The Limit Shape of the Leaky Abelian Sandpile Model Ian M. Alevy Department of Mathematics University of Rochester Joint work with Sevak Mkrtchyan December 2, 2020
The Abelian Sandpile Model (ASM) is a cellular automaton defined on a graph G = ( V , E ) . An initial sandpile distribution s : V → N If s ( v ) > deg( v ) then v is unstable and topples distributing sand to its neighbors: � s ( v ) �→ s ( v ) − deg( v ) s ( u ) �→ s ( u ) + 1 if u ∼ v . The sandpile evolves through toppling unstable sites. Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
� � � � The Abelian Sandpile Model (ASM) is a cellular automaton defined on a graph G = ( V , E ) . An initial sandpile distribution s : V → N If s ( v ) > deg( v ) then v is unstable and topples distributing sand to its neighbors: � s ( v ) �→ s ( v ) − deg( v ) s ( u ) �→ s ( u ) + 1 if u ∼ v . The sandpile evolves through toppling unstable sites. In this talk G = Z 2 but we will consider different toppling rules: Uniform ASM + 1 + 1 − 4 + 1 + 1 Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
� � � � � The Abelian Sandpile Model (ASM) is a cellular automaton defined on a graph G = ( V , E ) . An initial sandpile distribution s : V → N If s ( v ) > deg( v ) then v is unstable and topples distributing sand to its neighbors: � s ( v ) �→ s ( v ) − deg( v ) s ( u ) �→ s ( u ) + 1 if u ∼ v . The sandpile evolves through toppling unstable sites. In this talk G = Z 2 but we will consider different toppling rules: Directed ASM Uniform ASM + 1 + 1 � + 1 − 2 + 1 − 4 + 1 + 1 Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
� � � � � � � The Abelian Sandpile Model (ASM) is a cellular automaton defined on a graph G = ( V , E ) . An initial sandpile distribution s : V → N If s ( v ) > deg( v ) then v is unstable and topples distributing sand to its neighbors: � s ( v ) �→ s ( v ) − deg( v ) s ( u ) �→ s ( u ) + 1 if u ∼ v . The sandpile evolves through toppling unstable sites. In this talk G = Z 2 but we will consider different toppling rules: 1D ASM Directed ASM Uniform ASM + 1 − 2 + 1 + 1 + 1 � + 1 − 2 + 1 − 4 + 1 + 1 Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
� � 1-Dimensional ASM Start with initial sandpile s ( v ) = n δ ( 0 , 0 ) ( v ) topple until reaching a stable sandpile s ∞ . Question Toppling rule What is the stable sandpile? + 1 − 2 + 1 Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
� � 1-Dimensional ASM Start with initial sandpile s ( v ) = n δ ( 0 , 0 ) ( v ) topple until reaching a stable sandpile s ∞ . Question Toppling rule What is the stable sandpile? + 1 − 2 + 1 0 0 0 0 7 0 0 0 0 Figure: Initial sandpile with n = 7. 0 0 0 1 5 1 0 0 0 Figure: Result after toppling at the origin. Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
Sequence of topplings 0 0 0 2 3 2 0 0 0 Figure: Origin toppled again. 0 0 1 1 3 1 1 0 0 Figure: All unstable sites topple once more. some more topples.... 0 1 1 0 3 0 1 1 0 and the stable sandpile: 0 1 1 1 1 1 1 1 0 Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
Limit Shape of 1D ASM Let v = ( x , y ) . Proposition If s ( x , y ) = n δ ( 0 , 0 ) ( x , y ) then the stable sandpile for the 1D ASM is 1 if x = 0 and n is odd, 0 if x = 0 and n is even, s ∞ ( x , 0 ) = if 0 < | x | ≤ ⌊ n 1 2 ⌋ , if ⌊ n 0 2 ⌋ < | x | . s ∞ ( x , y ) = 0 if y > 0 . Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
Limit Shape of 1D ASM Let v = ( x , y ) . Proposition If s ( x , y ) = n δ ( 0 , 0 ) ( x , y ) then the stable sandpile for the 1D ASM is 1 if x = 0 and n is odd, 0 if x = 0 and n is even, s ∞ ( x , 0 ) = if 0 < | x | ≤ ⌊ n 1 2 ⌋ , if ⌊ n 0 2 ⌋ < | x | . s ∞ ( x , y ) = 0 if y > 0 . When d ≥ 2 the limit shape exhibits self-organization. Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
� � � � 2D ASM Let s ( x , y ) = n δ ( 0 , 0 ) ( x , y ) and topple until stable using the uniform toppling rule. The stable sandpile has a limit shape (Pegden-Smart 2013). Toppling rule + 1 + 1 − 4 + 1 + 1 Figure: Stable sandpile with n = 10 7 . Colors correspond to heights of sandpile.
