Logarithmic Fluctuations From Circularity Lionel Levine (MIT) AMS - PowerPoint PPT Presentation

Logarithmic Fluctuations From Circularity Lionel Levine (MIT) AMS Eastern Sectional Meeting April 9, 2011 Joint work with David Jerison and Scott Sheffield Lionel Levine Logarithmic Fluctuations From Circularity

From random walk to growth model Internal DLA ◮ Start with n particles at the origin in the square grid Z 2 . ◮ Each particle in turn performs a simple random walk until it finds an unoccupied site, stays there. ◮ A ( n ): the resulting random set of n sites in Z 2 . Growth rule: ◮ Let A (1) = { o } , and A ( n +1) = A ( n ) ∪{ X n ( τ n ) } where X 1 , X 2 ,... are independent random walks, and τ n = min { t | X n ( t ) �∈ A ( n ) } . Lionel Levine Logarithmic Fluctuations From Circularity

Internal DLA cluster in Z 2 . Closeup of the boundary. Lionel Levine Logarithmic Fluctuations From Circularity

Questions ◮ Limiting shape ◮ Fluctuations Lionel Levine Logarithmic Fluctuations From Circularity

Meakin & Deutch, J. Chem. Phys. 1986 ◮ “It is also of some fundamental significance to know just how smooth a surface formed by diffusion limited processes may be.” ◮ “Initially, we plotted ln( ξ ) vs ln( ℓ ) but the resulting plots were quite noticably curved. Figure 2 shows the dependence of ln( ξ ) on ln[ln( ℓ )].” Lionel Levine Logarithmic Fluctuations From Circularity

History of the Problem ◮ Diaconis-Fulton 1991 : Addition operation on subsets of Z d . ◮ Lawler-Bramson-Griffeath 1992 : w.p.1, B (1 − ε ) r ⊂ A ( π r 2 ) ⊂ B (1+ ε ) r eventually . ◮ Lawler 1995 : w.p.1, B r − r 1 / 3 log 2 r ⊂ A ( π r 2 ) ⊂ B r + r 1 / 3 log 4 r eventually . “A more interesting question... is whether the errors are o ( n α ) for some α < 1 / 3.” Lionel Levine Logarithmic Fluctuations From Circularity

Logarithmic Fluctuations Theorem Jerison - L. - Sheffield 2010 : with probability 1, B r − C log r ⊂ A ( π r 2 ) ⊂ B r + C log r eventually . Asselah - Gaudilli` ere 2010 independently obtained B r − C log r ⊂ A ( π r 2 ) ⊂ B r + C log 2 r eventually . Lionel Levine Logarithmic Fluctuations From Circularity

Logarithmic Fluctuations in Higher Dimensions In dimension d ≥ 3, let ω d be the volume of the unit ball in R d . Then with probability 1, B r − C √ log r ⊂ A ( ω d r d ) ⊂ B r + C √ log r eventually for a constant C depending only on d . (Jerison - L. - Sheffield 2010; Asselah - Gaudilli` ere 2010) Lionel Levine Logarithmic Fluctuations From Circularity

Elements of the proof ◮ Thin tentacles are unlikely. ◮ Martingales to detect fluctuations from circularity. ◮ “Self-improvement” Lionel Levine Logarithmic Fluctuations From Circularity

Thin tentacles are unlikely B ( z, m ) z A ( n ) A thin tentacle. Lemma. If 0 / ∈ B ( z , m ), then � � z ∈ A ( n ) , #( A ( n ) ∩ B ( z , m )) ≤ bm d � Ce − cm 2 / log m , d = 2 ≤ P Ce − cm 2 , d ≥ 3 for constants b , c , C > 0 depending only on the dimension d . Lionel Levine Logarithmic Fluctuations From Circularity

Early and late points in A ( n ) , for n = π r 2 ∂ B r + m ∂ B r ∂ B r − ℓ m -early point A ( n ) ℓ -late point Lionel Levine Logarithmic Fluctuations From Circularity

Early and late points Definition 1. z is an m -early point if: n < π ( | z |− m ) 2 z ∈ A ( n ) , Definition 2. z is an ℓ -late point if: n > π ( | z | + ℓ ) 2 z / ∈ A ( n ) , E m [ n ] = event that some point in A ( n ) is m -early L ℓ [ n ] = event that some point in B √ n / π − ℓ is ℓ -late Lionel Levine Logarithmic Fluctuations From Circularity

Structure of the argument: Self-improvement LEMMA 1. No ℓ -late points implies no m -early points: If m ≥ C ℓ , then P ( E m [ n ] ∩ L ℓ [ n ] c ) < n − 10 . LEMMA 2. No m -early points implies no ℓ -late points: � If ℓ ≥ C (log n ) m , then P ( L ℓ [ n ] ∩ E m [ n ] c ) < n − 10 . � Iterate, ℓ �→ C (log n ) C ℓ , which is decreasing until ℓ = C 2 log n . Lionel Levine Logarithmic Fluctuations From Circularity

