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◆✐❦♦❧❛♦s ❋♦✉♥t♦✉❧❛❦✐s ✶ ❑♦♥st❛♥t✐♥♦s P❛♥❛❣✐♦t♦✉ ✶ ✸✲❝♦♥♥❡❝t❡❞ ❝♦r❡s ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶ ▼❛①✲P❧❛♥❝❦✲■♥st✐t✉t ❢ür ■♥❢♦r♠❛t✐❦✱ ●❡r♠❛♥② ▲❛❜♦r❛t♦✐r❡ ❞✬■♥❢♦r♠❛t✐q✉❡ ❞❡ P❛r✐s✲◆♦r❞✱ ✶✻ ◆♦✈❡♠❜❡r ✷✵✶✵

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❣r❛♣❤s ❊r❞➤s ❛♥❞ ❘é♥②✐ ♣r♦✈❡❞ t❤❛t ❛r♦✉♥❞ ♣ ❝r = ✶ ♥ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ str✉❝t✉r❡ ♦❢ ● ( ♥ , ♣ ) ♦❝❝✉rs✿ ✐❢ ♣ < ✶ − ε ♥ ✱ t❤❡♥ ❛❧❧ ❝♦♠♣♦♥❡♥ts ♦❢ ● ♥ , ♣ ❝♦♥t❛✐♥ ❖ ( ❧♦❣ ♥ ) ✈❡rt✐❝❡s❀ ✐❢ ♣ > ✶ + ε ♥ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝♦♠♣♦♥❡♥t ✇✐t❤ Θ( ♥ ) ✈❡rt✐❝❡s✱ ✇❤❡r❡❛s ❡✈❡r② ♦t❤❡r ❝♦♠♣♦♥❡♥t ❤❛s ❖ ( ❧♦❣ ♥ ) ✈❡rt✐❝❡s✳ ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❣r❛♣❤s ❊r❞➤s ❛♥❞ ❘é♥②✐ ♣r♦✈❡❞ t❤❛t ❛r♦✉♥❞ ♣ ❝r = ✶ ♥ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ str✉❝t✉r❡ ♦❢ ● ( ♥ , ♣ ) ♦❝❝✉rs✿ ✐❢ ♣ < ✶ − ε ♥ ✱ t❤❡♥ ❛❧❧ ❝♦♠♣♦♥❡♥ts ♦❢ ● ♥ , ♣ ❝♦♥t❛✐♥ ❖ ( ❧♦❣ ♥ ) ✈❡rt✐❝❡s❀ ✐❢ ♣ > ✶ + ε ♥ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝♦♠♣♦♥❡♥t ✇✐t❤ Θ( ♥ ) ✈❡rt✐❝❡s✱ ✇❤❡r❡❛s ❡✈❡r② ♦t❤❡r ❝♦♠♣♦♥❡♥t ❤❛s ❖ ( ❧♦❣ ♥ ) ✈❡rt✐❝❡s✳ ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❣r❛♣❤s ❊r❞➤s ❛♥❞ ❘é♥②✐ ♣r♦✈❡❞ t❤❛t ❛r♦✉♥❞ ♣ ❝r = ✶ ♥ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ str✉❝t✉r❡ ♦❢ ● ( ♥ , ♣ ) ♦❝❝✉rs✿ ✐❢ ♣ < ✶ − ε ♥ ✱ t❤❡♥ ❛❧❧ ❝♦♠♣♦♥❡♥ts ♦❢ ● ♥ , ♣ ❝♦♥t❛✐♥ ❖ ( ❧♦❣ ♥ ) ✈❡rt✐❝❡s❀ ✐❢ ♣ > ✶ + ε ♥ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝♦♠♣♦♥❡♥t ✇✐t❤ Θ( ♥ ) ✈❡rt✐❝❡s✱ ✇❤❡r❡❛s ❡✈❡r② ♦t❤❡r ❝♦♠♣♦♥❡♥t ❤❛s ❖ ( ❧♦❣ ♥ ) ✈❡rt✐❝❡s✳ ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s ❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s ▲❡t P ♥ ❜❡ t❤❡ s❡t ♦❢ ❧❛❜❡❧❡❞ ❛❧❧ ♣❧❛♥❛r ❣r❛♣❤s ♦♥ ♥ ✈❡rt✐❝❡s ❛♥❞ ❧❡t P ♥ ❞❡♥♦t❡ ❛ ❣r❛♣❤ t❛❦❡♥ ❛t r❛♥❞♦♠ ❢r♦♠ P ♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ / |P ♥ | ✳ ❍♦✇ ♠❛♥② ❡❞❣❡s ❞♦❡s P ♥ t②♣✐❝❛❧❧② ❤❛✈❡❄ ●✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ s❤♦✇❡❞ t❤❛t ❡ ( P ♥ ) − κ ♥ ❞ √ → ◆ ( ✵ , ✶ ) , ✷ πλ ♥ ✇❤❡r❡ κ ≈ ✷ . ✷✶✸✷✻ ❛♥❞ λ ≈ ✵ . ✹✸✵✸✹ ❛r❡ ❝♦♥st❛♥ts✳ ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s ❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s ▲❡t P ♥ ❜❡ t❤❡ s❡t ♦❢ ❧❛❜❡❧❡❞ ❛❧❧ ♣❧❛♥❛r ❣r❛♣❤s ♦♥ ♥ ✈❡rt✐❝❡s ❛♥❞ ❧❡t P ♥ ❞❡♥♦t❡ ❛ ❣r❛♣❤ t❛❦❡♥ ❛t r❛♥❞♦♠ ❢r♦♠ P ♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ / |P ♥ | ✳ ❍♦✇ ♠❛♥② ❡❞❣❡s ❞♦❡s P ♥ t②♣✐❝❛❧❧② ❤❛✈❡❄ ●✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ s❤♦✇❡❞ t❤❛t ❡ ( P ♥ ) − κ ♥ ❞ √ → ◆ ( ✵ , ✶ ) , ✷ πλ ♥ ✇❤❡r❡ κ ≈ ✷ . ✷✶✸✷✻ ❛♥❞ λ ≈ ✵ . ✹✸✵✸✹ ❛r❡ ❝♦♥st❛♥ts✳ ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s ❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s ▲❡t P ♥ ❜❡ t❤❡ s❡t ♦❢ ❧❛❜❡❧❡❞ ❛❧❧ ♣❧❛♥❛r ❣r❛♣❤s ♦♥ ♥ ✈❡rt✐❝❡s ❛♥❞ ❧❡t P ♥ ❞❡♥♦t❡ ❛ ❣r❛♣❤ t❛❦❡♥ ❛t r❛♥❞♦♠ ❢r♦♠ P ♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ / |P ♥ | ✳ ❍♦✇ ♠❛♥② ❡❞❣❡s ❞♦❡s P ♥ t②♣✐❝❛❧❧② ❤❛✈❡❄ ●✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ s❤♦✇❡❞ t❤❛t ❡ ( P ♥ ) − κ ♥ ❞ √ → ◆ ( ✵ , ✶ ) , ✷ πλ ♥ ✇❤❡r❡ κ ≈ ✷ . ✷✶✸✷✻ ❛♥❞ λ ≈ ✵ . ✹✸✵✸✹ ❛r❡ ❝♦♥st❛♥ts✳ ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s ❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s ❲❤❛t ✐s t❤❡ t②♣✐❝❛❧ ❝♦♠♣♦♥❡♥t str✉❝t✉r❡ ♦❢ P ♥ ❄ ●✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ ♣r♦✈❡❞ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ P ♥ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ✶ + ❳ , ✇❤❡r❡ ❳ L = P♦ ( ν ) ✱ ❛♥❞ ν ≈ ✵ . ✵✸✼✹✸✾✳ ❚②♣✐❝❛❧ str✉❝t✉r❡ ♦❢ ❛ r❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❣✐❛♥t ❝♦♠♣♦♥❡♥t ❝♦♥t❛✐♥✐♥❣ ♥ − ❖ ❈ ( ✶ ) ✈❡rt✐❝❡s✱ ✇❤❡r❡❛s t❤❡ r❡♠❛✐♥✐♥❣ ❝♦♠♣♦♥❡♥ts ❛r❡ ✏s♠❛❧❧✧✳ ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✹

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s ❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s ❲❤❛t ✐s t❤❡ t②♣✐❝❛❧ ❝♦♠♣♦♥❡♥t str✉❝t✉r❡ ♦❢ P ♥ ❄ ●✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ ♣r♦✈❡❞ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ P ♥ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ✶ + ❳ , ✇❤❡r❡ ❳ L = P♦ ( ν ) ✱ ❛♥❞ ν ≈ ✵ . ✵✸✼✹✸✾✳ ❚②♣✐❝❛❧ str✉❝t✉r❡ ♦❢ ❛ r❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❣✐❛♥t ❝♦♠♣♦♥❡♥t ❝♦♥t❛✐♥✐♥❣ ♥ − ❖ ❈ ( ✶ ) ✈❡rt✐❝❡s✱ ✇❤❡r❡❛s t❤❡ r❡♠❛✐♥✐♥❣ ❝♦♠♣♦♥❡♥ts ❛r❡ ✏s♠❛❧❧✧✳ ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✹