getdoc9259. 269KB Jun 04 2011 12:04:41 AM

❊❧❡❝t✳ ❈♦♠♠✳ ✐♥ Pr♦❜❛❜✳ ✾ ✭✷✵✵✹✮✱ ✶✶✾✕✶✸✶

❊▲❊❈❚❘❖◆■❈
❈❖▼▼❯◆■❈❆❚■❖◆❙
✐♥ P❘❖❇❆❇■▲■❚❨

❊❘●❖❉■❈■❚❨ ❖❋ P❈❆ ✿ ❊◗❯■❱❆▲❊◆❈❊ ❇❊❚❲❊❊◆
❙P❆❚■❆▲ ❆◆❉ ❚❊▼P❖❘❆▲ ▼■❳■◆● ❈❖◆❉■❚■❖◆❙
P■❊❘❘❊✲❨❱❊❙ ▲❖❯■❙✶

■♥st✐t✉t ❢ür ▼❛t❤❡♠❛t✐❦✷ ✱ P♦ts❞❛♠ ❯♥✐✈❡rs✐tät✱ ❆♠ ♥❡✉❡♥ P❛❧❛✐s✲❙❛♥s ❙♦✉❝✐✱
P♦st❢❛❝❤ ✻✵ ✶✺ ✺✸✱ ❉✲✶✹ ✹✶✺ P♦ts❞❛♠
❡♠❛✐❧✿ ❧♦✉✐s❅♠❛t❤✳✉♥✐✲♣♦ts❞❛♠✳❞❡

❙✉❜♠✐tt❡❞ ✶✶ ❋❡❜r✉❛r② ✷✵✵✹✱ ❛❝❝❡♣t❡❞ ✐♥ ✜♥❛❧ ❢♦r♠ ✷✽ ❙❡♣t❡♠❜❡r ✷✵✵✹
❆▼❙ ✷✵✵✵ ❙✉❜❥❡❝t ❝❧❛ss✐✜❝❛t✐♦♥✿ ✻✵●✻✵ ❀ ✻✵❏✶✵ ❀ ✻✵❑✸✺ ❀ ✽✷❈✷✵ ❀ ✽✷❈✷✻ ❀ ✸✼❇✶✺
❑❡②✇♦r❞s✿ Pr♦❜❛❜✐❧✐st✐❝ ❈❡❧❧✉❧❛r ❆✉t♦♠❛t❛✱ ■♥t❡r❛❝t✐♥❣ P❛rt✐❝❧❡ ❙②st❡♠s✱ ❲❡❛❦ ▼✐①✐♥❣ ❈♦♥✲
❞✐t✐♦♥✱ ❊r❣♦❞✐❝✐t②✱ ❊①♣♦♥❡♥t✐❛❧ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✱ ●✐❜❜s ♠❡❛s✉r❡

❆❜str❛❝t
d

❋♦r ❛ ❣❡♥❡r❛❧ ❛ttr❛❝t✐✈❡ Pr♦❜❛❜✐❧✐st✐❝ ❈❡❧❧✉❧❛r ❆✉t♦♠❛t❛ ♦♥ S Z ✱ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ✭t✐♠❡✲✮
❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞s ❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤✐s ▼❛r❦♦✈✐❛♥ ♣❛r❛❧❧❡❧ ❞②♥❛♠✐❝s✱ ❡①♣♦♥❡♥t✐❛❧❧② ❢❛st ✐♥ t❤❡
✉♥✐❢♦r♠ ♥♦r♠✱ ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ ❝♦♥❞✐t✐♦♥ (A)✳ ❚❤✐s ❝♦♥❞✐t✐♦♥ ♠❡❛♥s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛②
♦❢ t❤❡ ✐♥✢✉❡♥❝❡ ❢r♦♠ t❤❡ ❜♦✉♥❞❛r② ❢♦r t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ♦❢ t❤❡ s②st❡♠ r❡str✐❝t❡❞ t♦ ✜♥✐t❡
d
❜♦①❡s✳ ❋♦r ❛ ❝❧❛ss ♦❢ r❡✈❡rs✐❜❧❡ P❈❆ ❞②♥❛♠✐❝s ♦♥ {−1, +1}Z ✱ ✇✐t❤ ❛ ♥❛t✉r❛❧❧② ❛ss♦❝✐❛t❡❞
●✐❜❜s✐❛♥ ♣♦t❡♥t✐❛❧ ϕ✱ ✇❡ ♣r♦✈❡ t❤❛t ❛ ✭s♣❛t✐❛❧✲✮ ✇❡❛❦ ♠✐①✐♥❣ ❝♦♥❞✐t✐♦♥ (WM) ❢♦r ϕ ✐♠♣❧✐❡s
t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❛ss✉♠♣t✐♦♥ (A)❀ t❤✉s ❡①♣♦♥❡♥t✐❛❧ ✭t✐♠❡✲✮ ❡r❣♦❞✐❝✐t② ♦❢ t❤❡s❡ ❞②♥❛♠✐❝s
t♦✇❛r❞s t❤❡ ✉♥✐q✉❡ ●✐❜❜s ♠❡❛s✉r❡ ❛ss♦❝✐❛t❡❞ t♦ ϕ ❤♦❧❞s✳ ❖♥ s♦♠❡ ♣❛rt✐❝✉❧❛r ❡①❛♠♣❧❡s ✇❡
st❛t❡ t❤❛t ❡①♣♦♥❡♥t✐❛❧ ❡r❣♦❞✐❝✐t② ❤♦❧❞s ❛s s♦♦♥ ❛s t❤❡r❡ ✐s ♥♦ ♣❤❛s❡ tr❛♥s✐t✐♦♥✳

✶ ■♥tr♦❞✉❝t✐♦♥
❚❤❡ ♠❛✐♥ ❢❡❛t✉r❡ ♦❢ Pr♦❜❛❜✐❧✐st✐❝ ❈❡❧❧✉❧❛r ❆✉t♦♠❛t❛ ❞②♥❛♠✐❝s ✭✉s✉❛❧❧② ❛❜❜r❡✈✐❛t❡❞ ✐♥ P❈❆✮
✐s t❤❡ ♣❛r❛❧❧❡❧✱ ♦r s②♥❝❤r♦♥♦✉s✱ ❡✈♦❧✉t✐♦♥ ♦❢ ❛❧❧ ✐♥t❡r❛❝t✐♥❣ ❡❧❡♠❡♥t❛r② ❝♦♠♣♦♥❡♥ts✳ ❚❤❡②
❛r❡ ♣r❡❝✐s❡❧② ❞✐s❝r❡t❡✲t✐♠❡ ▼❛r❦♦✈ ❝❤❛✐♥s ♦♥ ❛ ♣r♦❞✉❝t s♣❛❝❡ S Λ ✭❝♦♥✜❣✉r❛t✐♦♥ s♣❛❝❡✮ ✇❤♦s❡
tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② ✐s ❛ ♣r♦❞✉❝t ♠❡❛s✉r❡✳ ■♥ t❤✐s ♣❛♣❡r✱ S ✭s♣✐♥ s♣❛❝❡✮ ✐s ❛ss✉♠❡❞ t♦ ❜❡
❛ ✜♥✐t❡ s❡t ✇✐t❤ t♦t❛❧ ♦r❞❡r ❞❡♥♦t❡❞ ❜② 6 ❛♥❞ Λ ✭s❡t ♦❢ s✐t❡s✮ ❛ s✉❜s❡t✱ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡✱
♦❢ Zd ✳ ❚❤❡ ❢❛❝t t❤❛t t❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② ❦❡r♥❡❧ P (dσ|σ ′ ) ✭σ, σ ′ ∈ S Λ ✮ ✐s ❛ ♣r♦❞✉❝t
♠❡❛s✉r❡ ♠❡❛♥s t❤❛t ❛❧❧ s♣✐♥s {σk : k ∈ Λ} ❛r❡ s✐♠✉❧t❛♥❡♦✉s❧② ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ✉♣❞❛t❡❞✳

❚❤✐s tr❛♥s✐t✐♦♥ ♠❡❝❤❛♥✐s♠ ❞✐✛❡rs ❢r♦♠ t❤❡ ♦♥❡ ✐♥ t❤❡ ♠♦st ❝♦♠♠♦♥ ●✐❜❜s s❛♠♣❧❡rs✱ ✇❤❡r❡
♦♥❧② ♦♥❡ s✐t❡ ✐s ✉♣❞❛t❡❞ ❛t ❡❛❝❤ t✐♠❡ st❡♣✳ ■♥ ♦♣♣♦s✐t✐♦♥ t♦ t❤❡s❡ ❞②♥❛♠✐❝s ✇✐t❤ s❡q✉❡♥t✐❛❧
✶ P✳✲❨✳

▲❖❯■❙ ❆❈❑◆❖❲▲❊❉●❊❙ ❋■◆❆◆❈■❆▲ ❙❯PP❖❘❚ ❇❨ ❉❊❯❚❙❈❍❊ ❋❖❘❙❈❍❯◆●❙●❊▼❊■◆✲

❙❈❍❆❋❚ ❱■❆ ●❘❆❉❯■❊❘❚❊◆❑❖▲▲❊● ✷✺✶ ❵❙❚❖❈❍❆❙❚■❙❈❍❊ P❘❖❩❊❙❙❊ ❯◆❉ P❘❖❇❆❇■▲■❙❚■❙❈❍❊
❆◆❆▲❨❙■❙✬

✷ ♦♥

❧❡❛✈❡ ❢r♦♠ ❉❋● ●r❛❞✉✐❡rt❡♥❦♦❧❧❡❣ ✷✺✶✿ ✬❙t♦❝❤❛st✐s❝❤❡ Pr♦③❡ss❡ ✉♥❞ ♣r♦❜❛❜✐❧✐st✐s❝❤❡ ❆♥❛❧②s✐s✬✱ ❚❡❝❤✲

♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ❇❡r❧✐♥

✶✶✾

✶✷✵

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②

✉♣❞❛t✐♥❣✱ ✐t ✐s s✐♠♣❧❡ t♦ ❞❡✜♥❡ P❈❆✬s ♦♥ t❤❡ ✐♥✜♥✐t❡ s❡t S Z ✇✐t❤♦✉t ♣❛ss✐♥❣ t♦ ❝♦♥t✐♥✉♦✉s
t✐♠❡✳
d

