱ ♠❛①✐♠❛❧ ❝✉t✱ ❡t❝✳✳✳✮✱ ✲ t❤❡ t②♣✐❝❛❧ ❡✐❣❡♥✈❛❧✉❡s ❣✐✈❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ❧♦❝❛❧ ❣r❛♣❤ ♣r♦♣❡rt✐❡s ✭t②♣✐❝❛❧ ❞❡❣r❡❡✱ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ s♣❛♥♥✐♥❣ tr❡❡s✱ ♠❛t❝❤✐♥❣s✱ ❡t❝✳✳✳✮✳

P❛rt ■■ ✿ ❙♣❡❝tr✉♠ ❛♥❞ ●r❛♣❤ Pr♦♣❡rt✐❡s ❚②♣✐❝❛❧ ❊✐❣❡♥✈❛❧✉❡s ❛♥❞ P❛rt✐t✐♦♥ ❋✉♥❝t✐♦♥s

❊♠♣✐r✐❝❛❧ s♣❡❝tr❛❧ ❞✐str✐❜✉t✐♦♥ ✭❊❙❉✮ ❚❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✴ ❞❡♥s✐t② ♦❢ st❛t❡s ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ C✱ µA = 1 n

n

• i=1

δλi(A), ✐✳❡✳ ❢♦r ❛♥② s❡t I ⊂ C µA(I) = 1 n

n

• i=1

1(λi(A) ∈ I) ✐s t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ ❡✐❣❡♥✈❛❧✉❡s ✐♥ I ♦r ❡q✉✐✈❛❧❡♥t❧②✱ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ t②♣✐❝❛❧ ❡✐❣❡♥✈❛❧✉❡ ✐s ✐♥ I✳

• fdµA = 1

n

n

• i=1

f(λi(A)).

❑✐r❝❤♦❢❢ ▼❛tr✐①✲❚r❡❡ ❚❤❡♦r❡♠ ■❢ G ✐s ❛♥ ✉♥❞✐r❡❝t❡❞ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ t❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ s♣❛♥♥✐♥❣ tr❡❡s ♦❢ G ✐s ❡q✉❛❧ t♦ t(G) = 1 n

• λi=0

|λi|, ✇❤❡r❡ λi = λi(L)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ 1 n log t(G) = 0−

−∞

log |λ|dµL(λ) − 1 n log n.

❈❧♦s❡❞ ♣❛t❤s ❋♦r t ✐♥t❡❣❡r✱ ❧❡t St = |{❝❧♦s❡❞ ♣❛t❤s ♦❢ ❧❡♥❣t❤ t ✐♥ G}| ❲❡ ❤❛✈❡ St = Tr{Xt} =

n

• i=1

λi(X)t = n

• λtdµX.

■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r z ∈ C✱ Im(z) > 0✱ 1 n

• t0

St zt+1 =

• t0
• λt

zt+1 dµX =

• 1

z − λdµX ✐s t❤❡ ❈❛✉❝❤②✲❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ ♦❢ µX✳

❘❡t✉r♥ t✐♠❡s ■❢ Zt ✐s t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✇✐t❤ tr❛♥s✐t✐♦♥ ♠❛tr✐① P✱ 1 n

n

• v=1

P(Zt = v|Z0 = v) = 1 nTr{P t} =

• xtdµP .

❙✐♠✐❧❛r❧②✱ ❢♦r t > 0 r❡❛❧✱ ✐❢ Zt ✐s t❤❡ ▼❛r❦♦✈ ♣r♦❝❡ss ✇✐t❤ ❣❡♥❡r❛t♦r L✱ 1 n

n

• v=1

P(Zt = v|Z0 = v) =

• etLdµL.

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

▲♦❝❛❧✐③❛t✐♦♥✴❉❡❧♦❝❛❧✐③❛t✐♦♥ ♦❢ ❡✐❣❡♥✈❡❝t♦rs ❚❛❦❡ X ♦r L ✐♥ t❤❡ r❡✈❡rs✐❜❧❡ ❝❛s❡✳ ▲❡t ψ1, . . . ψn ❜❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ❚❤❡♥✱ ❢♦r ❛♥② k 1✱

• |ψk(1)|2, . . . , |ψk(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 ♦♥ Sn−1 = {x ∈ Rn : x = 1} ❄ ❋♦r ❡①❛♠♣❧❡✱ ❞♦ ✇❡ ❤❛✈❡ q✉❛♥t✉♠ ❡r❣♦❞✐❝✐t②✱ ✐✳❡✳ 1

• k 1(λk ∈ I)
• k

1(λk ∈ I)

n

• v=1

f(v)|ψk(v)|2 ≃ 1 n

n

• v=1

f(v).

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 ψ1, . . . ψn ❜❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ❚❛❦❡ 1 x n✱ ❞❡✜♥❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✱ µx

A = n

• k=1

|ψk(x)|2δλk. ❲❡ ❤❛✈❡

• λtdµx

A = (At)xx,

❛♥❞ 1 n

n

• x=1

µx

A = 1

n

n

• x=1

n

• k=1

|ψk(x)|2δλk = 1 n

n

• k=1

δλk

n

• x=1

|ψk(x)|2 = µA.

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) =

• x∈X

deg(x). E(X, Y ) =

• x∈X,y∈Y

1(uv ∈ E).

X Y

■s♦♣❡r✐♠❡tr✐❝ ✴ ❊①♣❛♥s✐♦♥ ❝♦♥st❛♥t ✿ h(G) = min

X⊂V

E(X, Xc) min (vol(X), vol(Xc)).