� � � � 2D ASM Let s ( x , y ) = n δ ( 0 , 0 ) ( x , y ) and topple until stable using the uniform toppling rule. The stable sandpile has a limit shape (Pegden-Smart 2013). Toppling rule + 1 + 1 − 4 + 1 + 1 Theorem (Levine-Peres 2008) The limit shape is bounded √ n between circles of radii c 1 √ n with c 2 / c 1 = √ Figure: Stable sandpile with n = 10 7 . 3 and c 2 2 . √ Colors correspond to heights of sandpile.
What is the limit shape of the ASM? The boundary of the limit shape is a Lipschitz graph (Aleksanyan-Shahgholian 2019) Figure: Stable sandpile with n = 10 7 . Colors correspond to heights of sandpile. Is the limit shape convex? Is it a circle, a polygon, or neither?
� Directed ASM The toppling rule determines the limit shape: Toppling rule + 1 � + 1 − 2 Figure: Stable sandpile with n = 10 5 . Black sites have one grain of sand. Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
Leaky Abelian Sandpile Model (Leaky-ASM) We compute the limit shape in the presence of dissipation. An initial sandpile distribution s : V → R ≥ 0 Dissipation d > 1 If s ( v ) > d · deg( v ) then v is unstable and topples distributing sand to its neighbors: � s ( v ) �→ s ( v ) − d · deg( v ) s ( u ) �→ s ( u ) + 1 if u ∼ v .
� � � � Leaky Abelian Sandpile Model (Leaky-ASM) We compute the limit shape in the presence of dissipation. An initial sandpile distribution s : V → R ≥ 0 Dissipation d > 1 If s ( v ) > d · deg( v ) then v is Uniform ASM with unstable and topples dissipation distributing sand to its neighbors: + 1 � s ( v ) �→ s ( v ) − d · deg( v ) + 1 − 4 d + 1 s ( u ) �→ s ( u ) + 1 if u ∼ v . + 1
Main Results Let s ( v ) = n δ ( 0 , 0 ) ( v ) and topple until stable using the uniform toppling rule. D n , d is the set of sites which have toppled. Theorem (A.- Mkrtchyan 2020) Let d > 1 and r = log n − 1 2 log log n. The boundary of r − 1 D n , d converges to the dual of the boundary of the gaseous phase in the amoeba of the spectral curve for the toppling rule.
Main Results Let s ( v ) = n δ ( 0 , 0 ) ( v ) and topple until stable using the uniform toppling rule. D n , d is the set of sites which have toppled. Theorem (A.- Mkrtchyan 2020) Let d > 1 and r = log n − 1 2 log log n. The boundary of r − 1 D n , d converges to the dual of the boundary of the gaseous phase in the amoeba of the spectral curve for the toppling rule. Theorem (A.- Mkrtchyan 2020) Let d n = 1 + t n . √ t n 1 If t n ≍ log( n ) then the boundary of log( n ) D n , d converges to a circle. 1 If t n ≍ n 1 − α with 0 < α < 1 , then the boundary of √ t n log( n ) D n , d is between circles of radii c 1 and c 2 with c 1 c 2 → α .
(a) d = 1 . 05 (b) d = 2 (c) d = 1000 Figure: Simulations of the Leaky-ASM with n ≈ 10 500 . Figure: Limit shapes from theorem.
Vanishing dissipation limit (a) d − 1 = 2 . 5 · 10 − 4 (b) d − 1 = 2 . 5 · 10 − 5 (c) d − 1 = 2 . 5 · 10 − 6 (d) d − 1 = 2 . 5 · 10 − 7 Figure: Leaky-ASM simulations with n = 10 7 .
Limiting sandpile Figure: Uniform ASM with background height − 1 and n = 10 7 . Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
Vanishing dissipation limit converges to uniform ASM Theorem (A.- Mkrtchyan (2020)) As d → 1 the Leaky-ASM converges to the ASM with background height − 1 . Sketch of proof: Couple the leaky-ASM to a modified ASM in which sites topple if they have 5 or more grains of sand. Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
Background ASM introduced by Bak-Tang-Wiesenfeld in 1987 as a model for fractals and self-organized criticality. At each time step a site is chosen randomly and one grain of sand is added. All unstable sites topple. The distribution of avalanches has a power law tail (Dhar 2006?). Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
Background ASM introduced by Bak-Tang-Wiesenfeld in 1987 as a model for fractals and self-organized criticality. At each time step a site is chosen randomly and one grain of sand is added. All unstable sites topple. The distribution of avalanches has a power law tail (Dhar 2006?). Dhar-Sadhu (2013) proposed using sandpiles to model pattern formation and proportionate growth. The odometer is piecewise quadratic (Ostojic 2003). A limit pattern exists (Pegden-Smart 2013). The internal fractal structure is connected to Apollonian circle packings (Levine-Pegden-Smart 2016 and Pegden-Smart 2020). Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model
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