Iterating Lemmas 1 and 2 ◮ Fix n and let ℓ, m be the maximal lateness and earliness occurring by time n . Iterate starting from m 0 = n : ◮ ( ℓ, m ) unlikely to belong to a vertical rectangle by Lemma 1. ◮ ( ℓ, m ) unlikely to belong to a horizontal rectangle by Lemma 2. Lionel Levine Logarithmic Fluctuations From Circularity

Early and late point detector To detect early points near ζ ∈ Z 2 , we use the martingale M ζ ( n ) = ∑ ( H ζ ( z ) − H ζ (0)) z ∈ � A ( n ) � � ζ / | ζ | where H ζ is a discrete harmonic function approximating Re . ζ − z ζ ∂ B | ζ | The fine print: ◮ Discrete harmonicity fails at three points z = ζ , ζ +1 , ζ +1+ i . ◮ Modified growth process � A ( n ) stops at ∂ B | ζ | (0). Lionel Levine Logarithmic Fluctuations From Circularity

Time change of Brownian motion ◮ To get a continuous time martingale, we use Brownian motions on the grid Z × R ∪ R × Z instead of random walks. ◮ Then there is a standard Brownian motion B ζ such that M ζ ( t ) = B ζ ( s ζ ( t )) where N ( M ( t i ) − M ( t i − 1 )) 2 ∑ s ζ ( t ) = lim i =1 is the quadratic variation of M ζ . Lionel Levine Logarithmic Fluctuations From Circularity

LEMMA 1. No ℓ -late implies no m = C ℓ -early Event Q [ z , k ]: ◮ z ∈ A ( k ) \ A ( k − 1). ◮ z is m -early ( z ∈ A ( π r 2 ) for r = | z |− m ). ◮ E m [ k − 1] c : No previous point is m -early. ◮ L ℓ [ n ] c : No point is ℓ -late. We will use M ζ for ζ = (1+4 m / r ) z to show for 0 < k ≤ n , P ( Q [ z , k ]) < n − 20 . Lionel Levine Logarithmic Fluctuations From Circularity

Main idea: Early but no late would be a large deviation! ◮ Recall there is a Brownian motion B ζ such that M ζ ( n ) = B ζ ( s ζ ( n )) . ◮ On the event Q [ z , k ] P ( M ζ ( k ) > c 0 m ) > 1 − n − 20 (1) and P ( s ζ ( k ) < 100log n ) > 1 − n − 20 . (2) ◮ On the other hand, ( s = 100log n ) � � ≤ e − s / 2 = n − 50 . B ζ ( s ′ ) ≥ s sup P s ′ ∈ [0 , s ] Lionel Levine Logarithmic Fluctuations From Circularity

Proof of (1) On the event Q [ z , k ] P ( M ζ ( k ) > c 0 m ) > 1 − n − 20 . ◮ Since z ∈ A ( k ) and thin tentacles are unlikely, we have with high probability, #( A ( k ) ∩ B ( z , m )) ≥ bm 2 . ◮ For each of these bm 2 points, the value of H ζ is order 1 / m , so their total contribution to M ζ ( k ) is order m . ◮ No ℓ -late points means that points elsewhere cannot compensate. Lionel Levine Logarithmic Fluctuations From Circularity

Proof of (2) : Controlling the Quadratic Variation On the event Q [ z , k ] P ( s ζ ( k ) < 100log n ) > 1 − n − 20 . ◮ Lemma: There are independent standard Brownian motions B 1 , B 2 ,... such that s ζ ( i +1) − s ζ ( i ) ≤ τ i where τ i is the first exit time of B i from the interval ( a i , b i ). a i = min H ζ ( z ) − H ζ (0) z ∈ ∂ ˜ A ( i ) b i = max H ζ ( z ) − H ζ (0) . z ∈ ∂ ˜ A ( i ) Lionel Levine Logarithmic Fluctuations From Circularity

Proof of (2) : Controlling the Quadratic Variation On the event Q [ z , k ] P ( s ζ ( k ) < 100log n ) > 1 − n − 20 . ◮ By independence of the τ i , E e s ζ ( k ) ≤ E e ( τ 1 + ··· + τ k ) = ( E e τ 1 ) ··· ( E e τ k ) . ◮ By large deviations for Brownian exit times, E e τ ( − a , b ) ≤ 1+10 ab . ◮ Easy to estimate a i , and use the fact that no previous point is m -early to bound b i . Conclude that � � e s ζ ( k ) 1 Q ≤ n 50 . E Lionel Levine Logarithmic Fluctuations From Circularity

What changes in higher dimensions? ◮ In dimension d ≥ 3 the quadratic variation s ζ ( n ) is constant order instead of log n . ◮ So the fluctuations are instead dominated by thin tentacles, which can grow to length √ log n . ◮ Still open: prove matching lower bounds on the fluctuations of order log n in dimension 2 and √ log n in dimensions d ≥ 3. Lionel Levine Logarithmic Fluctuations From Circularity

Thank You! Reference: ◮ D. Jerison, L. Levine and S. Sheffield, Logarithmic fluctuations for internal DLA. arXiv:1010.2483 Lionel Levine Logarithmic Fluctuations From Circularity