❈❤❡❡❣❡r✬s ■♥❡q✉❛❧✐t② ▲❡t 1 = λ1 > λ2 · · · λn −1 ❜❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ P✳ 1 − λ2 ✐s ❝❛❧❧❡❞ t❤❡ s♣❡❝tr❛❧ ❣❛♣ ♦❢ P✳ ❚❤❡♦r❡♠ h(G)2 2 1 − λ2 2h(G).

❈❤❡❡❣❡r✬s ■♥❡q✉❛❧✐t② ❙✐♥❝❡ P = D−1X = D−1/2(D−1/2XD−1/2)D1/2, t❤❡ λi ✬s ❛r❡ ❛❧s♦ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ S ✇✐t❤ S = D−1/2XD−1/2✳ ❙✐♥❝❡ P1 = 1✱ χ = D1/21 ✐s t❤❡ ❡✐❣❡♥✈❡❝t♦r ♦❢ S ❛ss♦❝✐❛t❡❞ t♦ λ1 = 1✳ ❍❡♥❝❡✱ ❢r♦♠ ❈♦✉r❛♥t✲❋✐s❤❡r ❢♦r♠✉❧❛✱ λ2 = max

x:x,χ=0

Sx, x x2 . ❖r ❡q✉✐✈❛❧❡♥t❧②✱ 1 − λ2 = min

x:x,χ=0

(I − S)x, x x2 .

❈❤❡❡❣❡r✬s ■♥❡q✉❛❧✐t② ❲❡ s❡t π(x) = deg(x) = (D1)(x)✳ ❙✐♥❝❡ π = D1/2χ ✇✐t❤ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡ f = D−1/2x✱ ❛❢t❡r s♦♠❡ ❛❧❣❡❜r❛✱ 1 − λ2 = min

f:f,π=0

• u∼v(f(u) − f(v))2
• v deg(v)f(v)2

. ▲❡t X s✉❝❤ t❤❛t h(G) = E(X, Xc) min (vol(X), vol(Xc)). ❲❡ t❛❦❡ f(v) = 1(v ∈ X) vol(X) − 1(v / ∈ X) vol(Xc) .

❈❤❡❡❣❡r✬s ■♥❡q✉❛❧✐t② ❲❡ ❤❛✈❡ f, π =

• x∈X

deg(x) vol(X) −

• x∈Xc

deg(x) vol(Xc) = 0, ❛♥❞ 1 − λ2

• u∼v(f(u) − f(v))2
• v deg(v)f(v)2

= 2E(X, Xc)(1/vol(X) − 1/vol(Xc))2 1/vol(X) + 1/vol(Xc)

• 2

E(X, Xc) min(vol(X), vol(Xc))

• 2h(G).

❆❧♦♥✲❇♦♣♣❛♥❛ ❜♦✉♥❞ ■❢ G ✐s ❛ d✲r❡❣✉❧❛r ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ ♦♥ n ✈❡rt✐❝❡s✱ t❤❡♥ λ1(X) = d ❛♥❞ λ2(X) 2 √ d − 1 + o(1). ❈♦♥s❡q✉❡♥t❧②✱ 1 − λ2(P) 1 − 2 √ d − 1 d + o(1).

❘❛♠❛♥✉❥❛♥ ❣r❛♣❤s ▲❡t G ❜❡ ❛ d✲r❡❣✉❧❛r ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ G ✐s ❘❛♠❛♥✉❥❛♥ ✐❢ λ2(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 G1, G2, · · · ✱ ✇✐t❤ |V (Gn)| ❣r♦✇✐♥❣ t♦ ✐♥✜♥✐t②✱ ❛r❡ ❦♥♦✇♥ t♦ ❡①✐st ✇❤❡♥ ✲ d = q + 1 ✇✐t❤ q = pk ❛♥❞ 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) λ1(X) ∆. ■❢ χ ✐s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ❛♥❞ λn(X) . . . λ1(X)✱ 1 − λ1(X) λn(X) χ 1 + λ1(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 Reλ(L)

1 − max

λ=1 Reλ(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 Zt ❜❡ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✇✐t❤ ❣❡♥❡r❛t♦r L✱ P x

t = etLex

✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦❢ Zt ❣✐✈❡♥ Z0 = x✳ ▲❡t λ1 = 0 > λ2 · · · λn t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ L ❛♥❞ ψ1 = 1/√n, . . . , ψn ❛♥ ♦rt❤♦❣♦♥❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ❋r♦♠ t❤❡ s♣❡❝tr❛❧ t❤❡♦r❡♠ etL =

n

• i=1

etλiψiψ∗

i

P x

t

= 1 n +

n

• i=2

etλiψiψi(x)

❙♣❡❝tr❛❧ ❣❛♣ ❘❡❝❛❧❧ t❤❛t Π = 1/n ✐s t❤❡ ✐♥✈❛r✐❛♥t ❞✐str✐❜✉t✐♦♥✳ ❲❡ ❣❡t P x

t − Π2 = n

• i=2

e2tλi|ψi(x)|2 e−2|λ2|t ❘❡❝❛❧❧ x

• i

|xi| √nx. ❙♦✱ |ψ2(x)|e−|λ2|t 2P x

t − ΠTV √ne−|λ2|t.

✇❤❡r❡ t❤❡ t♦t❛❧ ✈❛r✐❛t✐♦♥ ♥♦r♠ ✐s µ − νTV = 1 2

• x

|µ(x) − ν(x)|.

