Circles in the sand Lionel Levine (Cornell University) Wesley Pegden (Carnegie Mellon) Charles Smart (Cornell University) Harvard, April 15, 2015
The Abelian Sandpile (BTW 1987, Dhar 1990) ◮ Start with a pile of n chips at the origin in Z d . ◮ Each site x = ( x 1 , . . . , x d ) ∈ Z d has 2 d neighbors x ± e i , i = 1 , . . . , d . ◮ Any site with at least 2 d chips is unstable, and topples by sending one chip to each neighbor.
The Abelian Sandpile (BTW 1987, Dhar 1990) ◮ Start with a pile of n chips at the origin in Z d . ◮ Each site x = ( x 1 , . . . , x d ) ∈ Z d has 2 d neighbors x ± e i , i = 1 , . . . , d . ◮ Any site with at least 2 d chips is unstable, and topples by sending one chip to each neighbor. ◮ This may create further unstable sites, which also topple. ◮ Continue until there are no more unstable sites.
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 1 16 1 12 1 ��� 1
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 2 16 2 8 2 ��� 2
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 3 16 3 4 3 ��� 3
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 4 16 4 0 4 ��� 4
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 4 16 4 4 0 4 ��� 4
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 4 1 4 16 4 4 0 4 1 0 1 4 ��� ��� 4 1 4
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 4 1 4 4 16 4 0 4 1 0 1 4 ��� ��� 4 1 4
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 1 4 2 0 1 16 4 0 4 1 0 2 4 ��� ��� 4 1 4
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 1 4 2 0 1 16 4 0 4 1 0 2 4 4 ��� ��� 4 1 4
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 1 4 2 0 2 16 4 0 4 1 0 3 0 1 ��� ��� 4 1 4 1
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 1 4 2 0 2 16 4 0 4 1 0 3 0 1 ��� ��� 4 1 4 4 1
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 1 4 2 0 2 16 4 0 4 1 0 4 0 1 ��� ��� 4 2 0 2 1
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 1 4 2 0 2 16 4 0 4 1 0 4 4 0 1 ��� ��� 4 2 0 2 1
Toppling to Stabilize A Sandpile ◮ Example: n =16 chips in Z 2 . ◮ Sites with 4 or more chips are unstable. 1 4 2 1 2 16 4 0 4 1 1 0 1 1 ��� ��� 4 2 1 2 1 Stable.
Abelian Property ◮ The final stable configuration does not depend on the order of topplings. ◮ Neither does the number of times a given vertex topples.
Sandpile of 1 , 000 , 000 chips in Z 2 ◮ Ostojic 2002, Fey-Redig 2008, Dhar-Sadhu-Chandra 2009, L.-Peres 2009, Fey-L.-Peres 2010, Pegden-Smart 2011 ◮ Open problem: Determine the limit shape! (It exists.)
Limit shape 1: The sandpile computes an area-minimizing tropical curve through n given points Caracciolo-Paoletti-Sportiello 2010, Kalinin-Shkolnikov 2015
Limit shape 2: Identity element of the sandpile group of an n × n square grid Le Borgne-Rossin 2002. Sportiello 2015+
Sandpiles of the form h + n δ 0 h = 2 h = 1 h = 0
What about h = 3?
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 4 0 4 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 5 0 5 3 3 3 4 0 4 0 4 3 3 3 5 0 5 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3
3 3 3 4 3 3 3 3 3 5 0 5 3 3 3 5 1 4 1 5 3 4 0 4 0 4 0 4 3 5 1 4 1 5 3 3 3 5 0 5 3 3 3 3 3 4 3 3 3
3 3 5 0 5 3 3 3 5 1 4 1 5 3 5 1 5 0 5 1 5 0 4 0 4 0 4 0 5 1 5 0 5 1 5 3 5 1 4 1 5 3 3 3 5 0 5 3 3 . . . Never stops toppling!
3 5 1 4 1 5 3 5 1 5 0 5 1 5 1 5 1 4 1 5 1 4 0 4 0 4 0 4 1 5 1 4 1 5 1 5 1 5 0 5 1 5 3 5 1 4 1 5 3 . . . Never stops toppling!
5 1 5 0 5 1 5 1 5 1 4 1 5 1 5 1 5 0 5 1 5 0 4 0 4 0 4 0 5 1 5 0 5 1 5 1 5 1 4 1 5 1 5 1 5 0 5 1 5 . . . Never stops toppling!
1 5 1 4 1 5 1 5 1 5 0 5 1 5 1 5 1 4 1 5 1 4 0 4 0 4 0 4 1 5 1 4 1 5 1 5 1 5 0 5 1 5 1 5 1 4 1 5 1 . . . Never stops toppling!
