the limit shape of the leaky abelian sandpile modelmain results let s(x) = n (0;0)(x) and topple...

The Limit Shape of the Leaky AbelianSandpile Model

Ian M. Alevy

Department of MathematicsUniversity of Rochester

Joint work with Sevak Mkrtchyan

December 2, 2020

The Abelian Sandpile Model (ASM) is a cellular automatondefined on a graph G = (V ,E).

An initial sandpile distribution s : V → NIf s(x) > deg(x) then x is unstable and topples distributingsand to its neighbors:{

s(x) 7→ s(x)− deg(x)s(y) 7→ s(y) + 1 if y ∼ x .

The sandpile evolves through toppling unstable sites.

In this talk G = Z2 but we will consider different toppling rules:

1D ASM+1 −2 //oo +1

Directed ASM+1

−2

OO

// +1

Uniform ASM+1

+1 −4

OO

//

��

oo +1

+1

Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model

1D ASM+1 −2 //oo +1

Directed ASM+1

−2

OO

// +1

Uniform ASM+1

+1 −4

OO

//

��

oo +1

+1

1D ASM+1 −2 //oo +1

Directed ASM+1

−2

OO

// +1

Uniform ASM+1

+1 −4

OO

//

��

oo +1

+1

1D ASM+1 −2 //oo +1

Directed ASM+1

−2

OO

// +1

Uniform ASM+1

+1 −4

OO

//

��

oo +1

+1

1-Dimensional ASM

Start with initial sandpile s(x) = nδ(0,0)(x) topple untilreaching a stable sandpile s∞.

QuestionWhat is the stable sandpile?

Toppling rule

+1 −2 //oo +1

0 0 0 0 7 0 0 0 0Figure: Initial sandpile with n = 7.

0 0 0 1 5 1 0 0 0Figure: Result after toppling at the origin.

1-Dimensional ASM

Start with initial sandpile s(x) = nδ(0,0)(x) topple untilreaching a stable sandpile s∞.

QuestionWhat is the stable sandpile?

Toppling rule

+1 −2 //oo +1

0 0 0 0 7 0 0 0 0Figure: Initial sandpile with n = 7.

0 0 0 1 5 1 0 0 0Figure: Result after toppling at the origin.

Sequence of topplings

0 0 0 2 3 2 0 0 0Figure: Origin toppled again.

0 0 1 1 3 1 1 0 0Figure: All unstable sites topple once more.

some more topples....

0 1 1 0 3 0 1 1 0and the stable sandpile:

0 1 1 1 1 1 1 1 0Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model

Limit Shape of 1D ASM

Let x = (x1, x2).

Proposition

If s(x1, x2) = nδ(0,0)(x1, x2) then the stable sandpile for the 1DASM is

s∞(x1,0) =

1 if x1 = 0 and n is odd,0 if x1 = 0 and n is even,1 if 0 < |x1| ≤ bn2c,0 if bn2c < |x2|.

s∞(x1, x2) = 0 if x2 > 0.

When d ≥ 2 the limit shape exhibits self-organization.

Limit Shape of 1D ASM

Let x = (x1, x2).

Proposition

If s(x1, x2) = nδ(0,0)(x1, x2) then the stable sandpile for the 1DASM is

s∞(x1,0) =

1 if x1 = 0 and n is odd,0 if x1 = 0 and n is even,1 if 0 < |x1| ≤ bn2c,0 if bn2c < |x2|.

s∞(x1, x2) = 0 if x2 > 0.

When d ≥ 2 the limit shape exhibits self-organization.

2D ASMLet s(x1, x2) = nδ(0,0)(x1, x2) and topple until stable usingthe uniform toppling rule.The stable sandpile has a limit shape (Pegden-Smart2013).

Toppling rule

+1

+1 −4

OO

//

��

oo +1

+1

Theorem (Levine-Peres 2008)The limit shape is boundedbetween circles of radii c1

√n

and c2√

n with c2/c1 =√

3√2.

Figure: Stable sandpile with n = 107.Colors correspond to heights of sandpile.

2D ASMLet s(x1, x2) = nδ(0,0)(x1, x2) and topple until stable usingthe uniform toppling rule.The stable sandpile has a limit shape (Pegden-Smart2013).

Toppling rule

+1

+1 −4

OO

//

��

oo +1

+1

Theorem (Levine-Peres 2008)The limit shape is boundedbetween circles of radii c1

√n

and c2√

n with c2/c1 =√

3√2. Figure: Stable sandpile with n = 107.