❙♣❡❝tr❛❧ ❣❛♣ ❚❤❡ ♠✐①✐♥❣ t✐♠❡ ♦❢ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✐s ✉s✉❛❧❧② ❞❡✜♥❡❞ ❛s τ = inf

t>0 max x

P x

t − ΠTV 1

2. maxx |ψ2(x)| |λ2| τ log n 2|λ2|. ✭◆♦t❡ t❤❛t maxx |ψ2(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| =

• v∈V

deg+(v) =

• v∈V

deg−(v) = −Tr{L}. ❍❡♥❝❡ 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✐❜❧❡ ♠♦❞❡❧ ✿ (Xij)1i<jn ❛r❡ ✐♥❞❡♣❡♥❞❡♥t Ber(p) ✈❛r✐❛❜❧❡s ❛♥❞ Xij = Xji✳ ✲ ◆♦♥✲r❡✈❡rs✐❜❧❡ ♠♦❞❡❧ ✿ (Xij)1i=jn ❛r❡ ✐♥❞❡♣❡♥❞❡♥t Ber(p) ✈❛r✐❛❜❧❡s✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❝♦✉❧❞ ❝♦♥s✐❞❡r ❛ ❣❡♥❡r❛❧ ❞❡♣❡♥❞❡♥❝❡ ♦❢ (Xij, Xji)✳

❊r❞➤s✲❘é♥②✐ r❛♥❞♦♠ ❣r❛♣❤ ❲❡ ❤❛✈❡ E deg(v) = E

• u=v

Xuv = (n − 1)p. 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([0, 1]2) = W11 W12 W21 W22

• .

❋♦r t❤❡ ♥♦♥✲r❡✈❡rs✐❜❧❡ ❝❛s❡✱ ✇❡ s❡t (Xij)1i,jn ❛r❡ ✐♥❞❡♣❡♥❞❡♥t Ber(pij) ✈❛r✐❛❜❧❡s ✇✐t❤ pij = p W i n, j 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✐❜❧❡ ♠♦❞❡❧ ✿ (Xij)1i<jn ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s Ber(p) ❛♥❞ Xij = Xji✳ ❊✐❣❡♥✈❛❧✉❡s ♦❢ X ❢♦r n = 100✱ p = 1/2✳

❉❡♥s❡ ❊r❞➤s✲❘é♥②✐ ❍✐st♦❣r❛♠ ♦❢ ❡✐❣❡♥✈❛❧✉❡s λ2(X) . . . λn(X) ❢♦r n = 100 ❛♥❞ p = 1/2✳

❉❡♥s❡ ❊r❞➤s✲❘é♥②✐ ❍✐st♦❣r❛♠ ♦❢ ❡✐❣❡♥✈❛❧✉❡s λ2(X) . . . λn(X) ❢♦r n = 1000 ❛♥❞ p = 1/2✳

❲✐❣♥❡r✬s s❡♠✐✲❝✐r❝❧❡ ❧❛✇ ❲✐❣♥❡r ♠❛tr✐① ✿ Y = (Yij)1i,jn ✇✐t❤ (Yij)i>j ✐✐❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ (Yii)i1 ✐✐❞ ❛♥❞ Yji = Yij✱ ❚❤❡♦r❡♠ ■❢ EY11 = EY12 = 0✱ EY 2

12 = 1 t❤❡♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❢♦r ❛♥②

✐♥t❡r✈❛❧ I ⊂ R✱ µ Y

√n (I) = 1

n

n

• i=1

1 λi(Y ) √n ∈ I

• → µsc(I).

✇❤❡r❡ µsc(dx) = 1|x|2 √ 4 − x2dx✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛✳s✳ µY/√n ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ µsc✳

❙❡❝♦♥❞ ♠♦♠❡♥t ❆ss✉♠❡ Yii = 0✳

• x2dµ Y

√n

= 1 nTr Y √n 2 = 1 n2

• i,j

Y 2

ij

= 2 n2

• 1i<jn

Y 2

ij

❚❛❦✐♥❣ ❡①♣❡❝t❛t✐♦♥✱ ❢r♦♠ EY 2

12 = 1✱

E

• x2dµ Y

√n = n(n − 1)

n2 = 1 + o(1). ❍❡♥❝❡✱ ❢r♦♠ t❤❡ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱

• x2dµ Y

√n → 1 + o(1).

▼❡t❤♦❞ ♦❢ ♠♦♠❡♥ts ❲❡ ❤❛✈❡

• x2k+1dµsc = 0

❛♥❞

• x2kdµsc = ck,

✇❤❡r❡ ck ✐s t❤❡ k✲t❤ ❈❛t❛❧❛♥ ♥✉♠❜❡r ck = 1 k + 1 2k k

• .

❇② ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥t✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t

• xkdµ Y

√n = 1

nTr Y √n k = 1 n1+k/2

• i1,··· ,ik

k

• ℓ=1

Yiℓiℓ+1 =

• xkdµsc + o(1).

❙❡♠✐✲❝✐r❝❧❡ ❧❛✇ ❢♦r ❛❞❥❛❝❡♥❝② ♠❛tr✐① ❚❤❡♦r❡♠ ❋✐① p ∈ (0, 1) ❛♥❞ ❧❡t σ2 = p(1 − p)✳ ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❢♦r ❛♥② ✐♥t❡r✈❛❧ I ⊂ R✱ µ

X σ√n (I) = 1

n

n

• i=1

1 λi(X) σ√n ∈ I

• → µsc(I).

