t trs st t - PowerPoint PPT Presentation

A locking pattern Supporting images for a talk by Hari Bercovici Free probability and the N → ∞ 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 ✷ ,..., α 1 + ··· + α 6 , Tr ( A ) , Tr ( A )+ β ✶ ,..., Tr ( A )+ β 1 + ··· + β 6 , 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 ✺ − ✼