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 = 107. Colors correspond to heights ofsandpile.

Is the limit shape convex? Is it a circle, a polygon, or neither?

Directed ASM

The toppling rule determines the limit shape:

Figure: Stable sandpile with n = 105. Black sites haveone grain of sand.

Toppling rule

+1

−2

OO

// +1

Leaky Abelian Sandpile Model (Leaky-ASM)

We compute the limit shape in the presence of dissipation.

An initial sandpile distribution s : V → R≥0Dissipation d > 1

If s(x) > d · deg(x) then x isunstable and topplesdistributing sand to itsneighbors:{

s(x) 7→ s(x)− d · deg(x)s(y) 7→ s(y) + 1 if y ∼ x .

Uniform ASM withdissipation

+1

+1 −4d

OO

//

��

oo +1

+1

Leaky Abelian Sandpile Model (Leaky-ASM)

We compute the limit shape in the presence of dissipation.

An initial sandpile distribution s : V → R≥0Dissipation d > 1

If s(x) > d · deg(x) then x isunstable and topplesdistributing sand to itsneighbors:{

s(x) 7→ s(x)− d · deg(x)s(y) 7→ s(y) + 1 if y ∼ x .

Uniform ASM withdissipation

+1

+1 −4d

OO

//

��

oo +1

+1

Main ResultsLet s(x) = nδ(0,0)(x) and topple until stable using theuniform toppling rule.Dn,d is the set of sites which have toppled.

Theorem (A.- Mkrtchyan 2020)

Let d > 1 and r = log n − 12 log log n. The boundary of r−1Dn,d

converges to the dual of the boundary of the gaseous phase inthe amoeba of the spectral curve for the toppling rule.

Theorem (A.- Mkrtchyan 2020)Let dn = 1 + tn.

If tn � 1log(n) then the boundary of√

tnlog(n)

Dn,d converges to

a circle.If tn � 1n1−α with 0 < α < 1, then the boundary of√

tnlog(n)

Dn,d is between circles of radii c1 and c2 withc1c2→α.

Main ResultsLet s(x) = nδ(0,0)(x) and topple until stable using theuniform toppling rule.Dn,d is the set of sites which have toppled.

Theorem (A.- Mkrtchyan 2020)

Let d > 1 and r = log n − 12 log log n. The boundary of r−1Dn,d

converges to the dual of the boundary of the gaseous phase inthe amoeba of the spectral curve for the toppling rule.

Theorem (A.- Mkrtchyan 2020)Let dn = 1 + tn.

If tn � 1log(n) then the boundary of√

tnlog(n)

Dn,d converges to

a circle.If tn � 1n1−α with 0 < α < 1, then the boundary of√

tnlog(n)

Dn,d is between circles of radii c1 and c2 withc1c2→α.

(a) d = 1.05 (b) d = 2 (c) d = 1000

Figure: Simulations of the Leaky-ASM with n ≈ 10500.

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 = 107.

Limiting sandpile

Figure: Uniform ASM with background height −1 and n = 107.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 stable sandpile of the Leaky-ASM convergespointwise to the stable sandpile of the ASM with backgroundheight −1.

Sketch of proof:Couple the leaky-ASM to a modified ASM in which sites toppleif they have 5 or more grains of sand.

Background

ASM introduced by Bak-Tang-Wiesenfeld in 1987 as amodel for fractals and self-organized criticality.

At each time step a site is chosen randomly and one grainof sand is added. All unstable sites topple. The distributionof avalanches has a power law tail (Dhar 2006?).

Dissipative sandpiles introduced by Manna-Kiss-Kertész in1990 to model systems in which the average transfer ratiois a parameter or random quantity

Avalanches of thermal neutrons in a nuclear reactorcontrolled by cadmium rods are one example.

Dhar-Sadhu (2013) proposed using sandpiles to modelpattern 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 Apolloniancircle packings (Levine-Pegden-Smart 2016 andPegden-Smart 2020).

The ASM is a discrete model of a free boundary problem.

Background

The ASM is a discrete model of a free boundary problem.Ian M. Alevy The Limit Shape of the Leaky Abelian Sandpile Model

Computing the limit shape of the Leaky-ASM

Outline of our proof:Relate the Leaky-ASM to a killed random walk.

Use the steepest descent method to compute theasymptotic death probability.

Level curves of4n

and4(d − 1)

nin the death probability

bound the Leaky-ASM with n chips started at the origin.