❋r♦♠ ❲✐❣♥❡r ♠❛tr✐① t♦ ❆❞❥❛❝❡♥❝② ♠❛tr✐① ❲❡ ❤❛✈❡ σ2 = Var(X12) = E(X12 − EX12)2 = E(X2

12) − (EX12)2 = p(1 − p).

❆❧s♦✱ ✇✐t❤ J = 11∗✱ EX = p(J − I). ◆♦✇✱ X σ√n = (X − EX) σ√n + pJ σ√n − pI σ√n.

❋r♦♠ ❲✐❣♥❡r ♠❛tr✐① t♦ ❆❞❥❛❝❡♥❝② ♠❛tr✐① X σ√n = (X − EX) σ√n + pJ σ√n − pI σ√n. ✲

(X−EX) σ√n

✐s ❛ ❲✐❣♥❡r ♠❛tr✐①✱ ✐ts ❊❙❉ ❝♦♥✈❡r❣❡ t♦ t❤❡ s❡♠✐✲❝✐r❝✉❧❛r ❧❛✇✳ ✲

pJ σ√n ❤❛s r❛♥❦ ♦♥❡ ✭❜✉t ♥♦r♠ ♦❢ ♦r❞❡r √n ✦✮✳

✲

pI σ√n ❤❛s ♥♦r♠ ♦❢ ♦r❞❡r 1/√n ✭❜✉t ❢✉❧❧ r❛♥❦ ✦✮✳

■♥t❡r❧❛❝✐♥❣ ■♥❡q✉❛❧✐t✐❡s ❚❛❦❡ A ∈ Mn(R) s②♠♠❡tr✐❝ ❛♥❞ A′ ∈ Mn−1(R) ❛ ♠✐♥♦r✱ A = A′ v v∗ u

• ❚❤❡♥✱ ✐❢ λi+1 λi✱

λi+1(A) λi(A′) λi(A). ❚❛❦❡ A, B ∈ Mn(R) s②♠♠❡tr✐❝ ❛♥❞ rank(A − B) = k t❤❡♥ λi+k(A) λi(B) λi−k(A).

■♥t❡r❧❛❝✐♥❣ ■♥❡q✉❛❧✐t✐❡s ❉❡✜♥❡✱ t❤❡ ❑♦❧♠♦❣♦r♦✈✲❙♠✐r♥♦✈ ❞✐st❛♥❝❡ ❛s dKS(µ, ν) = sup

t

|µ((−∞, t)) − ν((−∞, t))|. ■❢ dKS(µn, µ) → 0 t❤❡♥ µn ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ µ✳ ▲❡♠♠❛ ■❢ A, B ❛r❡ ❍❡r♠✐t✐❛♥ n × n ♠❛tr✐❝❡s✱ t❤❡♥ dKS(µA, µB) rank(A − B) n .

❋r♦♠ ❲✐❣♥❡r ♠❛tr✐① t♦ ❆❞❥❛❝❡♥❝② ♠❛tr✐① X σ√n = (X − EX) σ√n + pJ σ√n − pI σ√n. ✲

(X−EX) σ√n

✐s ❛ ❲✐❣♥❡r ♠❛tr✐①✱ ✐ts ❊❙❉ ❝♦♥✈❡r❣❡ t♦ t❤❡ s❡♠✐✲❝✐r❝✉❧❛r ❧❛✇✳ ✲

pJ σ√n ❤❛s r❛♥❦ ♦♥❡ ✭❜✉t ♥♦r♠ ♦❢ ♦r❞❡r √n ✦✮✳

✲

pI σ√n ❤❛s ♥♦r♠ ♦❢ ♦r❞❡r 1/√n ✭❜✉t ❢✉❧❧ r❛♥❦ ✦✮✳

■♥❤♦♠♦❣❡♥❡♦✉s r❛♥❞♦♠ ❣r❛♣❤ ❆ s✐♠✐❧❛r st✉❞② ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r t❤❡ ❊❙❉ ♦❢ ❞❡♥s❡ ✉♥❞✐r❡❝t❡❞ ✐♥❤♦♠♦❣❡♥❡♦✉s ❣r❛♣❤s✱ ✇❤❡r❡ Xij

d

= Ber(W(i/n, j/n)), ❛♥❞ W : [0, 1]2 → [0, 1] ✐s ❛ ❝♦♥st❛♥t ❜② ❜❧♦❝❦ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❧✐♠✐t ✐s t❤❡ s❡♠✐✲❝✐r❝❧❡ ✐❢ σ2(x) = 1 W(x, y)(1 − W(x, y))dy ✐s ❛ ❝♦♥st❛♥t ❢✉♥❝✐t♦♥ ♦❢ x✳

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

❋➯r❡❞✐✲❑♦♠❧ós ❚❤❡♦r❡♠ ❲✐❣♥❡r ♠❛tr✐① Y = (Yij)1i,jn ✇✐t❤ (Yij)i>j ✐✐❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ (Yii)i1 ✐✐❞ ❛♥❞ Yji = Yij✱ ❚❤❡♦r❡♠ ■❢ EY11 = EY12 = 0✱ EY 2

12 = 1✱ EY 2 11 < ∞✱ EY 4 11 < ∞✱ t❤❡♥ ✇✐t❤

♣r♦❜❛❜✐❧✐t② ♦♥❡✱ λ1 Y √n

• = 2 + o(1) = −λn

Y √n

• .