❚❤❡ ♠❛✐♥ ♣✉r♣♦s❡ ♦❢ t❤✐s ❛rt✐❝❧❡ ✐s t♦ st✉❞② r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ❝♦♥❞✐t✐♦♥s
✇❤✐❝❤ ✐♥s✉r❡ t❤❡ ❢❛st❡st ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞s ❛♥ ❡q✉✐❧✐❜r✐✉♠ st❛t❡ ✭νP = ν ✮ ♦❢ P❈❆ ❞②♥❛♠✐❝s
d
♦♥ S Z ✳ ▲❡t ✉s ❡♠♣❤❛s✐s❡ t❤❛t t❤❡ ♥♦♥✲❞❡❣❡♥❡r❛❝② ❤②♣♦t❤❡s✐s ✇❡ ✇✐❧❧ ❛ss✉♠❡ ✐♠♣❧✐❡s t❤❛t
t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ P❈❆ ❞②♥❛♠✐❝s ♦♥ S Λ ✇❤❡r❡ Λ ✐s ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ Zd ✭❝❛❧❧❡❞
✜♥✐t❡ ✈♦❧✉♠❡ P❈❆ ❞②♥❛♠✐❝s✮ ✐s ✇❡❧❧✲❦♥♦✇♥✳ ■t ✐s ❛ ❝❧❛ss✐❝❛❧ r❡s✉❧t ❢r♦♠ t❤❡ t❤❡♦r② ♦❢ ✜♥✐t❡
st❛t❡ s♣❛❝❡ ❛♣❡r✐♦❞✐❝ ✐rr❡❞✉❝✐❜❧❡ ▼❛r❦♦✈ ❈❤❛✐♥s✳ ❙✉❝❤ ❞✐s❝r❡t❡ t✐♠❡ ♣r♦❝❡ss❡s ❛❞♠✐t ❛ ✉♥✐q✉❡
st❛t✐♦♥❛r② ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✱ ❛♥❞ ❛r❡ ❡r❣♦❞✐❝✳ ❍♦✇❡✈❡r✱ ✐❢ t❤❡ P❈❆ ❞②♥❛♠✐❝s ✐s ❝♦♥s✐❞❡r❡❞
d
♦♥ S Z ✭✐♥✜♥✐t❡ ✈♦❧✉♠❡ ❞②♥❛♠✐❝s✮✱ s♦♠❡ ♥♦♥✲❡r❣♦❞✐❝ ❜❡❤❛✈✐♦✉r ♠❛② ❛r✐s❡ ✭s❡❡ ❢♦r ✐♥st❛♥❝❡
❡①❛♠♣❧❡ ✷ s❡❝t✐♦♥ III ✐♥ ❬✽❪✮✳ ❚❤❡ ♠♦st ❢❛♠♦✉s ❝♦♥❞✐t✐♦♥ ✇❤✐❝❤ ✐♥s✉r❡s ❡r❣♦❞✐❝✐t② ♦❢ t❤❡ P❈❆
d
❞②♥❛♠✐❝s ♦♥ S Z ✐s ❞✉❡ t♦ ❉♦❜r✉s❤✐♥ ❛♥❞ ❱❛s❡rs❤t❡✐♥✬s ✇♦r❦ ✭s❡❡ ❬✶✺❪✮✱ ❛♥❞ ❛♣♣❧✐❡s ✐♥ t❤❡
❤✐❣❤✲t❡♠♣❡r❛t✉r❡ r❡❣✐♠❡✳ ❖t❤❡rs ❝♦♥❞✐t✐♦♥s ♦❢ ❡r❣♦❞✐❝✐t② ❢♦r ❣❡♥❡r❛❧ P❈❆ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡
❢♦❧❧♦✇✐♥❣ ✇♦r❦s✿ ❬✹✱ ✼✱ ✾✱ ✶✷✱ ✶✸❪✳ ❙❡❡ ❢♦r ✐♥st❛♥❝❡ ❙❡❝t✐♦♥s ✻✳✶✳✷ ❛♥❞ ✻✳✶✳✸ ✐♥ ❬✶✵❪ ❢♦r ❞❡t❛✐❧s✳
❚❤❡② ❛❧❧ ❛r❡ ❡✛❡❝t✐✈❡ ♦♥❧② ✇❤❡♥ s♦♠❡ ❤✐❣❤✲t❡♠♣❡r❛t✉r❡ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s ♦r ✐♥ ♣❡rt✉r❜❛t✐✈❡

❝❛s❡s✳
❲❡ ✇✐❧❧ ❤❡r❡ ❛❞♦♣t ❛♥♦t❤❡r ❛♣♣r♦❛❝❤✱ ♣❛rt✐❛❧❧② ✐♥s♣✐r❡❞ ❜② ▼❛rt✐♥❡❧❧✐ ❛♥❞ ❖❧✐✈✐❡r✐✬s ✇♦r❦ ❢♦r
❛ ❝❧❛ss ♦❢ ❝♦♥t✐♥✉♦✉s t✐♠❡ ■♥t❡r❛❝t✐♥❣ P❛rt✐❝❧❡ ❙②st❡♠s ❝❛❧❧❡❞ ●❧❛✉❜❡r ❞②♥❛♠✐❝s ✭s❡❡ ❬✶✹❪✮✱
❛♥❞ ❜❛s❡❞ ♦♥ ❛ ❢❛♠♦✉s st❛t❡♠❡♥t ♦❢ ❍♦❧❧❡② ❛❜♦✉t r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✭❬✻❪✮✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛
❝♦♥❞✐t✐♦♥ (A) ✇❤✐❝❤ ♠❡❛♥s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♦❢ t❤❡ ✐♥✢✉❡♥❝❡ ❢r♦♠ t❤❡ ❜♦✉♥❞❛r② ❢♦r t❤❡
✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ t❤❡ s②st❡♠ r❡str✐❝t❡❞ t♦ ❛♥② ✜♥✐t❡ ❜♦①✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❤❡r❡ ♣r♦✈❡❞ t♦ ❜❡
❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❡①♣♦♥❡♥t✐❛❧❧② ❢❛st ❡r❣♦❞✐❝✐t② ✭❚❤❡♦r❡♠ ✶✮✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ (A) ✇❡ ✉s❡ ✐s ♥♦t
❛ ❝♦♥str✉❝t✐✈❡ ❝r✐t❡r✐♦♥ ❧✐❦❡ t❤❡ ❉♦❜r✉s❤✐♥✲❱❛s❡rs❤t❡✐♥ ❝♦♥❞✐t✐♦♥✱ ♦r ✐ts ❣❡♥❡r❛❧✐s❡❞ ✈❡rs✐♦♥
❞❡✈❡❧♦♣❡❞ ✐♥ ❬✶✷❪ ❛♥❞ ♥✉♠❡r✐❝❛❧❧② st✉❞✐❡❞ ✐♥ ❬✷❪✳ ❇✉t✱ t❤❡♦r❡t✐❝❛❧❧②✱ ❝♦♠♣❛r✐s♦♥ ♦❢ s♣❛t✐❛❧ ❛♥❞
t✐♠❡ ♠✐①✐♥❣ ❛r❡ ❛❧✇❛②s ✐♥t❡r❡st✐♥❣ ✭❝❢✳ ❬✶✹✱ ✸❪✮✳ ❋✉rt❤❡r♠♦r❡ ✇❡ ♣r❡s❡♥t ❞✐✛❡r❡♥t ❡①❛♠♣❧❡s
✐♥ ✇❤✐❝❤ (A) ✐s s❛t✐s✜❡❞ ♦♥ ❛ ❧❛r❣❡r ❞♦♠❛✐♥ t❤❛♥ ❉♦❜r✉s❤✐♥✲❱❛s❡rs❤t❡✐♥ ❝♦♥❞✐t✐♦♥✱ ❛♥❞ ✐s
♠♦r❡♦✈❡r ♦♣t✐♠❛❧ ❢♦r t❤❡s❡ ♠♦❞❡❧s✳
■♥ s❡❝t✐♦♥ ✷ ✇❡ st❛t❡ ♦✉r ♠❛✐♥ r❡s✉❧ts✳ ❚❤❡ ✜rst ❛♥❞ ♠♦r❡ ❣❡♥❡r❛❧ ♦♥❡ ✭❚❤❡♦r❡♠ ✶✮ ✐s t❤❡
❢♦❧❧♦✇✐♥❣✿ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞s ❡q✉✐❧✐❜r✐✉♠ ✐♥ t❤❡ ✉♥✐❢♦r♠ ♥♦r♠ ✇✐t❤ ❛♥ ❡①♣♦♥❡♥t✐❛❧ r❛t❡ ✐s
❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝♦♥❞✐t✐♦♥ (A)✳ ■♥ ♦t❤❡r ✇♦r❞s ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ✐♥ s♣❛❝❡ ✐s ❡q✉✐✈❛❧❡♥t
t♦ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ✐♥ t✐♠❡✳ ■t ✇✐❧❧ t❤❡♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❛ ❝❧❛ss ♦❢ r❡✈❡rs✐❜❧❡ P❈❆ ❞②♥❛♠✐❝s
d
♦♥ {−1, +1}Z ✱ ❛ss♦❝✐❛t❡❞ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛② t♦ ❛ ●✐❜❜s✐❛♥ ♣♦t❡♥t✐❛❧ ϕ✳ ❲❡ ♣r♦✈❡ t❤❛t t❤❡ ✉s✉❛❧
✇❡❛❦ ♠✐①✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ϕ ✐♠♣❧✐❡s t❤❡ ✈❛❧✐❞✐t② ♦❢ (A)✱ t❤✉s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❡r❣♦❞✐❝✐t② ♦❢ t❤❡
❞②♥❛♠✐❝s t♦✇❛r❞s t❤❡ ✉♥✐q✉❡ ●✐❜❜s ♠❡❛s✉r❡ ❛ss♦❝✐❛t❡❞ t♦ ϕ ❤♦❧❞s ✭❚❤❡♦r❡♠ ✷✮✳ ❋♦r s♦♠❡

