Upper tails of self-intersection local times: survey of proof - PowerPoint PPT Presentation

Weierstraß-Institut für Angewandte Analysis und Stochastik Upper tails of self-intersection local times: survey of proof techniques Wolfgang König TU Berlin and WIAS Berlin Mohrenstraße 39 · 10117 Berlin · Tel. 030 20372 0 · www.wias-berlin.de · Luminy, 6 December 2010

The Model � Simple random walk ( S n ) n ∈ N 0 on Z d � Local times ℓ n ( z ) = ∑ n i = 0 1 l { S i = z } for n ∈ N , z ∈ Z d � p -norm of local times � ℓ n � p = ( ∑ z ∈ Z d ℓ n ( z ) p ) 1 / p For p ∈ N , we have the p -fold self-intersection local time (SILT): n ∑ � ℓ n � p 1 l { S i 1 = ··· = S ip } , p = i 1 ,..., i p = 0 Typical behaviour [C ERNY 2007] for d = 2 and [B ECKER /K ÖNIG 2009] for d ≥ 3 :  n ( p + 1 ) / 2 if d = 1 ,   E [ � ℓ n � p  n ( log n ) p − 1 p ] ∼ Ca ( n ) , a ( n ) = if d = 2 , where  if d ≥ 3 . n   Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 2 (15)

Goal Goal: Asymptotics of 1 n log P ( � 1 n ℓ n � p ≥ r n ) , n → ∞ , for ( nr n ) p − E [ � ℓ n � p p ] → ∞ . � very large deviations: ( nr n ) p ≫ a ( n ) � large deviations: ( nr n ) p ∼ γ a ( n ) with γ > C What is the best path strategy to produce many self-intersections? Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 3 (15)

Rough Heuristics (1) (only very large-deviations case ( nr n ) p ≫ a ( n ) ) Strategy to meet {� 1 n ℓ n � p ≥ r n } : The path fills a ball B α n of radius 1 ≪ α n ≪ n 1 / d within a time interval [ 0 , t n ] ⊂ [ 0 , n ] in order to produce ( nr n ) p self-intersections, and runs freely afterwards. Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 4 (15)

Rough Heuristics (1) (only very large-deviations case ( nr n ) p ≫ a ( n ) ) Strategy to meet {� 1 n ℓ n � p ≥ r n } : The path fills a ball B α n of radius 1 ≪ α n ≪ n 1 / d within a time interval [ 0 , t n ] ⊂ [ 0 , n ] in order to produce ( nr n ) p self-intersections, and runs freely afterwards. Then ℓ n ( z ) ≈ ℓ t n ( z ) ≍ t n α − d for z ∈ B α n n Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 4 (15)

Rough Heuristics (1) (only very large-deviations case ( nr n ) p ≫ a ( n ) ) Strategy to meet {� 1 n ℓ n � p ≥ r n } : The path fills a ball B α n of radius 1 ≪ α n ≪ n 1 / d within a time interval [ 0 , t n ] ⊂ [ 0 , n ] in order to produce ( nr n ) p self-intersections, and runs freely afterwards. Then ℓ n ( z ) ≈ ℓ t n ( z ) ≍ t n α − d for z ∈ B α n n and p ≈ ∑ n α d ( 1 − p ) t n ≍ nr n α d ( p − 1 ) / p ( nr n ) p ≍ � ℓ n � p ℓ t n ( z ) p ≍ α d n ( t n α − d n ) p = t p i.e., , . n n z ∈ B α n Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 4 (15)

Rough Heuristics (1) (only very large-deviations case ( nr n ) p ≫ a ( n ) ) Strategy to meet {� 1 n ℓ n � p ≥ r n } : The path fills a ball B α n of radius 1 ≪ α n ≪ n 1 / d within a time interval [ 0 , t n ] ⊂ [ 0 , n ] in order to produce ( nr n ) p self-intersections, and runs freely afterwards. Then ℓ n ( z ) ≈ ℓ t n ( z ) ≍ t n α − d for z ∈ B α n n and p ≈ ∑ n α d ( 1 − p ) t n ≍ nr n α d ( p − 1 ) / p ( nr n ) p ≍ � ℓ n � p ℓ t n ( z ) p ≍ α d n ( t n α − d n ) p = t p i.e., , . n n z ∈ B α n and − log P ( S [ 0 , t n ] ⊂ B α n ) ≍ t n d p ( p − 1 ) − 2 ≍ nr n α . n α 2 n Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 4 (15)

Rough Heuristics (1) (only very large-deviations case ( nr n ) p ≫ a ( n ) ) Strategy to meet {� 1 n ℓ n � p ≥ r n } : The path fills a ball B α n of radius 1 ≪ α n ≪ n 1 / d within a time interval [ 0 , t n ] ⊂ [ 0 , n ] in order to produce ( nr n ) p self-intersections, and runs freely afterwards. Then ℓ n ( z ) ≈ ℓ t n ( z ) ≍ t n α − d for z ∈ B α n n and p ≈ ∑ n α d ( 1 − p ) t n ≍ nr n α d ( p − 1 ) / p ( nr n ) p ≍ � ℓ n � p ℓ t n ( z ) p ≍ α d n ( t n α − d n ) p = t p i.e., , . n n z ∈ B α n and − log P ( S [ 0 , t n ] ⊂ B α n ) ≍ t n p ( p − 1 ) − 2 d ≍ nr n α . n α 2 n Optimal choices: p 2 p  2 p � n if d < d ( 1 − p ) r if d < p − 1 , p − 1 ,  n t n ≍ α n ≍ and , 2 p 2 p nr n if d > 1 if d > p − 1 , p − 1 .  Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 4 (15)