❘❡❝❛❧❧ t❤❛t supp(µsc) = [−2, 2]✳

❊①tr❡♠❛❧ ❊✐❣❡♥✈❛❧✉❡s ♦❢ ❉❡♥s❡ ❊r❞➤s✲❘é♥②✐ ❚❤❡♦r❡♠ ❋✐① p ∈ (0, 1) ❛♥❞ ❧❡t σ2 = p(1 − p)✳ ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ λ1(X) = pn + O(√n), λ2(X) = 2σ√n + o(√n) = −λn(X), ❛♥❞✱ ✐❢ ψ1 ✐s t❤❡ P❡rr♦♥ ❡✐❣❡♥✈❡❝t♦r✱ ψ1 − 1/√n = O(1/√n).

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

• λi(A + B) − λσ(i)(A)
• B.

❊①tr❡♠❛❧ ❊✐❣❡♥✈❛❧✉❡s ♦❢ ❉❡♥s❡ ❊r❞➤s✲❘é♥②✐ X = p(J − I) + (X − EX). ❚❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ J − I ❛r❡ ✿ n − 1 ❛♥❞ −1 ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② n − 1✳ ❋r♦♠ ❋➯r❡❞✐✲❑♦♠❧ós ❚❤❡♦r❡♠✱ X − EX = 2σ√n + o(√n) ❍❡♥❝❡✱ |λ1(X) − p(n − 1)| = O(√n) |λ2(X) + p|

• (2 + o(1))σ√n.

❍♦✇❡✈❡r✱ ❢r♦♠ t❤❡ s❡♠✐✲❝✐r❝❧❡ ❧❛✇✱ ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t λ2(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)−1u = 0 ❛♥❞ ψ = (A − λ)−1u. ✐s ❛♥ ❡✐❣❡♥✈❡❝t♦r✳

❘❛♥❦ ♦♥❡ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❊✐❣❡♥✈❡❝t♦r ❆♣♣❧② t❤✐s X = (X − EX − pI) + p11∗, ❛♥❞ λ = λ1(X) = pn + O(√n). ❋r♦♠ X − EX − pI = O(√n)✱ ✇❡ ❣❡t ψ1 − 1/√n = O(1/√n).

▼❛r❦♦✈ ❚r❛♥s✐t✐♦♥ ▼❛tr✐① ❆ s✐♠✐❧❛r ❛♥❛❧②s✐s ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r P = D−1X. λ2(P) = 2σ p√n + o 1 √n

• = −λn(P).

❊✐❣❡♥✈❛❧✉❡s ♦❢ P ❢♦r n = 100✱ p = 1/2✳

▼❛r❦♦✈ ❚r❛♥s✐t✐♦♥ ▼❛tr✐① ❍✐st♦❣r❛♠ ♦❢ λn(P) . . . λ2(P) ❢♦r n = 1000✱ p = 1/2✳

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

▲❛♣❧❛❝✐❛♥ L = X − D. ❍✐st♦❣r❛♠ ♦❢ λn(X) . . . λ2(L) ❢♦r n = 1000✱ p = 1/2✳

▲❛♣❧❛❝✐❛♥ ❚❤❡♦r❡♠ ❋✐① p ∈ (0, 1) ❛♥❞ ❧❡t σ2 = p(1 − p)✳ ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❢♦r ❛♥② ✐♥t❡r✈❛❧ I ⊂ R✱ µ L+npI

σ√n (I) → µ(I).

✇❤❡r❡ µ = µsc ⊞ N(0, 1)✳ ▼♦r❡♦✈❡r✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ λ2(L) = −np + σ(1 + o(1))

• 2n log n.

✭❉✐♥❣✴❏✐❛♥❣ ✷✵✶✵✮✱ ✭❏✐❛♥❣ ✷✵✶✷✮

❍❡✉r✐st✐❝s L + npI σ√n = X − EX σ√n − D − ED σ√n + pJ σ√n. ✲

X−EX σ√n

✿ ❲✐❣♥❡r ♠❛tr✐① ✿ ✐ts ❊❙❉ ❝♦♥✈❡r❣❡s t♦ t❤❡ s❡♠✐✲❝✐r❝✉❧❛r ❧❛✇✳ ✲

D−ED σ√n

✿ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ✇✐t❤ ❛♣♣r♦①✐♠❛t❡❧② ✐✐❞ ●❛✉ss✐❛♥ N(0, 1) ❝♦❡✣❝✐❡♥ts✱ Dii =

j Xij✳

✲

pJ σ√n ✿ ♦♥❡ ❡✐❣❡♥✈❛❧✉❡ p√n σ

→ ∞✱ ❛❧❧ ♦t❤❡rs 0✳

❋r❡❡ ❝♦♥✈♦❧✉t✐♦♥ ▲❡t An ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❞❡t❡r♠✐♥✐st✐❝ ❍❡r♠✐t✐❛♥ n × n ♠❛tr✐❝❡s s✉❝❤ t❤❛t ❢♦r ❛♥② ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ f✱

• fµAn →
• fdµ.

❚❤❡♥✱ ✐❢ Y ✐s ❛ ❲✐❣♥❡r ♠❛tr✐①✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱

• fµ Y

√n +An →

• fdν.