♣❛rt✐❝✉❧❛r P❈❆ ♦❢ t❤✐s ❝❧❛ss✱ ✇❡ ❛❧s♦ ♣r♦✈❡ t❤❛t (A) ✐s ✇❡❛❦❡r t❤❛♥ t❤❡ ❉♦❜r✉s❤✐♥✲❱❛s❡rs❤t❡✐♥
❡r❣♦❞✐❝✐t② ❝♦♥❞✐t✐♦♥ ❛♥❞ ♥♦t❡ t❤❛t t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❡r❣♦❞✐❝✐t② ❤♦❧❞s ❛s s♦♦♥ ❛s t❤❡r❡ ✐s ♥♦ ♣❤❛s❡
tr❛♥s✐t✐♦♥✳ ❖✉r r❡s✉❧t ❛r❡ t❤❡♥ t❤❡ ✜rst ♦♣t✐♠❛❧ ♦♥❡s ✐♥ t❤✐s ❝♦♥t❡①t✳ ❙❡❝t✐♦♥s ✸ ❛♥❞ ✹ ❛r❡
r❡s♣❡❝t✐✈❡❧② ❞❡✈♦t❡❞ t♦ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❚❤❡♦r❡♠s ❛♥❞ ✉s❡❢✉❧ ▲❡♠♠❛s✳

✷ ▼❛✐♥ r❡s✉❧ts
▲❡t P ❞❡♥♦t❡s ❛ P❈❆ ❞②♥❛♠✐❝s ♦♥ S Z ✳ ❚❤✐s ♠❡❛♥s ❛ ▼❛r❦♦✈ ❈❤❛✐♥ ♦♥ S Z ✇❤♦s❡ tr❛♥s✐t✐♦♥
d
d
♣r♦❜❛❜✐❧✐t② ❦❡r♥❡❧ P ✈❡r✐✜❡s ❢♦r ❛❧❧ ❝♦♥✜❣✉r❛t✐♦♥ η ∈ S Z ✱ σ = (σk )k∈Zd ∈ S Z ✱ P ( dσ | η ) =
d

⊗

k∈Zd

d

pk ( dσk | η )✱ ✇❤❡r❡ ❢♦r ❛❧❧ s✐t❡ k ∈ Zd ✱ ❢♦r ❛❧❧ η ✱ pk ( . |η) ✐s ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ S ✱

❊r❣♦❞✐❝✐t② ♦❢ P❈❆

✶✷✶

❝❛❧❧❡❞ ✉♣❞❛t✐♥❣ r✉❧❡✳ ❋♦r ❛♥② s✉❜s❡t ∆ ♦❢ Zd ✱ ❛♥❞ ❢♦r ❛❧❧ ❝♦♥✜❣✉r❛t✐♦♥s σ ❛♥❞ η ♦❢ S Z ✱ t❤❡
❝♦♥✜❣✉r❛t✐♦♥ σ∆ η∆c ✐s ❞❡✜♥❡❞ ❜② σk ✐❢ k ∈ ∆✱ ❡❧s❡ ηk ✳ ▲❡t t❤❡ ♥♦t❛t✐♦♥ σ∆ ❞❡s✐❣♥ (σk )k∈∆
t♦♦✳ ▲❡t Λ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ Zd ✭❞❡♥♦t❡❞ ❜② Λ ⋐ Zd ✮✳ ❲❡ ❝❛❧❧ ✜♥✐t❡ ✈♦❧✉♠❡ P❈❆ ❞②♥❛♠✐❝s
d
c
✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ τ ✭τ ∈ S Z ♦r τ ∈ S Λ ✮✱ t❤❡ ▼❛r❦♦✈ ❈❤❛✐♥ ♦♥ S Λ ✇❤♦s❡ tr❛♥s✐t✐♦♥
d

♣r♦❜❛❜✐❧✐t② PΛτ ✐s ❞❡✜♥❡❞ ❜②✿ PΛτ (dσΛ | ηΛ ) =

⊗ pk ( dσk | ηΛ τΛc ). ■t ♠❛② ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤

k∈Λ

t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥✜♥✐t❡ ✈♦❧✉♠❡ P❈❆ ❞②♥❛♠✐❝s ♦♥ S Z ✿ PΛτ (dσ | ηΛ ) =
d

⊗ pk ( dσk | ηΛ τΛc ) ⊗

k∈Λ

❞❡♥♦t❡ t❤❡ st❛t✐♦♥❛r② ♠❡❛s✉r❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ✜♥✐t❡ ✈♦❧✉♠❡ ❞②♥❛♠✐❝s
δτΛc (dσΛc )✳ ▲❡t
d
✇✐t❤ t❤❡ ❇♦r❡❧ σ ✲✜❡❧❞ ❛ss♦❝✐❛t❡❞ t♦ t❤❡
PΛτ ✳ ❋♦r ν ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ S Z ✭❡q✉✐♣♣❡❞
R
♣r♦❞✉❝t t♦♣♦❧♦❣②✮✱ νP r❡❢❡rs t♦ νP (dσ) = P (dσ|η)ν(dη)✳ ❘❡❝✉rs✐✈❡❧② νP (n) = (νP (n−1) )P ✳
R
d
❋♦r ❡❛❝❤ ❢✉♥❝t✐♦♥ f ♦♥ S Z ✱ P (f ) ✐s t❤❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② P (f )(η) = f (σ)P (dσ|η)✳ ❆❧❧ t❤❡
νΛτ

♠❡❛s✉r❡s ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ♣❛♣❡r ❛r❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s✳

P❈❆ ❞②♥❛♠✐❝s ❝♦♥s✐❞❡r❡❞ ❤❡r❡ ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ♥♦♥ ❞❡❣❡♥❡r❛t❡✿ ∀k ∈ Zd , ∀η ∈ S Z ✱ ∀s ∈ S ✱

pk ( s | η ) > 0❀ t❤❡② ❛r❡ ❛❧s♦ ❧♦❝❛❧✱ ✇❤✐❝❤ ♠❡❛♥s✿ ∀k ∈ Zd , ∃ Vk ⋐ Zd , pk ( . |η) = pk ( . |ηVk ) ❛♥❞
d
t❤❡② ❛r❡ ❛❧s♦ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ∀k ∈ Zd , ∀s ∈ S, ∀η ∈ S Z , pk ( s | η ) = p0 ( s | θ−k η )✱
d
✇❤❡r❡ θk0 (σ) ❞❡✜♥❡s t❤❡ tr❛♥s❧❛t✐♦♥ ♦❢ ❛ ❝♦♥✜❣✉r❛t✐♦♥ σ ♦❢ S Z ✇✐t❤ θk0 (σ) = (σk−k0 )k∈Zd ✳
d

❆ttr❛❝t✐✈✐t② ♦❢ P❈❆ ❞②♥❛♠✐❝s ✐s ♠♦r❡♦✈❡r ❛ss✉♠❡❞ ❤❡r❡✿ ❖♥❡ ❝❛♥ ♦r❞❡r t✇♦ ❝♦♥✜❣✉r❛t✐♦♥s ❜②
❞❡✜♥✐♥❣ σ 4 η ✐❢ ∀k ∈ Λ, σk 6 ηk ✳ ❆ r❡❛❧ ❢✉♥❝t✐♦♥ f ♦♥ S Λ ✇✐❧❧ t❤❡♥ ❜❡ s❛✐❞ t♦ ❜❡ ✐♥❝r❡❛s✐♥❣
✐❢ σ 4 η ✐♠♣❧✐❡s f (σ) 6 f (η)✳ ❚❤✉s t✇♦ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ν1 ❛♥❞ ν2 s❛t✐s❢② t❤❡ st♦❝❤❛st✐❝
♦r❞❡r✐♥❣ Rν1 4 ν2 ✐❢✱ ❢♦r ❛❧❧ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s f ♦♥ S Λ ✱ ν1 (f ) 6 ν2 (f )✱ ✇✐t❤ t❤❡ ♥♦t❛t✐♦♥
νi (f ) = f (σ)νi (dσ)✳ ❆s ▼❛r❦♦✈ ❝❤❛✐♥✱ ❛ P❈❆ ❞②♥❛♠✐❝s P ♦♥ S Λ ✭Λ ⊂ Zd ✮ ✐s ❛ttr❛❝t✐✈❡ ✐❢
❢♦r ❛❧❧ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ f ✱ P (f ) ✐s st✐❧❧ ✐♥❝r❡❛s✐♥❣✳ ▲❡t ✉s ❞❡✜♥❡ t♦♦✱ ❢♦r s ∈ S, σ ∈ S Λ ✱ t❤❡
❢✉♥❝t✐♦♥ Gk (s, σ) ❜②✿
X
Gk (s, . ) =
pk (s′ | . ).
✭✶✮
s′ >s

❘❡❝❛❧❧ t❤❛t ❛ P❈❆ ❞②♥❛♠✐❝s ✐s ❛ttr❛❝t✐✈❡ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ❢♦r ❛❧❧ k ✐♥ Λ✱ ❛♥❞ ❛❧❧ ✈❛❧✉❡ s ∈ S ✱
t❤❡ ❢✉♥❝t✐♦♥ Gk (s, .) ✐s ✐♥❝r❡❛s✐♥❣ ✭✐♥ σ ✮✳
❆ r❡❛❧ ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ f ♦♥ S Z ✐s s❛✐❞ ❧♦❝❛❧ ✐❢ ∃Λf ⋐ Zd , ∀σ ∈ S Z , f (σ) = f (σΛf )✳ ❲❡ ❞❡✜♥❡✱
d
❢♦r ❡❛❝❤ f ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❝♦♠♣❛❝t S Z ❛♥❞ ❢♦r ❛❧❧ k ✐♥ Zd ✱
d

d


n
o
d


∆f (k) = sup f (σ) − f (η) : (σ, η) ∈ (S Z )2 , σ{k}c ≡ η{k}c ,

❛♥❞ t❤❡ s❡♠✐✲♥♦r♠ |k f |k= k∈Zd ∆f (k)✳ ❋♦r L ✐♥t❡❣❡r✱ B(L) ✐s t❤❡ ❜❛❧❧ B(0, L) ✇✐t❤ r❡s♣❡❝t
P
t♦ t❤❡ ♥♦r♠ kkk1 = di=1 |ki |✱ k = (k1 , k2 , . . . , kd ) ∈ Zd ✳