Outline of our proof:Relate the Leaky-ASM to a killed random walk.Use the steepest descent method to compute theasymptotic death probability.

Level curves of4n

and4(d − 1)

Outline of our proof:Relate the Leaky-ASM to a killed random walk.Use the steepest descent method to compute theasymptotic death probability.

Level curves of4n

and4(d − 1)

Killed random walkLet X1,X2, . . . be i.i.d random variables with

P{Xj = (1,0)} =1

4d, P{Xj = (−1,0)} =

14d

,

P{Xj = (0,1)} =1

4d, P{Xj = (0,−1)} =

14d

,

P{Xj = (0,0)} = 1−4

4d= 1− 1

d.

The killed random walk (KRW) started at x ∈ Z2 is thesequence S1,S2, . . . where

Sn = x +n∑

i=1

KiXi

and

Ki =

{1 if the walker is alive at step i0 else.

Connection to sandpiles

Let Gd (x) = P(walker dies at x) be the death probability.

DefinitionThe odometer function u(x) = total sand emitted from x .

Start with initial sandpile s(x) = nδ0,0(x) and topple untilreaching the stable sandpile s∞(x).

Proposition (A.-Mkrtchyan 2020)For the operator

T =1d

∆−(

d − 1d

)I

we have

T (u(x)−Gd (x)) =d − 1

dns∞(x).

T =1d

∆−(

d − 1d

)I

we have

T (u(x)−Gd (x)) =d − 1

dns∞(x).

T =1d

∆−(

d − 1d

)I

we have

T (u(x)−Gd (x)) =d − 1

dns∞(x).

Key lemma

“Invert”

T =1d

∆−(

d − 1d

)I

and use inequality0 ≤ s∞(x) < 4d

to obtain the key lemma:

Lemma (A.-Mkrtchyan 2020)

1 If Gd (x) <4(d − 1)

n, then u(x) = 0, i.e. x 6∈ Dn,d .

2 If Gd (x) ≥4dn

, then u(x) ≥ 4d, i.e. x ∈ Dn,d .

Dn,d is the set of sites which topple.

Consequence

Asymptotics of Gd (x) give the boundary of the limit shape.

Key lemma

“Invert”

T =1d

∆−(

d − 1d

)I

and use inequality0 ≤ s∞(x) < 4d

to obtain the key lemma:

Lemma (A.-Mkrtchyan 2020)

1 If Gd (x) <4(d − 1)

n, then u(x) = 0, i.e. x 6∈ Dn,d .

2 If Gd (x) ≥4dn

, then u(x) ≥ 4d, i.e. x ∈ Dn,d .

Dn,d is the set of sites which topple.

Consequence

Asymptotics of Gd (x) give the boundary of the limit shape.

Massive Laplacian

The spectral curve of the massive Laplacian can be used tocompute asymptotics of Gd (x).

Definition

The massive Laplacian ∆m : CV → CV is defined by

(∆mf )(x) =∑y∼x

P(x → y)(f (y)− f (x))− P(dies)f (x)

=∑y∼x

P(x → y)f (y)− f (x)

where P(x → y) is the probability that the KRW moves fromvertex x to y and P(dies) is the probability it is killed.

Spectral curve of KRW

When the probabilities are periodic the spectral curve is

P(z,w) = det ∆m(z,w).

Probabilities are modified by z or w when crossing afundamental domain.

For the KRW the fundamental domain has size 1× 1 and

(∆mf )(x) =∑y∼x

14d

f (y)− f (x).

∆m is a 1× 1 matrix with spectral curve

P(z,w) = 4d −(

z + z−1 + w + w−1).

Spectral curve of KRW

When the probabilities are periodic the spectral curve is

P(z,w) = det ∆m(z,w).

Probabilities are modified by z or w when crossing afundamental domain.For the KRW the fundamental domain has size 1× 1 and

(∆mf )(x) =∑y∼x

14d

f (y)− f (x).

∆m is a 1× 1 matrix with spectral curve

P(z,w) = 4d −(

z + z−1 + w + w−1).

Asymptotic death probability

Expand the normalized spectral curve in a power seriesconvergent near (1,1) to compute probabilities:

4(d − 1)P(z,w)

=4(d − 1)

4d − (z + z−1 + w + w−1)

=d − 1

d

∞∑k=0

(z + z−1 + w + w−1

4d

)k=∑

k ,l∈ZGd (k , l)zkw l ,

where Gd (k , l) is the probability the KRW dies at (k , l).