❛♥❞ ν := µsc ⊞ µ. ■♥ ❤✐❣❤ ❞✐♠❡♥s✐♦♥✱ t❤❡ s♣❡❝tr❛ ❛❞❞ ✉♣ ✦ ✦

▼❛①✐♠✉♠ ♦❢ ●❛✉ss✐❛♥ ✈❛r✐❛❜❧❡s L + npI σ√n = X − EX σ√n − D − ED σ√n + pJ σ√n. ■❢ (Zi)i1 ❛r❡ ✐✐❞ N(0, 1) ✈❛r✐❛❜❧❡s t❤❡♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ max

1in Zi = (1 + o(1))

• 2 log n.

■♥ ♣❛rt✐❝✉❧❛r✱

• D − ED

σ√n

• = (1 + o(1))
• 2 log n,

❛♥❞ ✳ ✳ ✳ λ2(L) = −np + σ(1 + o(1))

• 2n 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✳ ◆♦t❡ t❤❛t σ =

• p(1 − p) ∼ √p ✇❤❡♥ p = o(1)✳

❙♣❛rs❡ ❊r❞➤s✲❘é♥②✐ ❆ ❦❡② t❡❝❤♥✐❝❛❧ st❛t❡♠❡♥t ✐s ❑❤♦r✉♥③❤② ✭✷✵✵✶✮✱ ❱✉ ✭✷✵✵✼✮ ✿ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱

• X − EX

σ√n

• = 2 + o(1)

✇❤❡♥ np log n → ∞. ❍♦✇❡✈❡r✱ ✇❤❡♥ p = o

• log n

n

• ✱
• X−EX

σ√n

• ≫ 1✳

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

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

❈✐r❝✉❧❛r ▲❛✇ ❚❤❡♦r❡♠ ❆ss✉♠❡ t❤❛t (Yij)i,j1 ❛r❡ ✐✐❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s EY12 = 0✱ ❛♥❞ E|Y12|2 = 1✱ ❝♦♥s✐❞❡r t❤❡ ♠❛tr✐① Y = (Yij)1i,jn. ❚❤❡♥✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❢♦r ❛♥② ❇♦r❡❧ s❡t I ⊂ C✱ µY/√n(I) → µc(I), ✇✐t❤ µc(dxdy) = 1 π1|z|1dz.

✭▼❡❤t❛ ✶✾✻✼✮✱ ✭●✐r❦♦ ✶✾✽✹✮✱ ✭❇❛✐ ✶✾✾✼✮✱ ✭P❛♥✴❩❤♦✉ ✷✵✶✵✮✱ ✭●öt③❡✴❚✐❦❤♦♠✐r♦✈ ✷✵✶✵✮ . . . ✱ ✭❚❛♦✴❱✉ ✷✵✶✵✮✳

❈✐r❝✉❧❛r ❧❛✇ ❢♦r ❛❞❥❛❝❡♥❝② ♠❛tr✐① ❚❤❡♦r❡♠ ❆ss✉♠❡ (log n)6/n ≪ p 1 − δ ❛♥❞ ❧❡t σ2 = p(1 − p)✳ ❋♦r ❛♥② ❇♦r❡❧ I ⊂ C✱ ✐♥ ♣r♦❜❛❜✐❧✐t②✱ µ

X σ√n (I) → µc(I).

✭❇♦r❞❡♥❛✈❡✱ ❈❛♣✉t♦✱ ❈❤❛❢❛ï ✷✵✶✹✮

❍❡✉r✐st✐❝s X σ√n = (X − EX) σ√n + pJ σ√n − pI σ√n. ✲

(X−EX) σ√n

✐s ❛ r❛♥❞♦♠ ✐✐❞ ♠❛tr✐①✱ ✐ts ❊❙❉ ❝♦♥✈❡r❣❡s t♦ t❤❡ ❝✐r❝✉❧❛r ❧❛✇✳ ✲

pJ σ√n ❤❛s r❛♥❦ ♦♥❡ ✭❜✉t ♥♦r♠ ♦❢ ♦r❞❡r √n ✦ ✮✳

✲

pI σ√n ❤❛s ♥♦r♠ ♦❢ ♦r❞❡r 1/√n ✭❜✉t ❢✉❧❧ r❛♥❦ ✦ ✮✳

P❡rt✉r❜❛t✐♦♥ ♦❢ ♥♦♥✲❍❡r♠✐t✐❛♥ ♠❛tr✐❝❡s ❚❛❦❡ N =      1 · · · 1 · · · ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳✳✳ . . .      ❛♥❞ C =      1 · · · 1 · · · ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳✳✳ ε . . .      . ❆❧❧ ❡✐❣❡♥✈❛❧✉❡s ♦❢ N ❛r❡ 0✱ µN = δ0. ❋♦r ε = 1✱ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ C ❛r❡ e2iπk/n✱ 1 k n ✿

• fdµC → 1

2π 2π f(e2iθ)dθ. ❚r✉❡ ❛s s♦♦♥ ❛s ε1/n → 1 ✦ ✦

▼❛r❦♦✈ ❚r❛♥s✐t✐♦♥ ▼❛tr✐① ❋♦r (log n)6/n ≪ p 1 − δ✱ t❤❡r❡ ✐s ❛ ❝✐r❝✉❧❛r ❧❛✇ ❢♦r P = D−1X ✇✐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❡❞✉❝✐❜❧❡ ❛♥❞ Π − 1/n❚❱ = O

• σ

p

• log n

n

• + O

√σ p log n n3/4

• .