P

❚❤❡♦r❡♠ ✶ ▲❡t S ❜❡ ❛ t♦t❛❧❧② ♦r❞❡r❡❞ ✜♥✐t❡ s❡t ✇✐t❤ ♠❛①✐♠❛❧ ✭r❡s♣✳ ♠✐♥✐♠❛❧✮ ❡❧❡♠❡♥t ❞❡♥♦t❡❞
❜② +✭r❡s♣✳ −✮✳ + ✭r❡s♣✳ − ✮ ❞❡♥♦t❡s ❝♦♥✜❣✉r❛t✐♦♥s ❡q✉❛❧ t♦ + ✭r❡s♣✳ −✮ ✐♥ ❛❧❧ s✐t❡s✳ ▲❡t P ❜❡
d
+
❛♥ ❛ttr❛❝t✐✈❡✱ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✱ ♥♦♥ ❞❡❣❡♥❡r❛t❡✱ ❧♦❝❛❧ P❈❆ ❞②♥❛♠✐❝s ♦♥ S Z ✳ ▲❡t νB(L)
−
+
−
✭r❡s♣✳ νB(L)
✮ ❜❡ t❤❡ st❛t✐♦♥❛r② ♠❡❛s✉r❡ ♦❢ PB(L)
✭r❡s♣✳ PB(L)
✮✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ s♣❛t✐❛❧ ♠✐①✐♥❣
❝♦♥❞✐t✐♦♥✿ ∃C > 0, ∃M > 0, ∃L1 ∈ N∗ , ∀L ∈ N∗ , L > L1 ,
Z

+
σ0 dνB(L)
−

Z

−
σ0 dνB(L)
6 Ce−M L

(A)

✶✷✷

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②
P t♦✇❛r❞s t❤❡ ✉♥✐q✉❡
d
∃λ > 0✱ ∃n1 ✱ ∀n > n1 ✱ ∀f ❧♦❝❛❧ ❢✉♥❝t✐♦♥ ♦♥ S Z :




sup δσ P (n) (f ) − ν(f ) 6 2|k f |k e−λn .

✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❞②♥❛♠✐❝s
❡①♣♦♥❡♥t✐❛❧ r❛t❡✿

❡q✉✐❧✐❜r✐✉♠ st❛t❡

σ

ν

✇✐t❤

✭✷✮

■♥ ♦r❞❡r t♦ ❜❡tt❡r ✐♥t❡r♣r❡t t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❝♦♥❞✐t✐♦♥ (A) ❛♥❞ t❤❡ r❡❧❡✈❛♥❝❡ ♦❢ ❚❤❡♦r❡♠ ✶✱
d
✇❡ t❤❡♥ ❛♣♣❧② ✐t t♦ ❛ ✇✐❞❡ ❝❧❛ss ♦❢ r❡✈❡rs✐❜❧❡ P❈❆ ❞②♥❛♠✐❝s ♦♥ {−1, +1}Z ✳ ❋✐rst✱ ❧❡t ✉s
r❡❝❛❧❧ s♦♠❡ ❦♥♦✇♥ ❢❛❝ts ❛❜♦✉t r❡✈❡rs✐❜❧❡ P❈❆ ❞②♥❛♠✐❝s ✭t❤❛t ✐s t♦ s❛② P❈❆ ❞②♥❛♠✐❝s ✇❤♦s❡
s❡t ♦❢ r❡✈❡rs✐❜❧❡ ♠❡❛s✉r❡s R ✐s ♥♦t ❡♠♣t②✮✳ ❚❤❡ st✉❞② ♦❢ t❤❡ q✉❛❧✐t❛t✐✈❡ ♥❛t✉r❡ ♦❢ t❤❡✐r
❡q✉✐❧✐❜r✐✉♠ st❛t❡s ❛s ●✐❜❜s ♠❡❛s✉r❡s ✇❛s ✐♥✐t✐❛t❡❞ ❜② ❑♦③❧♦✈ ❛♥❞ ❱❛s✐❧②❡✈ ✭s❡❡ ❬✽✱ ✶✻❪✮✳ ●✐❜❜s
♠❡❛s✉r❡s ✇✐t❤ r❡s♣❡❝t t♦ s♦♠❡ ❞②♥❛♠✐❝s✬ ♥❛t✉r❛❧❧② ❛ss♦❝✐❛t❡❞ ♣♦t❡♥t✐❛❧✱ ❛r❡ ✐♥❞❡❡❞ ♥❛t✉r❛❧
❝❛♥❞✐❞❛t❡s ❛s st❛t✐♦♥❛r② st❛t❡s✳ ■♥ ❬✶✱ ✶✵❪✱ ♣r❡❝✐s❡ r❡❧❛t✐♦♥s ✇❡r❡ ❡st❛❜❧✐s❤❡❞ ❜❡t✇❡❡♥ t❤❡ s❡ts ♦❢
st❛t✐♦♥❛r② ♠❡❛s✉r❡s✱ r❡✈❡rs✐❜❧❡ ♠❡❛s✉r❡s ❛♥❞ s♦♠❡ ●✐❜❜s ♠❡❛s✉r❡s ✭s❡❡ Pr♦♣♦s✐t✐♦♥ ✸✳✸ ✐♥ ❬✶❪✮✳
▼♦r❡♦✈❡r✱ ✉♥❧✐❦❡ ✇❤❛t ✐s ❞♦♥❡ ✭♦r ❡①♣❡❝t❡❞ t♦ ❤♦❧❞✮ ❢♦r ❝♦♥t✐♥✉♦✉s t✐♠❡ ■♥t❡r❛❝t✐♥❣ P❛rt✐❝❧❡
❙②st❡♠s ❧✐❦❡ ●❧❛✉❜❡r ❞②♥❛♠✐❝s ♦r ❣r❛❞✐❡♥t ❞✐✛✉s✐♦♥s✱ ✐t ✐s s❤♦✇♥ t❤❛t ●✐❜❜s ♠❡❛s✉r❡s ♠❛②
❜❡ ♥♦♥ st❛t✐♦♥❛r② ❢♦r P❈❆✬s ❞②♥❛♠✐❝s✱ ✇❤✐❝❤ ✐s ❛ ❝❤❛r❛❝t❡r✐st✐❝ ♠❛♥✐❢❡st❛t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡t❡
t✐♠❡ ❝❛s❡✳
❆ss✉♠❡ ✉♥t✐❧ t❤❡ ❡♥❞ ♦❢ t❤✐s s❡❝t✐♦♥ ❛♥❞ ✐♥ s❡❝t✐♦♥ ✹ t❤❛t S = {−1, +1}✳ ❲❡ ❝❛❧❧ ❝❧❛ss C0 t❤❡
d
❢❛♠✐❧② ♦❢ P❈❆ ❞②♥❛♠✐❝s ♦♥ {−1, +1}Z ✇❤♦s❡ ✉♣❞❛t✐♥❣ r✉❧❡ (pk )k∈Zd ✐s ❣✐✈❡♥ ❜②✿ ∀k ∈ Zd ✱
d
∀η ∈ S Z ✱ ∀s ∈ S

X
1
1 + s tanh(β
✭✸✮
K(k ′ − k)ηk′ ) ,
pk (s | η) =
2

k′ ∈Zd

✇❤❡r❡ β ✐s ❛ ♣♦s✐t✐✈❡ r❡❛❧ ♣❛r❛♠❡t❡r ❛♥❞ K : Zd → R ✐s ❛♥ ✐♥t❡r❛❝t✐♦♥ ❢✉♥❝t✐♦♥ ❜❡t✇❡❡♥ s✐t❡s
✇❤✐❝❤ ✐s s②♠♠❡tr✐❝ ❛♥❞ ❤❛s ✜♥✐t❡ r❛♥❣❡ R > 0 ✭✐✳❡✳ ❢♦r ❛❧❧ k ♦❢ Zd s✉❝❤ t❤❛t kkk1 > R t❤❡♥
K(k) = 0✮✳ ❘❡♠❛r❦ t❤❛t β = 0 ✐s t❤❡ ✐♥❞❡♣❡♥❞❡♥t ❝❛s❡ ✭s✐t❡s ❞♦♥✬t ✐♥t❡r❛❝t✮✱ ❛♥❞ t❤❛t ✇❤❡♥
β ✐♥❝r❡❛s❡s✱ t❤❡ ❞②♥❛♠✐❝s ❜❡❝♦♠❡s ❧❡ss ❛♥❞ ❧❡ss r❛♥❞♦♠✳ ❙♦ β ♠❛② ❜❡ t❤♦✉❣❤t ❛s ❛ ❦✐♥❞
♦❢ ✐♥✈❡rs❡ t❡♠♣❡r❛t✉r❡ ♣❛r❛♠❡t❡r✳ ❙❡❡ s✉❜s❡❝t✐♦♥ ✹✳✶✳✶ ✐♥ ❬✶✵❪ ❢♦r t❤❡ ❣❡♥❡r❛❧✐t② ♦❢ t❤❡ ❝❧❛ss
d
C0 ❛♠♦♥❣ r❡✈❡rs✐❜❧❡ P❈❆ ❞②♥❛♠✐❝s ♦♥ {−1, +1}Z ✳ ❉✉❡ t♦ t❤❡✐r ❞❡✜♥✐t✐♦♥✱ P❈❆ ❞②♥❛♠✐❝s
✐♥ C0 ❛r❡ ❧♦❝❛❧✱ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✱ ♥♦♥ ❞❡❣❡♥❡r❛t❡✳ ■t ✐s ❦♥♦✇♥ ✭s❡❡ ❬✽✱ ✶❪✮ t❤❛t ❛♥② P❈❆
❞②♥❛♠✐❝s P ✐♥ C0 ❛❞♠✐ts ❛t ❧❡❛st ♦♥❡ r❡✈❡rs✐❜❧❡ ♠❡❛s✉r❡ ✇❤✐❝❤ ✐s ❛ ●✐❜❜s ♠❡❛s✉r❡ ❛ss♦❝✐❛t❡❞
t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♠✉❧t✐❜♦❞② ♣♦t❡♥t✐❛❧ ϕ✿
 P

ϕUk (σUk ) = − log cosh β j K(k − j)σj
ϕΛ (σΛ )
= 0 ♦t❤❡r✇✐s❡.