Contour integration gives the coefficients in the directionva = (1,a) for 0 < a < 1:

Gd (rva) =1

(2πi)2

∮Cw

∮Cz

4(d − 1)P(z,w)

dzzr+1

dwwar+1

=4(d − 1)

2πi

∮C

f (w)erS(w)dw

where

f (w) =1

w√

(4d − w − 1/w)2 − 4

S(w) = log

4d − w − 1w −√(

4d − w − 1w)2 − 4

2wa

.

Use the steepest descent method to compute the asymptotics.

Contour integration gives the coefficients in the directionva = (1,a) for 0 < a < 1:

Gd (rva) =1

(2πi)2

∮Cw

∮Cz

4(d − 1)P(z,w)

dzzr+1

dwwar+1

=4(d − 1)

2πi

∮C

f (w)erS(w)dw

where

f (w) =1

w√

(4d − w − 1/w)2 − 4

S(w) = log

4d − w − 1w −√(

4d − w − 1w)2 − 4

2wa

.Use the steepest descent method to compute the asymptotics.

Steepest descent methodLet w+ be the real critical point of S(w) with w+ > 1Deform the contour of integration to pass through thecritical point and make the change of variablew = w+ + i y√r :

Gd (rva) =4(d − 1)

2πi

∮C

f (w)erS(w)dw

=4(d − 1)

2π√

rf (w+)erS(w+)

∫ ∞−∞

e−S′′(w+)y2

2 (1 + o(1))dy .

=4(d − 1)√2πS′′(w+)r

f (w+)erS(w+)(1 + o(1)).

Solving

Gd (rova) =4(d − 1)

nand Gd (riva) =

4dn.

gives the boundaries for the limit shape.

Steepest descent methodLet w+ be the real critical point of S(w) with w+ > 1Deform the contour of integration to pass through thecritical point and make the change of variablew = w+ + i y√r :

Gd (rva) =4(d − 1)

2πi

∮C

f (w)erS(w)dw

=4(d − 1)

2π√

rf (w+)erS(w+)

∫ ∞−∞

e−S′′(w+)y2

2 (1 + o(1))dy .

=4(d − 1)√2πS′′(w+)r

f (w+)erS(w+)(1 + o(1)).

Solving

Gd (rova) =4(d − 1)

nand Gd (riva) =

4dn.

gives the boundaries for the limit shape.

The limit shape for initial sandpile s0 = nδ(0,0) is parametrizedby

− log(n)(

1S(w+)

,a

S(w+)

)for 0 ≤ a ≤ 1,

and its reflections with respect to the coordinate axes and theline y = x .

Figure: Limit shapes with d = 1.05,2, and 1000.

Limit Shapes and AmoebaeThe amoeba of a polynomial P(z,w) is the image of{(z,w) ∈ C2 : P(z,w) = 0} under the map

(z,w) 7→ (log |z|, log |w |).

Figure: The boundary of the amoeba ofP(z,w) = 4d −

(z + z−1 + w + w−1

)and its dual curve. The red

curve bounds the gaseous phase.

DefinitionThe bounded complementary component of an amoeba is thegaseous phase.

Theorem (A.-Mkrtchyan 2020)The limit shape of the Leaky-ASM is (up to scale) the dual ofthe boundary of the gaseous phase in the amoeba.

For P(z,w) = 4d −(z + z−1 + w + w−1

)the boundary of

the gaseous phase is given by the implicit equation

4d = ex + e−x + ey + e−y with x , y ∈ R.

The boundary of the gaseous phase is z,w ∈ R with zw > 0.

The other boundary components correspond to zw < 0.

For P(z,w) = 4d −(z + z−1 + w + w−1

)the boundary of

For P(z,w) = 4d −(z + z−1 + w + w−1

)the boundary of

Why do amoebae appear?

Asymptotic level curves of

Gd (rva) =4(d − 1)

2πi

∮C

f (w)erS(w)dw

correspond to the limit shape.If the model has a spectral curve P(z,w) andS(w) = − ln(zwa) for (z,w) satisfying P(z,w) = 0 then theasymptotic level curves of Pd (rva) are given by theboundary of the gaseous phase in the amoeba.

Thank you!I. Alevy and S. Mkrtchyan, The Limit Shape of the LeakyAbelian Sandpile Model,arXiv e-prints , arXiv:2010.01946 (October 2020),2010.01946.

the limit shape of the leaky abelian sandpile modelmain results let s(x) = n (0;0)(x) and topple...

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