Rough Heuristics (2) Hence, we conjecture Theorem A. 2 p  2 p − 1 n ℓ n � p ≥ r n ) ≍ 1 ≍ 1 2 p t n d ( p − 1 ) ∨ 1 d ( p − 1 ) r if d < p − 1 , n log P ( � 1  n ≍ r ≍ n α 2 α 2 n 2 p r n if d > n n p − 1 .  Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 5 (15)

Rough Heuristics (2) Hence, we conjecture Theorem A. 2 p  2 p − 1 n ℓ n � p ≥ r n ) ≍ 1 ≍ 1 2 p t n d ( p − 1 ) ∨ 1 d ( p − 1 ) r if d < p − 1 , n log P ( � 1  n ≍ r ≍ n α 2 α 2 n 2 p r n if d > n n p − 1 .  � Lower-critical dimension: homogeneous squeezing on a large area. � Upper-critical dimension: short-time clumping on finitely many sites. Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 5 (15)

Precise heuristics (1) 2 p First subcritical dimensions d < p − 1 . Scaled normalized version of ℓ n : L n ( x ) = α d n for x ∈ R d . ⌊ x α n ⌋ � � n ℓ n , Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 6 (15)

Precise heuristics (1) 2 p First subcritical dimensions d < p − 1 . Scaled normalized version of ℓ n : L n ( x ) = α d n for x ∈ R d . ⌊ x α n ⌋ � � n ℓ n , Weak large-deviation principle (in the spirit of D ONSKER -V ARADHAN ) with speed n α − 2 and rate function n I ( f ) = 1 � 2 � ∇ f � � 2 , 2 i.e., − n � � �� P ( L n ∈ · ) = exp f 2 ∈· I ( f )+ o ( 1 ) inf . α 2 n Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 6 (15)

Precise heuristics (1) 2 p First subcritical dimensions d < p − 1 . Scaled normalized version of ℓ n : L n ( x ) = α d n for x ∈ R d . ⌊ x α n ⌋ � � n ℓ n , Weak large-deviation principle (in the spirit of D ONSKER -V ARADHAN ) with speed n α − 2 and rate function n I ( f ) = 1 � 2 � ∇ f � � 2 , 2 i.e., − n � � �� P ( L n ∈ · ) = exp f 2 ∈· I ( f )+ o ( 1 ) inf . α 2 n Note that d ( 1 − p ) ℓ n ( z ) p � 1 / p � p � 1 / p � z � ∑ = n α − d � ∑ L n = n α p � L n � p = nr n � L n � p . � ℓ n � p = n n α n z ∈ Z d z ∈ Z d Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 6 (15)

Precise heuristics (1) 2 p First subcritical dimensions d < p − 1 . Scaled normalized version of ℓ n : L n ( x ) = α d n for x ∈ R d . ⌊ x α n ⌋ � � n ℓ n , Weak large-deviation principle (in the spirit of D ONSKER -V ARADHAN ) with speed n α − 2 and rate function n I ( f ) = 1 � 2 � ∇ f � � 2 , 2 i.e., − n � � �� P ( L n ∈ · ) = exp f 2 ∈· I ( f )+ o ( 1 ) inf . α 2 n Note that d ( 1 − p ) ℓ n ( z ) p � 1 / p � p � 1 / p � z � ∑ = n α − d � ∑ L n = n α p � L n � p = nr n � L n � p . � ℓ n � p = n n α n z ∈ Z d z ∈ Z d Hence, 2 p n {� 1 � L n � p ≥ 1 d ( p − 1 ) n ℓ n � p ≥ r n } = and = nr � � . n α 2 n Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 6 (15)

Precise Heuristics (2) 2 p Hence, we conjecture, for d < p − 1 , Theorem B. 2 p d ( 1 − p ) r n log P ( � 1 lim n ℓ n � p ≥ r n ) = − χ d , p , n → ∞ n where � 1 2 � ∇ f � 2 2 : f ∈ H 1 ( R d ) , � f 2 � p = 1 = � f � 2 � χ d , p = inf . Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 7 (15)

Precise Heuristics (2) 2 p Hence, we conjecture, for d < p − 1 , Theorem B. 2 p d ( 1 − p ) r n log P ( � 1 lim n ℓ n � p ≥ r n ) = − χ d , p , n → ∞ n where � 1 2 � ∇ f � 2 2 : f ∈ H 1 ( R d ) , � f 2 � p = 1 = � f � 2 � χ d , p = inf . Remark: χ d , p > 0 d ( p − 1 ) ≤ 2 p [G ANTERT /K ÖNIG /S HI 2004] ⇐ ⇒ . Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 7 (15)

Precise Heuristics (3) 2 p Now supercritical dimensions d > p − 1 . We approximate � 1 p ≥ 1 {� 1 �� � � n ℓ n � p ≥ r n } ≈ � ℓ st n � p ≥ nr n � � = ℓ st n . � � st n s � Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 8 (15)

Precise Heuristics (3) 2 p Now supercritical dimensions d > p − 1 . We approximate � 1 p ≥ 1 {� 1 �� � � n ℓ n � p ≥ r n } ≈ � ℓ st n � p ≥ nr n � � = ℓ st n . � � st n s � 1 st n ℓ st n satisfies a large-deviation principle with scale st n and some rate function Now I ( d ) . Upper tails of self-intersection local times · Luminy, 6 December 2010 · Seite 8 (15)