✇❤❡r❡ Uk = {j : K(k − j) 6= 0}

✭✹✮

▼♦r❡♦✈❡r Pr♦♣♦s✐t✐♦♥ ✸✳✸ ✐♥ ❬✶❪ st❛t❡❞ t❤❡ ♣r❡❝✐s❡ r❡❧❛t✐♦♥s R = S ∩ G(ϕ) ❛♥❞ Rs = Ss ✱
✇❤❡r❡ S ✭r❡s♣✳ R✮ ❞❡♥♦t❡s t❤❡ s❡t ♦❢ P ✲st❛t✐♦♥❛r② ✭r❡s♣✳ P ✲r❡✈❡rs✐❜❧❡✮ ♠❡❛s✉r❡s✱ Ss ❛♥❞ Rs
t❤❡✐r r❡s♣❡❝t✐✈❡ s♣❛❝❡✲tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s✬ ♣❛rts✱ ❛♥❞ G(ϕ) t❤❡ s❡t ♦❢ ●✐❜❜s ♠❡❛s✉r❡s
d
♦♥ S Z ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ♣♦t❡♥t✐❛❧ ϕ✳
❖♥❡ ❛❧s♦ ❝❤❡❝❦s t❤❛t s✉❝❤ ❛ P❈❆ ❞②♥❛♠✐❝s P ✐s ❛ttr❛❝t✐✈❡✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢✉♥❝t✐♦♥ K(.) ✐s
♥♦♥✲♥❡❣❛t✐✈❡ ✭s❡❡ Pr♦♣❡rt② ✹✳✶✳✷ ✐♥ ❬✶✵❪✮✳ ❋r♦♠ ♥♦✇ ♦♥✱ ❧❡t ✉s ❛ss✉♠❡ t❤❛t K ✐s ♥♦♥ ♥❡❣❛t✐✈❡✳
▼✐①✐♥❣ ❝♦♥❞✐t✐♦♥s ❢♦r ❛ ♣♦t❡♥t✐❛❧ ϕ ❞❡✜♥❡ ❞✐✛❡r❡♥t r❡❣✐♦♥s ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ ❛❜s❡♥❝❡ ♦❢ ♣❤❛s❡
tr❛♥s✐t✐♦♥ ❢♦r t❤❡ ❛ss♦❝✐❛t❡❞ ●✐❜❜s ♠❡❛s✉r❡s✳ ❙tr♦♥❣ ♠✐①✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ✉s✉❛❧❧② r❡❧❛t❡❞
t♦ t❤❡ ❞♦♠❛✐♥ ✇❤❡r❡ ❉♦❜r✉s❤✐♥✬s ✉♥✐q✉❡♥❡ss ❤♦❧❞s✱ ❛♥❞ ✇❡❛❦ ♠✐①✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡①✲
♣❡❝t❡❞ t♦ ❜❡ ✈❛❧✐❞ ✐♥ t❤❡ ♠❛✐♥ ♣❛rt ♦❢ t❤❡ ✉♥✐q✉❡♥❡ss ❞♦♠❛✐♥✿ ❙❡❡ ❬✶✹❪ ❢♦r ❛ r❡✈✐❡✇ ♦♥

❊r❣♦❞✐❝✐t② ♦❢ P❈❆

✶✷✸

t❤❡s❡ ❝♦♥❞✐t✐♦♥s✳ ❍❡r❡✱ ✇❡ ❝❛❧❧ ✇❡❛❦ ♠✐①✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♣♦t❡♥t✐❛❧ ϕ✱ t❤❡ ❝♦♥❞✐t✐♦♥✿
∃C > 0, ∃M > 0, ∀L > 2,
Z
Z
σ0 µ(dσB(L) |σB(L)c = +1) −
σ0 µ(dσB(L) |σB(L)c = −1) 6 Ce−M L

(WM)

✇❤❡r❡ µ ✐s t❤❡ ✉♥✐q✉❡ ●✐❜❜s ♠❡❛s✉r❡ ❛ss♦❝✐❛t❡❞ t♦ ϕ✳ ❋♦r ❢❡rr♦♠❛❣♥❡t✐❝ ♣♦t❡♥t✐❛❧s✱ ✐t ✐s ✐♥❞❡❡❞
t❤❡ ❡q✉✐✈❛❧❡♥t ❢♦r♠ ♦❢ ♠♦r❡ ❣❡♥❡r❛❧ ✇❡❛❦ ♠✐①✐♥❣ ❝♦♥❞✐t✐♦♥✳
❚❤❡♦r❡♠ ✷ ▲❡t P ❜❡ ❛♥ ❛ttr❛❝t✐✈❡ P❈❆ ❞②♥❛♠✐❝s ♦♥ {−1, +1}Z ♦❢ t❤❡ ❝❧❛ss C0 ❞❡✜♥❡❞ ❜② ✭✸✮✱
❧❡t ϕ ❞❡♥♦t❡ t❤❡ ♣♦t❡♥t✐❛❧ ❝❛♥♦♥✐❝❛❧❧② ❛ss♦❝✐❛t❡❞ ❞❡✜♥❡❞ ✐♥ ✭✹✮✱ ❛♥❞ G(ϕ) t❤❡ s❡t ♦❢ ●✐❜❜s
♠❡❛s✉r❡s ✇✳r✳t ϕ✳
d

• ■❢ t❤❡r❡ ✐s ♣❤❛s❡ tr❛♥s✐t✐♦♥ ✭✐✳❡✳ #G(ϕ) > 1✮ t❤❡♥ t❤❡ ❞②♥❛♠✐❝s P ✐s ♥♦♥✲❡r❣♦❞✐❝✳
• ❖t❤❡r✇✐s❡✱ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ✭✐✳❡✳ G(ϕ) = {µ}✮ t❤❡ ❞②♥❛♠✐❝s P ✐s ❡r❣♦❞✐❝
t♦✇❛r❞s t❤❡ ✉♥✐q✉❡ ●✐❜❜s ♠❡❛s✉r❡ µ✳
▼♦r❡♦✈❡r ✐❢ ✇❡ ❛ss✉♠❡ t❤❡ ♣♦t❡♥t✐❛❧ ϕ s❛t✐s✜❡s t❤❡ ✇❡❛❦ ♠✐①✐♥❣ ❝♦♥❞✐t✐♦♥ (WM)✱ t❤❡♥
t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞s µ ❤♦❧❞s ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ r❛t❡✳