A dichotomy Any sandpile τ : Z d → N is either ◮ stabilizing : every site topples finitely often ◮ or exploding : every site topples infinitely often
An open problem ◮ Given a probability distribution µ on N , decide whether the i.i.d. sandpile τ ∼ � x ∈ Z 2 µ is stabilizing or exploding. ◮ For example, find the smallest λ such that i.i.d. Poisson( λ ) is exploding.
How to prove an explosion ◮ Claim : If every site in Z d topples at least once, then every site topples infinitely often.
How to prove an explosion ◮ Claim : If every site in Z d topples at least once, then every site topples infinitely often. ◮ Otherwise, let x be the first site to finish toppling.
How to prove an explosion ◮ Claim : If every site in Z d topples at least once, then every site topples infinitely often. ◮ Otherwise, let x be the first site to finish toppling. ◮ Each neighbor of x topples at least one more time, so x receives at least 2 d additional chips. ◮ So x must topple again. ⇒⇐
The Odometer Function ◮ u ( x ) = number of times x topples.
The Odometer Function ◮ u ( x ) = number of times x topples. ◮ Discrete Laplacian: � ∆ u ( x ) = u ( y ) − 2 d u ( x ) y ∼ x
The Odometer Function ◮ u ( x ) = number of times x topples. ◮ Discrete Laplacian: � ∆ u ( x ) = u ( y ) − 2 d u ( x ) y ∼ x = chips received − chips emitted
The Odometer Function ◮ u ( x ) = number of times x topples. ◮ Discrete Laplacian: � ∆ u ( x ) = u ( y ) − 2 d u ( x ) y ∼ x = chips received − chips emitted = τ ∞ ( x ) − τ ( x ) where τ is the initial unstable chip configuration and τ ∞ is the final stable configuration.
Stabilizing Functions ◮ Given a chip configuration τ on Z d and a function u 1 : Z d → Z , call u 1 stabilizing for τ if τ + ∆ u 1 ≤ 2 d − 1 .
Stabilizing Functions ◮ Given a chip configuration τ on Z d and a function u 1 : Z d → Z , call u 1 stabilizing for τ if τ + ∆ u 1 ≤ 2 d − 1 . ◮ If u 1 and u 2 are stabilizing for τ , then τ + ∆min( u 1 , u 2 ) ≤ τ + max(∆ u 1 , ∆ u 2 ) ≤ 2 d − 1 so min( u 1 , u 2 ) is also stabilizing for τ .
Least Action Principle ◮ Let τ be a sandpile on Z d with odometer function u . ◮ Least Action Principle: If v : Z d → Z ≥ 0 is stabilizing for τ , then u ≤ v .
Least Action Principle ◮ Let τ be a sandpile on Z d with odometer function u . ◮ Least Action Principle: If v : Z d → Z ≥ 0 is stabilizing for τ , then u ≤ v . ◮ So the odometer is minimal among all nonnegative stabilizing functions: u ( x ) = min { v ( x ) | v ≥ 0 is stabilizing for τ } . ◮ Interpretation: “Sandpiles are lazy.”
The Green function of Z d ◮ G : Z d → R and ∆ G = − δ 0 . ◮ In dimensions d ≥ 3, G ( x ) = E 0 # { k | X k = x } is the expected number of visits to x by simple random walk started at 0. ◮ As | x | → ∞ , � c d | x | 2 − d d ≥ 3 G ( x ) ∼ g ( x ) = c 2 log | x | d = 2 .
An integer obstacle problem ◮ The odometer function for n chips at the origin is given by u = nG + w where G is the Green function of Z d , and w is the pointwise smallest function on Z d satisfying w ≥ − nG ∆ w ≤ 2 d − 1 w + nG is Z -valued
An integer obstacle problem ◮ The odometer function for n chips at the origin is given by u = nG + w where G is the Green function of Z d , and w is the pointwise smallest function on Z d satisfying w ≥ − nG ∆ w ≤ 2 d − 1 w + nG is Z -valued ◮ What happens if we replace Z by R ?
Abelian sandpile Divisible sandpile (Integrality constraint) (No integrality constraint)
Scaling limit of the abelian sandpile in Z d ◮ Consider s n = n δ 0 + ∆ u n , the sandpile formed from n chips at the origin. ◮ Let r = n 1 / d and ¯ s n ( x ) = s n ( rx ) (rescaled sandpile) w n ( x ) = r − 2 u n ( rx ) − nG ( rx ) ¯ (rescaled odometer)
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