t trs st t - - PowerPoint PPT Presentation

A locking pattern

Supporting images for a talk by Hari Bercovici Free probability and theN→ ∞ limit Berkeley, March 1916

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣✐❝t✉r❡s✱ ✇❡ ♠✉st t❤✐♥❦ ♦❢ t❤r❡❡ ✼ ×✼ ♠❛tr✐❝❡s A,B,C ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s α✶ = α✷ > α✸ = α✹ > α✺ = α✻ = α✼, β✶ > β✷ = β✸ = β✹ > β✺ = β✻ = β✼✱ γ✶ = γ✷ > γ✸ = γ✹ > γ✺ = γ✻ = γ✼ s✉❝❤ t❤❛t A+B = C. ❆ ❤✐✈❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡s❡ ✈❛❧✉❡s ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ ♦❢ s✐③❡ ✼✱ ❛✣♥❡ ♦♥ ❡❛❝❤ ✉♥✐t ✏❧❛tt✐❝❡✑ tr✐❛♥❣❧❡✱ ❛♥❞ ✇✐t❤ ❜♦✉♥❞❛r② ✈❛❧✉❡s ❛t ❧❛tt✐❝❡ ♣♦✐♥ts ✭st❛rt✐♥❣ ❛t t❤❡ t♦♣ ❛♥❞ ❝♦♥t✐♥✉✐♥❣ ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✮ ❡q✉❛❧ t♦ ✵,α✶,α✶ +α✷,...,α1 +···+α6, Tr(A),Tr(A)+β✶,...,Tr(A)+β1 +···+β6, Tr(C) = Tr(A)+Tr(B),Tr(C)−γ✼,Tr(C)−γ✼ −γ✻,...,γ✶

❚❤❡ ♣✐❝t✉r❡s ✐♥❞✐❝❛t❡ s✉❝❝❡ss✐✈❡❧② t❤❡ ❧❛r❣❡r ❛r❡❛s ♦♥ ✇❤✐❝❤ t❤❡ ❤✐✈❡ ♠✉st ❜❡ ❛♥ ❛✣♥❡ ❢✉♥❝t✐♦♥✳ ❚❤❡♥ t❤❡ r❡❞ ❛rr♦✇s ✐♥❞✐❝❛t❡ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ ❤✐✈❡ ✈❛❧✉❡s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❛t t❤❡ ❝♦r♥❡rs ♦❢ t❤❡s❡ r❡❣✐♦♥s✳ ❚❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ ❤✐✈❡ ✈❛❧✉❡s ❛t t❤❡ ❡♥❞ ♣♦✐♥ts ♦❢ t❤❡ r❡❞ ❛rr♦✇s ✐s t❤❡♥ ❝❛❧❝✉❧❛t❡❞✳ ✭❚❤❡ ❤✐✈❡ ✐s h ❛♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡ ✈❛❧✉❡s s❤♦✉❧❞ ❜❡ str❛✐❣❤t❢♦r✇❛r❞✳✮ ❚❤❡ ❧❛st t✇♦ ✜❣✉r❡s ✐♥❞✐❝❛t❡ t❤❡ ✏❝❛♥♦♥✐❝❛❧✑ ❤♦♥❡②❝♦♠❜ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤✐s ❧♦❝❦✐♥❣ ♣❛t❡r♥✳ ❚❤❡ ❤♦♥❡②❝♦♠❜ s❤♦✇s ❤♦✇ t♦ ♦❜t❛✐♥ t❤❡ ❤✐✈❡ ❞✐✛❡r❡♥❝❡s ♠♦r❡ ❡✣❝✐❡♥t❧②✳ ❚❤❡s❡ ❞✐✛❡r❡♥❝❡s ❛r❡ t❤❡♥ ✉s❡❞ ✐♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❝❡rt❛✐♥ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥s ✇❤♦s❡ ❡①✐st❡♥❝❡ ✇❛s ❝♦♥❥❡❝t✉r❡❞ ❜② ❉❛♥✐❧♦✈ ❛♥❞ ❑♦s❤❡✈♦②✳ ❚❤✐s ✐s ♣❛rt ♦❢ ❛ ❥♦✐♥t ♣r♦❥❡❝t ✇✐t❤ ❲✳ ❙✳ ▲✐ ✇❤✐❝❤ ✇❡ ❤♦♣❡ ❜r✐♥❣s ♥❡✇ ✐♥s✐❣❤t ✐♥ t❤❡ ❈♦♥♥❡s ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠✳

h(✻,✶)−h(✹,✶) = ✷α✺−✼

h(✸,✷)−h(✹,✶) = β✷−✹

h(✷,✷)−h(✸,✷) = β✺−✼ −γ✺−✼

h(✸,✵)−h(✸,✶) = α✶−✷ +α✺−✼ +β✷−✹ +β✺−✼ −γ✶−✷ −γ✸−✹ −γ✺−✼