■♥ ❬✶❪✱ ✇❡ ❡st❛❜❧✐s❤❡❞ t❤❛t✱ ❢♦r ♥❡❛r❡st ♥❡✐❣❤❜♦✉r ✐♥t❡r❛❝t✐♦♥ ❢✉♥❝t✐♦♥ K✱ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❤♦❧❞s
❢♦r β ❧❛r❣❡✳ ❋♦r ✐♥st❛♥❝❡✱ ✇❤❡♥ d = 2✱ ❧❡t PJ ❜❡ t❤❡ P❈❆ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❝❧❛ss C0 ♦❜t❛✐♥❡❞
t❛❦✐♥❣✿ K(±e1 ) = K(±e2 ) = J > 0, K(k) = 0 ♦t❤❡r✇✐s❡✱ ✇❤❡r❡ (e1 , e2 ) ✐s ❛ ❜❛s✐s ♦❢ R2 ❛♥❞ J ❛
♣♦s✐t✐✈❡ ❝♦♥st❛♥t✳ ❚❤❡ ❝❛♥♦♥✐❝❛❧❧② ❛ss♦❝✐❛t❡❞
♣♦t❡♥t✐❛❧ ϕJ ✭❝❢✳ ✭✹✮ ✮ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦✉r✲❜♦❞②
P
♣♦t❡♥t✐❛❧✿ ϕJ,Vk (σVk ) = − log cosh(βJ j∈Uk σj ) ✇❤❡r❡ Uk = {k − e1 , k + e1 , k − e2 , k + e2 }✳
❋r♦♠ ❚❤❡♦r❡♠ ✷ ✇❡ ❝♦♥❝❧✉❞❡ ❤❡r❡ t❤❛t ❢♦r β ❧❛r❣❡✱ t❤❡ P❈❆ PJ ✐s ♥♦♥✲❡r❣♦❞✐❝ s✐♥❝❡ ✐t ❤❛s ❛t
❧❡❛st t✇♦ ❞✐✛❡r❡♥t st❛t✐♦♥❛r② st❛t❡s ν − ❛♥❞ ν + ✳
▲❡t ✉s ♥♦✇ ❞✐s❝✉ss ❤♦✇ ❧❛r❣❡ ✐s t❤❡ ❞♦♠❛✐♥ ✇❤❡r❡ ❝♦♥❞✐t✐♦♥ (WM) ❤♦❧❞s✳ ❖♥❡ ❝♦♥❥❡❝t✉r❡s
❲❡❛❦ ▼✐①✐♥❣ ❝♦♥❞✐t✐♦♥ ❢♦r ●✐❜❜s ♠❡❛s✉r❡ ✐s ✈❛❧✐❞ ✉♣ t♦ t❤❡ ❝r✐t✐❝❛❧ t❡♠♣❡r❛t✉r❡✱ t❤❛t ✐s✱ ❛s
s♦♦♥ ❛s t❤❡r❡ ✐s ♥♦ ♣❤❛s❡ tr❛♥s✐t✐♦♥✳ ■♥ t❤❛t s❡♥s❡✱ ♦✉r ♠❛✐♥ r❡s✉❧t ✇♦✉❧❞ ❣✐✈❡ ❡r❣♦❞✐❝✐t② ✇✐t❤
❡①♣♦♥❡♥t✐❛❧ r❛t❡ ♦♥ ❛ ♠✉❝❤ ❧❛r❣❡r r❡❣✐♦♥ ❛s t❤❡ r❡❣✐♦♥ ✇❤❡r❡ t❤❡ ❉♦❜r✉s❤✐♥✲❱❛s❡rs❤t❡✐♥ ❝r✐t❡✲
r✐♦♥ ❤♦❧❞s✳ ■♥ ❢❛❝t✱ ❧❡t ✉s ♠❡♥t✐♦♥ t❤❡ r❡❢❡r❡♥❝❡ ❬✺❪✱ ✇❤❡r❡✱ ✉s✐♥❣ ♣❡r❝♦❧❛t✐♦♥ t❡❝❤♥✐q✉❡s✱ ✐t ✐s
♣r♦✈❡❞ t❤❛t ✐♥ ❞✐♠❡♥s✐♦♥ d = 2✱ ❢♦r ❛ ❢❡rr♦♠❛❣♥❡t✐❝ ♥❡❛r❡st ♥❡✐❣❤❜♦✉r ■s✐♥❣ ♠♦❞❡❧ ✇✐t❤♦✉t ❡①✲
tr❡♠❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞✱ t❤❡ ❛ss♦❝✐❛t❡❞ ●✐❜❜s ♠❡❛s✉r❡ ✐s ✇❡❛❦ ♠✐①✐♥❣ ❛s s♦♦♥ ❛s ✐t ✐s ✉♥✐q✉❡ ✭✐✳❡✳
∀β, β < βc ✮✳ ■♥ ♦r❞❡r t♦ ♣r❡❝✐s❡ t❤✐s ❛ss❡rt✐♦♥✱ ❧❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❞②♥❛♠✐❝s PJ ✳ ❆ ♣r♦❥❡❝t✐♦♥
❛r❣✉♠❡♥t r❡❧❛t❡s t❤❡ ♣♦t❡♥t✐❛❧ ϕJ ❛ss♦❝✐❛t❡❞ t♦ PJ ✇✐t❤ t❤❡ ✉s✉❛❧ ■s✐♥❣ ❢❡rr♦♠❛❣♥❡t✐❝ ♣❛✐r ♣♦✲
t❡♥t✐❛❧ ✇✐t❤ ✐♥t❡♥s✐t② ❝♦❡✣❝✐❡♥t J ✭s❡❡ ❬✶✻❪✮✳ ❉✉❡ t♦ ❍✐❣✉❝❤✐✬s r❡s✉❧t✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ●✐❜❜s
st❛t❡ ❛ss♦❝✐❛t❡❞ t♦ t❤✐s ♣♦t❡♥t✐❛❧ ϕJ ✐s ✇❡❛❦ ♠✐①✐♥❣ ❛s s♦♦♥ ❛s t❤❡r❡ ✐s ♥♦ ♣❤❛s❡ tr❛♥s✐t✐♦♥✱
✇❤✐❝❤ ❤❛♣♣❡♥s ❢♦r β ❧♦✇❡r t❤❛♥√ t❤❡ ❝r✐t✐❝❛❧ ✈❛❧✉❡ βc ✱ ✇❤✐❝❤ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ■s✐♥❣ ❝r✐t✐❝❛❧
2)
✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ♦❜t❛✐♥ t❤❛t t❤❡ P❈❆ ❞②♥❛♠✐❝s PJ ✐s
✐♥✈❡rs❡ t❡♠♣❡r❛t✉r❡ βc = log(1+
2J
❡r❣♦❞✐❝ ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ r❛t❡ ❢♦r β < βc ❛♥❞ ♥♦♥✲❡r❣♦❞✐❝ ❢♦r β > βc ✳ ❚❛❦✐♥❣ J = 1✱ βc ≃ 0.441❀
s✐♥❝❡ ❉♦❜r✉s❤✐♥✲❱❛s❡rs❤t❡✐♥ ❝r✐t❡r✐♦♥ ❛♣♣❧✐❡s ♦♥❧② ❢♦r β < 21 ❆r❣t❤( 12 ) ≃ 0.275 ✭❝❢✳ ♣❛rt ✻✳✶✳✷
✐♥ ❬✶✵❪✮✱ ♦✉rs ✐s ❜❡tt❡r✳

✸ Pr♦♦❢ ♦❢ t❤❡ ❚❤❡♦r❡♠ ✶
❚❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦♠❡ ❝♦✉♣❧✐♥❣ ♦❢ P❈❆ ❞②♥❛♠✐❝s ♣r❡s❡r✈✲
✐♥❣ t❤❡ st♦❝❤❛st✐❝ ♦r❞❡r✐♥❣✳ ▲❡t (P 1 , P 2 , . . . , P N ) ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ N ✲✉♣❧❡ ♦❢ P❈❆ ❞②♥❛♠✐❝s

✶✷✹

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②
✇❤✐❝❤ ♠❡❛♥s P❈❆ r❡❧❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦♥♦t♦♥✐❝✐t② ♣r♦♣❡rt② ∀k ∈ Zd ✱ ∀ζ 1 4 ζ 2 4
d
N
i
. . . 4 ζ N ∈ S Z , ∀s ∈ S ✱ G1k (s | ζ 1 ) 6 G2k (s | ζ 2 ) 6 . . . 6 GN
k (s | ζ ) ✇❤❡r❡ G ✐s t❤❡
❢✉♥❝t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ P i ❜② ✭✶✮✳ ❚❤❡r❡ ❡①✐sts ✭❝❢✳ ❬✶✶❪✮ ❛ ♠♦♥♦t♦♥❡ s②♥❝❤r♦♥♦✉s ❝♦✉♣❧✐♥❣ ♦♥
d
(S Z )N ❞❡♥♦t❡❞ ❜② P 1 ⊛ P 2 ⊛ . . . ⊛ P N ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✿ ❢♦r ❛❧❧ ✐♥✐t✐❛❧ ❝♦♥✜❣✉r❛✲
t✐♦♥ σ 1 4 σ 2 4 . . . 4 σ N ❛♥❞ ❢♦r ❛❧❧ t✐♠❡s n✱
P1 ⊛ ... ⊛ PN




ω 1 (n) 4 . . . 4 ω N (n)  (ω 1 , . . . , ω N )(0) = (σ 1 , . . . , σ N ) = 1.

❙✉❝❤ ❛ ❝♦✉♣❧✐♥❣ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ✐♥❝r❡❛s✐♥❣ s②♥❝❤r♦♥♦✉s ❝♦✉♣❧✐♥❣✳ ❚❤❡ ♥♦t❛t✐♦♥ IP ❞❡♥♦t❡s t❤❡
❝♦✉♣❧✐♥❣ P ⊛ P ⊛ . . . ⊛ P ♦❢ N t✐♠❡s t❤❡ s❛♠❡ P❈❆ ❞②♥❛♠✐❝s P ✱ ✇❤❡r❡ N ✇✐❧❧ ❜❡ ❛ ✜♥✐t❡ ❧❛r❣❡
❡♥♦✉❣❤ ♥✉♠❜❡r✳
❚❤✐s ❝♦✉♣❧✐♥❣ ❛❧❧♦✇s ✉s t♦ ❞❡✈❡❧♦♣ s♦♠❡ ♠♦♥♦t♦♥✐❝✐t② ❛r❣✉♠❡♥t ❛♥❞ t♦ st❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣
r❡s✉❧t✱ ✇❤♦s❡ ♣r♦♦❢ ✐s ✐♥ ❬✶✶❪✿

Pr♦♣♦s✐t✐♦♥ ✸ ❚❤❡
♠❡❛s✉r❡ νΛ+ ✭r❡s♣✳ νΛ− ✮ ✐s t❤❡ ♠❛①✐♠❛❧ ✭r❡s♣✳ ♠✐♥✐♠❛❧✮ ♠❡❛s✉r❡ ♦❢ t❤❡
τ
Λc
s❡t {νΛ : τ ∈ S } ♦❢ st❛t✐♦♥❛r② ♠❡❛s✉r❡s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ P❈❆ ❞②♥❛♠✐❝s PΛτ ♦♥ t❤❡ ✜①❡❞
✜♥✐t❡ ✈♦❧✉♠❡ Λ ❛♥❞ ✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ τ ✳ ▲❡t ν+ ❛♥❞ ν− ❞❡♥♦t❡ t❤❡ ♠❛①✐♠❛❧ ❛♥❞ t❤❡
♠✐♥✐♠❛❧ ❡❧❡♠❡♥ts ♦❢ t❤❡ s❡t S ♦❢ st❛t✐♦♥❛r② ♠❡❛s✉r❡s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ P❈❆ ❞②♥❛♠✐❝s P ✳
❋♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s ❤♦❧❞✿
+
(n)
ν+ = lim νB(L)
⊗ δ(+
+)B(L)c = lim δ+ P

✭✺✮

−
(n)
.
ν− = lim νB(L)
⊗ δ(−
−)B(L)c = lim δ− P

✭✻✮

n→∞

L→∞

n→∞

L→∞

■♥ ♣❛rt✐❝✉❧❛r✱ P ❛❞♠✐ts ❛ ✉♥✐q✉❡ st❛t✐♦♥❛r② ♠❡❛s✉r❡ ν ✐❢ ❛♥❞ ♦♥❧② ✐❢ ν− = ν+ ✳
◆♦t❡ t❤❛t P (n) ❞❡♥♦t❡s P ◦ P ◦ . . . ◦ P ✱ ❛♥❞ s♦ ✐s ❢♦r ✐♥st❛♥❝❡ δ+ P (n) t❤❡ ❧❛✇ ❛t t✐♠❡ n ♦❢ t❤❡
▼❛r❦♦✈ ❈❤❛✐♥ ✇✐t❤ tr❛♥s✐t✐♦♥ ❦❡r♥❡❧ P ❛♥❞ ✐♥✐t✐❛❧ ❞✐str✐❜✉t✐♦♥ δ+ ✳

❘❡♠❛r❦ ✹ ◆♦t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❛♥❣❡ ♦❢ ❞❡♣❡♥❞❡♥❝❡ ✇✳r✳t✳ t❤❡ ♣❛st ❢♦r ❧♦❝❛❧ P❈❆✳ ▲❡t ✉s
d
(1)
(n)
❞❡✜♥❡ Λ = ∪k∈Λ Vk = Λ , ❛♥❞ Λ = ∪k∈Λ(n−1) Vk ✳ ❚❤❡♥✿ ∀n, ∀Λ ⋐ Zd ✱ ∀(σ, η) ∈ (S Z )2