❙✐♠✐❧❛r❧②✱ ❢♦r t❤❡ ▼❛r❦♦✈ tr❛♥s✐t✐♦♥ ♠❛tr✐① P✱ π − 1/n❚❱ = O σ p√n

• + O

√σ log n pn3/4

• ,

✭❇♦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 = Gn ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ d✲r❡❣✉❧❛r ❣r❛♣❤s ♦♥ n ✈❡rt✐❝❡s s✉❝❤ t❤❛t ❢♦r ❛♥② k✱ |{❝②❝❧❡s ♦❢ ❧❡♥❣❤t k}| = ok(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 = Gn ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛❧❧② tr❡❡✲❧✐❦❡ d✲r❡❣✉❧❛r ❣r❛♣❤s ♦♥ n ✈❡rt✐❝❡s t❤❡♥ ❢♦r ❛♥② I ⊂ R✱ 1 n

n

• k=1

1(λk(X) ∈ I) = µX(I) → µKM(I). ✇❤❡r❡ µKM(dx) = d 2π

• 4(d − 1) − x2

d2 − x2 1|x|2

√ d−1dx.

❲❡ ❤❛✈❡ µKM(I √ d) → µsc(I) ✇❤❡♥ d → ∞✳

❙✐♠✉❧❛t✐♦♥ ❚❛❦❡ d = 4✱ n = 2000 ❛♥❞ G ❛ ✉♥✐❢♦r♠❧② s❛♠♣❧❡❞ d✲r❡❣✉❧❛r ❣r❛♣❤✳

■❞❡❛ ♦❢ Pr♦♦❢ ❙✐♥❝❡ G ✐s ❧♦❝❛❧❧② tr❡❡✲❧✐❦❡✱ T ✐s t❤❡ ✐♥✜♥✐t❡ d✲r❡❣✉❧❛r tr❡❡✳

• λkdµX = 1

nTrXk = 1 n

n

• v=1

|{❝❧♦s❡❞ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ k st❛rt✐♥❣ ❢r♦♠ v ✐♥ G}| = |{❝❧♦s❡❞ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ k st❛rt✐♥❣ ❢r♦♠ t❤❡ r♦♦t ♦❢ T}| + ok(1) =

• λkdµKM + ok(1).

■t ✐s t❤❡♥ ❢❛✐r❧② ❡❛s② t♦ ❝♦♠♣✉t❡ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛❧❦s ♦♥ t❤❡ ✐♥✜♥✐t❡ r❡❣✉❧❛r tr❡❡✳

❋r✐❡❞♠❛♥✬s ❚❤❡♦r❡♠ ❲❡ ❤❛✈❡ λ1(X) = d✱ supp(µKM) = [−2 √ d − 1, 2 √ d − 1] ❛♥❞ λ2(X) 2 √ d − 1 + o(1)✳ ❘❡❝❛❧❧ t❤❛t λ2(X) 2 √ d − 1 ❢♦r ❘❛♠❛♥✉❥❛♥ ❣r❛♣❤s✳ ❚❤❡♦r❡♠ ❋✐① ❡✈❡♥ ✐♥t❡❣❡r d 4✳ ▲❡t G = Gn ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ d✲r❡❣✉❧❛r ❣r❛♣❤s ♦♥ n ✈❡rt✐❝❡s✱ t❤❡♥ ✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✱ λ2(X) = 2 √ d − 1 + o(1) = −λn(X). ▼♦st r❡❣✉❧❛r ❣r❛♣❤s ❛r❡ ♥❡❛r❧② ❘❛♠❛♥✉❥❛♥ ✦ ✭❋r✐❡❞♠❛♥ ✷✵✵✹✮

❊r❞➤s✲❘é♥②✐ ❚❤❡♦r❡♠ ❋✐① ✐♥t❡❣❡r c > 0✳ ▲❡t G = Gn ❜❡ ❛♥ ❊r❞➤s✲❘é♥②✐ ❣r❛♣❤ ✇✐t❤ ♣❛r❛♠❡t❡r p = c/n✳ ❚❤❡♥✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✱ ❢♦r ❛♥② ✐♥t❡r✈❛❧ I ⊂ R✱ µX(I) = 1 n

n

• k=1

1(λk(X) ∈ I) → µc(I), ❢♦r s♦♠❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µc ✇✐t❤ s✉♣♣♦rt R✳

❊r❞➤s✲❘é♥②✐ ❍✐st♦❣r❛♠ ♦❢ ❡✐❣❡♥✈❛❧✉❡s ❢♦r c = 4 ❛♥❞ n = 500✳ ❚❤❡ ♠❛①✐♠✉♠ ❡✐❣❡♥✈❛❧✉❡ ✐s λ1(X) = (1 + o(1))

• log n

log log n

✭❙✉❞❛❦♦✈ ❛♥❞ ❑r✐✈❡❧❡✈✐❝❤ ✷✵✵✸✮✳

❊r❞➤s✲❘é♥②✐ ❚❤❡r❡ ✐s ♥♦ ❡①♣❧✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r µc✳ ▲❡t Λ = {λi, i 1}✱ ❜❡ t❤❡ ❛t♦♠s ♦❢ µc✱ ✐✳❡✳ µc({λ}) > 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ∈ Λ. ❚❤❡♥ Λ ✐s t❤❡ s❡t ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡rs

• λ∈Λ

µc({λ}) < 1 ✐❢ ❛♥❞ ♦♥❧② ✐❢ c > 1✳ ❆❧s♦ µc({0}) ❤❛s ❛ ❝❧♦s❡❞✲❢♦r♠ ❡①♣r❡ss✐♦♥✳

❇♦r❞❡♥❛✈❡✴▲❡❧❛r❣❡✴❙❛❧❡③ ✭✷✵✶✷✮✱ ❙❛❧❡③ ✭✷✵✶✸✮✱ ❇♦r❞❡♥❛✈❡✴❱✐rá❣✴❙❡♥ ✭✷✵✶✹✮✳

❊r❞➤s✲❘é♥②✐ ❊✐❣❡♥✈❛❧✉❡s ❢♦r c = 4 ❛♥❞ n = 500 ♦❢ t❤❡ ♥♦♥✲❜❛❝❦tr❛❝❦✐♥❣ ♠❛tr✐① B✳ ◆♦ t❤❡♦r❡♠ ②❡t ✦

■♥ ❙✉♠♠❛r②

■♥ ❙✉♠♠❛r② ❚❤❡ s♣❡❝tr❛ ♦❢ ✇❡❧❧✲❝❤♦s❡♥ ❣r❛♣❤ ♠❛tr✐❝❡s ❝♦♥t❛✐♥ ❛ ❧♦t ♦❢ ♠❡❛♥✐♥❣❢✉❧ ✐♥❢♦r♠❛t✐♦♥ ❛♥❞ s♣❡❝tr❛❧ ❛❧❣♦r✐t❤♠s ❛r❡ r❡❛s♦♥❛❜❧② ❢❛st✳

■♥ ❙✉♠♠❛r② ❯♥❞❡rst❛♥❞✐♥❣ t❤❡ s♣❡❝tr❛ ♦❢ r❛♥❞♦♠ ❣r❛♣❤s ♠❛② ♥♦t❛❜❧② ❤❡❧♣ t♦ s♦rt ♠❡❛♥✐♥❣❢✉❧ ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ ♥♦✐s❡✳

■♥ ❙✉♠♠❛r② ❆s ❧♦♥❣ ❛s t❤❡ ❛✈❡r❛❣❡ ❞❡❣r❡❡ ♦❢ ❛ ✈❡rt❡① ❣r♦✇s t♦ ✐♥✜♥✐t②✱ ❣❡♥❡r❛❧ s♣❡❝tr❛❧ ❣r❛♣❤ t❤❡♦r②✱ t❤❡♦r② ♦❢ ♣❡rt✉r❜❛t✐♦♥s ❛♥❞ r❛♥❞♦♠ ♠❛tr✐❝❡s ❛r❡ ✈❡r② ✉s❡❢✉❧✳

■♥ ❙✉♠♠❛r② ❚❤❡ s♣❡❝tr✉♠ ♦❢ ❞✐❧✉t❡❞ r❛♥❞♦♠ ❣r❛♣❤s ✐s ✈❡r② ❢❛r ❢r♦♠ ❜❡✐♥❣ ✉♥❞❡rst♦♦❞✳

❆ ❢❡✇ ❣❡♥❡r❛❧ r❡❢❡r❡♥❝❡s ❘❛♥❞♦♠ ♠❛tr✐❝❡s

• ✳ ❆♥❞❡rs♦♥✱ ❆✳ ●✉✐♦♥♥❡t ❛♥❞ ❖✳ ❩❡✐t♦✉♥✐✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦

❘❛♥❞♦♠ ▼❛tr✐❝❡s✱ ✷✵✵✾✳ ❩✳ ❇❛✐ ❛♥❞ ❏✳ ❙✐❧✈❡rst❡✐♥✱ ❙♣❡❝tr❛❧ ❆♥❛❧②s✐s ♦❢ ▲❛r❣❡ ❉✐♠❡♥s✐♦♥❛❧ ❘❛♥❞♦♠ ▼❛tr✐❝❡s✱ ✷✵✵✻✳ ❙♣❡❝tr❛❧ ❣r❛♣❤ t❤❡♦r② ▲✳ ▲♦✈ás③✱ ❊✐❣❡♥✈❛❧✉❡s ♦❢ ❣r❛♣❤s✱ ✷✵✵✼✳ ❋✳ ❈❤✉♥❣✱ ❙♣❡❝tr❛❧ ●r❛♣❤ ❚❤❡♦r② ✱ ✶✾✾✷✳ ❆ ❇r♦✉✇❡r ❛♥❞ ❲ ❍❛❡♠❡rs✱ ❙♣❡❝tr❛ ♦❢ ●r❛♣❤s✱ ✷✵✶✷✳ ❇✳ ▼♦❤❛r ❛♥❞ ❲✳ ❲♦❡ss✱ ❆ ❙✉r✈❡② ♦♥ ❙♣❡❝tr❛ ♦❢ ■♥✜♥✐t❡

• r❛♣❤s✱ ✶✾✽✾✳

▼❛r❦♦✈ ❝❤❛✐♥s ❛♥❞ r❛♥❞♦♠ ✇❛❧❦s ♦♥ ❣r❛♣❤ ❉✳ ❆❧❞♦✉s ❛♥❞ ❏✳ ❋✐❧❧✱ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❈❤❛✐♥s ❛♥❞ ❘❛♥❞♦♠ ❲❛❧❦s ♦♥ ●r❛♣❤s✳ ❉✳ ▲❡✈✐♥ ✱ ❨✳ P❡r❡s✱ ❊✳ ❲✐❧♠❡r✱ ▼❛r❦♦✈ ❈❤❛✐♥s ❛♥❞ ▼✐①✐♥❣ ❚✐♠❡s✱ ✷✵✵✽✳

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥ ✦