✇✐t❤ σΛ(n) ≡ ηΛ(n) ✱ IP ωΛ1 (n) ≡ ωΛ2 (n)(ω 1 , ω 2 )(0) = (σ, η) = 1✳

Pr♦♦❢✳ ✭✭✷✮ ✐♠♣❧✐❡s (A) ✐♥ ❚❤❡♦r❡♠ ✶✮
+
■t ✉s❡s ❛ ✉s✉❛❧ str❛t❡❣② ❛♥❞ t❛❦❡s ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❝♦✉♣❧✐♥❣ P ⊛ PB(L)
✳ ▲❡t L ❜❡ ❛ ✜①❡❞
✐♥t❡❣❡r✱ ❧❛r❣❡r t❤❛♥ L1 = n1 ✇❤❡r❡ n1 ✐s ❞❡✜♥❡❞ ✐♥ ✭✷✮✳ ❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ ✭st❛t❡❞ ✐♥ ❬✶✶❪✮

+
−
νB(L)
⊗ δ(−
+)B(L)c ;
−)B(L)c 4 ν 4 νB(L) ⊗ δ(+

✭✼✮

t❤❡ ♣♦s✐t✐✈✐t② ♦❢ ❡❛❝❤ ❢♦❧❧♦✇✐♥❣ t❡r♠ ✐s st❛t❡❞✳ ❲❡ ❤❛✈❡✿
06

Z

+
σ0 dνB(L)
−

Z

−
σ0 dνB(L)
=

Z

+
σ0 dνB(L)
−

Z

Z

 Z
−
σ0 dν −
σ0 dνB(L)
,
σ0 dν +

−λL
❛♥❞ ✇❡ ✇✐❧❧ st❛t❡
✭✇❤❡r❡ f0 (σ) = σ0 ✮✳ ❲❡ ♦♥❧② ❣✐✈❡
R t❤❛t+❡❛❝❤ ♣❛rt
R ✐s ❧♦✇❡r t❤❛♥ 2|k f0 |k e
t❤❡ ♣r♦♦❢ ❢♦r σ0 dνB(L) − σ0 dν s✐♥❝❡ t❤❡ ♣r♦♦❢ ❢♦r t❤❡ ♠✐♥✐♠❛❧ − ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s
❛♥❛❧♦❣♦✉s✳ ❋♦r ❛♥② n ∈ N∗ ✱

 


(n)
(n)
+
+
+
+
νB(L)
(σ0 ) − ν(σ0 ) = νB(L)
(σ0 ) − δ+ PB(L)
(f0 ) + δ+ PB(L)
(f0 ) − δ+ P (n) (f0 ) +


δ+ P (n) (f0 ) − ν(σ0 ) .

❊r❣♦❞✐❝✐t② ♦❢ P❈❆

✶✷✺

+
+
❯s✐♥❣ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ PB(L)
⊛ PB(L)
t❤❡ ✜rst t❡r♠ ✐s ♥♦♥ ♣♦s✐t✐✈❡✳ ❯s✐♥❣ t❤❡ ❛ss✉♠♣✲
t✐♦♥ ✭✷✮ t❤❡ t❤✐r❞ t❡r♠ ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❛❜♦✈❡ ❜② 2|k f0 |k e−λn ✭∀n > n1 ✮✳ ❈❤♦♦s❡ ♥♦✇
+
n = L✳ ❘❡✇r✐t❡ t❤❡ s❡❝♦♥❞ t❡r♠ ❛s Q
l + ,+
(ω02 (n) − ω01 (n)) ✇❤❡r❡



+
+
+, + ) .
. |(ω 1 , ω 2 )(0) = (+
Q
l + ,+
( . ) = P ⊛ PB(L)

+
❯s✐♥❣ ▲❡♠♠❛ ✺✱ ✇❡ ❜♦✉♥❞ t❤❡ s❡❝♦♥❞ t❡r♠ ❢r♦♠ ❛❜♦✈❡ ✇✐t❤ κ′ Q
l + ,+
(ω02 (n) 6= ω01 (n))✳ ❆❝✲
❝♦r❞✐♥❣ t♦ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❝♦✉♣❧✐♥❣ ❛♥❞ ✉s✐♥❣ ❘❡♠❛r❦ ✹✱ ♥♦t❡ t❤❛t ✇✐t❤ r❡s♣❡❝t t♦
+
Q
l + ,+
(.)✱ ω02 (n) 6= ω01 (n) ✐s ♣♦ss✐❜❧❡ ♦♥❧② ✐❢ ✐t ❡①✐sts ❛ ♣r❡✈✐♦✉s t✐♠❡ n′ ✭0 < n′ < n✮ ❛♥❞ ❛ s✐t❡ k
(n′ )
(n′ )
s✉❝❤ t❤❛t ωk2 (n′ ) = ωk1 (n′ ) 6= + ✳ ❇② t❛❦✐♥❣ n = L✱ ✇❡ ❤❛✈❡ {0}
⊂ B(L)❀
✐♥ B(L)c ∩ {0}
+
s♦ ✐s t❤✐s ❡✈❡♥t ❡♠♣t②✱ ✇❤✐❝❤ ❡♥s✉r❡s Q
l + ,+
(ω02 (n) 6= ω01 (n)) = 0✳ ❚❤✉s ✐s (A) ♣r♦✈❡❞✳
✷
Pr♦♦❢✳ ✭(A) ✐♠♣❧✐❡s ✭✷✮ ✐♥ ❚❤❡♦r❡♠ ✶✮
❚❤❡ ♠♦st ❞❡❧✐❝❛t❡ ♣❛rt ✐s t♦ ❡st❛❜❧✐s❤ t❤❡ ❡①♣♦♥❡♥t✐❛❧ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞s ❡q✉✐❧✐❜r✐✉♠✳
❖✉r ♣r♦♦❢ ✐s ✐♥s♣✐r❡❞ ❜② ▼❛rt✐♥❡❧❧✐ ❛♥❞ ❖❧✐✈✐❡r✐ ♣r♦♦❢ ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❡r❣♦❞✐❝✐t② ❢♦r ❝♦♥t✐♥✉♦✉s
d
t✐♠❡ ●❧❛✉❜❡r ❞②♥❛♠✐❝s ♦♥ {−1, +1}Z ✭s❡❡ ❬✶✹❪✮✳ ❋♦r ❛♥② t✐♠❡ n ∈ N✱ ❧❡t ✉s ❞❡✜♥❡ ❛ ❝♦❡✣❝✐❡♥t
✇❤✐❝❤ ❝♦♥tr♦❧s t❤❡ ❡r❣♦❞✐❝✐t②✿





−, + ) .
ρ(n) = IP ω01 (n) 6= ω02 (n)(ω 1 , ω 2 )(0) = (−

✭✽✮

■❢ ✇❡ ❛ss✉♠❡ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❜♦✉♥❞ (A)✱ t❤❛♥❦s t♦ ❢♦rt❤❝♦♠✐♥❣ ▲❡♠♠❛ ✽✱ ✇❡ ❞❡❞✉❝❡ t❤❛t
limn→∞ ρ(n) = 0✳ ❘❡♣♦rt✐♥❣ ❛ss✉♠♣t✐♦♥ (A) ✐♥ t❤❡ ✐♥❡q✉❛❧✐t② ✭✶✵✮✱ ✇❡ ❝❛♥ ✉s❡ ❢♦rt❤❝♦♠✐♥❣
▲❡♠♠❛ ✶✶ t♦ ❞❡❞✉❝❡ t❤❛t (ρ(n))n∈N∗ ❝♦♥✈❡r❣❡ t♦ 0 ❢❛st❡r t❤❛♥ n1d ✳ ❋✐♥❛❧❧②✱ ✉s✐♥❣ ✐♥❡q✉❛❧✐t② ✭✾✮
❛♥❞ ▲❡♠♠❛ ✶✷✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ρ(n) ❝♦♥✈❡r❣❡s t♦ 0 ❡①♣♦♥❡♥t✐❛❧❧② ❢❛st❀ t❤✉s✱ t❤❛♥❦s t♦
▲❡♠♠❛ ✼✱ ❝♦♥❝❧✉s✐♦♥ ❤♦❧❞s✳
✷

❚❡❝❤♥✐❝❛❧ ❧❡♠♠❛s✿ ❋✐rst r❡♠❛r❦ t❤❡ ❡❛s② ❢❛❝t✿
▲❡♠♠❛ ✺ ▲❡t (Ω, A, P) ❜❡ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✱ ❛♥❞ Z ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ✈❛❧✉❡s ✐♥ ❛ ✜♥✐t❡
s❡t {z1 < . . . < zm } ♦❢ R✱ s✉❝❤ t❤❛t P(Z > 0) = 1✳ ❚❤❡♥✱ ✐❢ κ = max{ z1i , zi > 0, 1 6 i 6 m}
❛♥❞ κ′ = max{z
R i , 1 6 i 6R m} ✭✇❤✐❝❤ ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❧❛✇ ♦❢ Z ✉♥❞❡r P ✮ ✇❡ ❤❛✈❡✿
P(Z 6= 0) 6 κ ZdP ❛♥❞ ZdP 6 κ′ P(Z 6= 0)✳
❯s✐♥❣ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♣r♦♣❡rt② ♦❢ t❤❡ ❝♦✉♣❧✐♥❣✱ t❤❡ t✇♦ ❢♦❧❧♦✇✐♥❣ ▲❡♠♠❛s ❛r❡ ❡❛s✐❧② ♣r♦✈❡❞✳

▲❡♠♠❛ ✻






d

∀σ, η ∈ S Z , σ 4 η ✱ IP ω01 (n) 6= ω02 (n)(ω 1 , ω 2 )(0) = (σ, η) 6 ρ(n)✳

Z
∀Λ ⋐ Zd ✱ ∀n

 ∈ N✱ ∀ξ ∈ S
✱





−)Λc 4 P ω(n) ∈ .ω(0) = ξ 4 PΛ+ ω(n) ∈ .ω(0) = ξΛ (+
+)Λc ✳
ω(n) ∈ .ω(0) = ξΛ (−
−, + ))✳
ρ(n) 6 PΛ− ⊛ PΛ+ (ω01 (n) 6= ω02 (n) |(ω 1 , ω 2 )(0) = (−
d



PΛ−


▲❡♠♠❛ ✼ ❚❤❡ s❡q✉❡♥❝❡ (ρ(n))n∈N∗ ✐s ❞❡❝r❡❛s✐♥❣✱ ❛♥❞ ∀f ✱ ∀σ, η ✱




P (f (ω(n))|ω(0) = σ) − P (f (ω(n))|ω(0) = η) 6 2 |k f |k ρ(n).




❚❤✉s✱ ✐❢ limn→∞ ρ(n) = 0✱ t❤❡ ❞②♥❛♠✐❝s P ✐s ❡r❣♦❞✐❝✱ ❛♥❞ supσ P (f (ω(n))|ω(0) = σ) − ν(f )
6 2 |k f |k ρ(n)✱ ✇❤❡r❡ ν ❞❡♥♦t❡s t❤❡ ✉♥✐q✉❡ st❛t✐♦♥❛r② ♠❡❛s✉r❡✳
◆♦t❡ t❤❛t ❞✉❡ t♦ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♦❢ ρ(.)✱ ✇❡ ❝❛♥ r❡str✐❝t ♦✉rs❡❧✈❡s t♦ t❤❡ ❝❛s❡ ρ(.) > 0✳

✶✷✻

❊❧❡❝tr♦♥✐❝ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ Pr♦❜❛❜✐❧✐t②
▲❡♠♠❛ ✽ ∃κ✱ ∀Λ ⋐ Zd ✱ limn→∞ ρ(n) 6 κ



R

+
−
σ0 dνΛ

R

−
σ0 dνΛ




✳




−, + ) = 1 s✐♥❝❡ t❤❡ ❝♦✉♣❧✐♥❣
ω01 (n) 6 ω02 (n))  (ω 1 (0), ω 2 (0)) = (−

◆♦t❡ PΛ− ⊛ PΛ+
♣r❡s❡r✈❡s t❤❡ ♦r❞❡r✳ ❙♦✱ t❤❛♥❦s t♦ ▲❡♠♠❛ ✺✱ ❛♣♣❧✐❡❞ ✇✐t❤
−, + )) ❛♥❞ Z = ω02 (n) − ω01 (n) ✇❡ ❤❛✈❡✿
P = PΛ− ⊛ PΛ+ ( . |(ω 1 (0), ω 2 (0)) = (−
Pr♦♦❢✳






−, + ) 6 κ PΛ+ (ω0 (n)|ω(0) = + )−PΛ− (ω0 (n)|ω(0) =
PΛ− ⊛PΛ+ ω01 (n) 6= ω02 (n)(ω 1 (0), ω 2 (0)) = (−

−)

✇❤❡r❡ κ = (min{s − s′ : s > s′ ; s, s′ ∈ S})−1 ✳ ❇② ▲❡♠♠❛ ✻✱ ρ(n) ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❛❜♦✈❡ ❜②
t❤❡ ❧✳❤✳s ♦❢ t❤❡ ♣r❡✈✐♦✉s ✐♥❡q✉❛❧✐t②✳ ❲❡ ❝♦♥❝❧✉❞❡ ❜② t❛❦✐♥❣ t❤❡ ❧✐♠✐t ✐♥ n✱ ❛♥❞ ✉s✐♥❣ t❤❡ ✜♥✐t❡
✈♦❧✉♠❡ ❡r❣♦❞✐❝✐t②✳
✷
❘❡♠❛r❦ ✾

❆s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ▲❡♠♠❛ ✽ ✇❡ ❣❡t

t❤❡ ❡r❣♦❞✐❝✐t② ♦❢

P

limn→∞ ρ(n) = 0✱

✇❤✐❝❤ ✐♠♣❧✐❡s

t❤❛♥❦s t♦ ▲❡♠♠❛ ✼✳

▲❡t ✉s ❞❡♥♦t❡ ❜② R = maxk′ ∈V0 kk′ k1 t❤❡ ✜♥✐t❡ r❛♥❣❡ ♦❢ t❤❡ ❧♦❝❛❧ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t P❈❆
❞②♥❛♠✐❝s P ✳
▲❡♠♠❛ ✶✵

❚❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ✐♥❡q✉❛❧✐t✐❡s ❤♦❧❞✿

∀n ∈ N∗ , ρ(2n) 6 (2nR + 1)d ρ2 (n) ;
Z

Z
+
−
.
∀n, ∀L ∈ N∗ , ρ(2n) 6 2(2L + 1)d ρ2 (n) + 2κ
σ0 dνB(L)
− σ0 dνB(L)
Pr♦♦❢✳

✭✾✮
✭✶✵✮

▲❡t n ❜❡ ❛ ✜①❡❞ ✐♥t❡❣❡r✳

Pr♦♦❢ ♦❢ ✐♥❡q✉❛❧✐t②
 ✭✾✮






+
−, + ) ✳ ❯s✐♥❣ ▼❛r❦♦✈ ♣r♦♣❡rt② ♦❢ IP ✿
▲❡t νn− ,+
( . ) = IP (ω 1 , ω 2 )(n) ∈ . (ω 1 , ω 2 )(0) = (−
ρ(2n) =

Z





+
IP ω01 (2n) 6= ω02 (2n)(ω 1 , ω 2 )(n) = (ξ− , ξ+ ) νn− ,+
(dξ− , dξ+ ) .

+
◆♦t❡ t❤❛t νn− ,+
✲❛❧♠♦st s✉r❡❧②✱ ξ− 4 ξ+ ✳ ▲❡t A = {(ξ− , ξ+ ) : ∃k ∈ Zd , kkk1 6 nR, ξk− 6= ξk+ }✳
❚❤❛♥❦s t♦ ❘❡♠❛r❦ ✹ ♦❜s❡r✈❡ t❤❛t t❤❡ ❡①❛❝t ❝♦♥tr♦❧ ♦❢ ✐♥t❡r❛❝t✐♦♥ ✐♥❢♦r♠❛t✐♦♥✬s ♣r♦♣❛❣❛✲
(n)
t✐♦♥ ❢♦r P❈❆ ✐♠♣❧✐❡s t❤❛t t❤❡ ❛❜♦✈❡ ✐♥t❡❣r❛❧ ✈❛♥✐s❤❡s ♦♥ Ac ❜❡❝❛✉s❡ B(nR) ⊃ {0} ✱ ❛♥❞ s♦
−
+
ξB(nR)
≡ ξB(nR)
✳ ❚❤❡♥✿

ρ(2n) =

Z

A





+
IP ω01 (n) 6= ω02 (n)(ω 1 , ω 2 )(0) = (ξ− , ξ+ ) νn− ,+
(dξ− , dξ+ ) .

+
❯s✐♥❣ ▲❡♠♠❛ ✻✱ ✇❡ ♦❜t❛✐♥ ρ(2n) 6 ρ(n) νn− ,+
(A)✳
−
− +
❲r✐t✐♥❣ A = ∪{k∈Zd : kkk1 6nR} {(ξ , ξ ) : ξk 6= ξk+ } ✇❡ ❞❡❞✉❝❡✿
+
νn− ,+
(A) 6

X

k∈Zd ,kkk 6nR
1





−, + ) .
IP ωk1 (n) 6= ωk2 (n)(ω 1 , ω 2 )(0) = (−

+
(A) 6 ρ(n)#B(nR) 6
❙✐♥❝❡ P ✐s tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s ❢r♦♠ νn− ,+
d
ρ(n)(2nR + 1) ✇❤❡r❡ #B(nR) ❞❡♥♦t❡s t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ B(nR)✳

❊r❣♦❞✐❝✐t② ♦❢ P❈❆

✶✷✼

Pr♦♦❢ ♦❢ ✐♥❡q✉❛❧✐t②
 ✭✶✵✮





−, η, + ) ν(dη) ✇❤❡r❡ ν ✐s ❛ P ✲st❛t✐♦♥❛r②
❲r✐t❡ ρ(2n) = IP ω01 (2n) 6= ω03 (2n)(ω 1 , ω 2 , ω 3 )(0) = (−
1
2
3
♠❡❛s✉r❡✳
◆♦t❡ t❤❛t ω0 (n) 6 ω0 (n) 6 ω0 (n)✱ 


1
2
−, η, + ) ✲❛❧♠♦st s✉r❡❧②✱ s♦ t❤❛t
IP (ω , ω , ω 3 ) ∈ .  (ω 1 , ω 2 , ω 3 )(0) = (−
{ω01 (n) 6= ω03 (n)} = {ω01 (n) 6= ω02 (n)} ∪ {ω02 (n) 6= ω03 (n)}, ✇❤❡r❡ t❤❡ ✉♥✐♦♥ ✐s ♥♦♥ ♥❡❝❡ss❛r✐❧②
❞✐s❥♦✐♥t ✭✉♥❧❡ss ❝❛r❞✐♥❛❧✐t② ♦❢ S ✐s 2✮✳ ❚❤✉s✱ ❢♦❧❧♦✇✐♥❣ ❞❡❝♦♠♣♦s✐t✐♦♥ ❤♦❧❞s✿
R

Z




−, η) ν(dη)
ρ(2n) 6 IP ω01 (2n) 6= ω02 (2n)(ω 1 , ω 2 )(0) = (−
Z




+ IP ω01 (2n) 6= ω02 (2n)(ω 1 , ω 2 )(0) = (η, +) ν(dη) .

✭✶✶✮

■t ✐s t❤❡♥ ❡♥♦✉❣❤ t♦ ♣r♦✈❡ t❤❛t ❡❛❝❤ ♦❢ t❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ❜♦✉♥❞❡❞ ❢r♦♠ ❛❜♦✈❡ ❜② ❤❛❧❢ t❤❡
q✉❛♥t✐t② ✇❛♥t❡❞✳
 ❈♦♥s✐❞❡r ✜rst t❤❡ s❡❝♦♥❞ t❡r♠ ✐♥ t❤❡
 r✳❤✳s✳ ✳
 1 2
+
η,+
1
2
▲❡t νn = IP (ω , ω )(n) = .  (ω , ω )(0) = (η, + ) ✳ ▲❡t ✉s ✇r✐t❡✿
Z





IP �