The Limit Shape of the Leaky Abelian Sandpile Model Ian M. Alevy - - PowerPoint PPT Presentation

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 = Z2 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 = Z2 but we will consider different toppling rules: Directed ASM +1 −2

• +1

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 = Z2 but we will consider different toppling rules: 1D ASM +1 −2

• +1

Directed ASM +1 −2

• +1

Uniform ASM +1 +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) = nδ(0,0)(v) topple until reaching a stable sandpile s∞. Question What is the stable sandpile? Toppling rule +1 −2

• +1

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

1-Dimensional ASM

Start with initial sandpile s(v) = nδ(0,0)(v) topple until reaching a stable sandpile s∞. Question What is the stable sandpile? Toppling rule +1 −2

• +1

7

Figure: Initial sandpile with n = 7.

1 5 1

Figure: Result after toppling at the origin.

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

Sequence of topplings

2 3 2

Figure: Origin toppled again.

1 1 3 1 1

Figure: All unstable sites topple once more.

some more topples....

1 1 3 1 1

and the stable sandpile:

1 1 1 1 1 1 1

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) = nδ(0,0)(x, y) then the stable sandpile for the 1D ASM is s∞(x, 0) =            1 if x = 0 and n is odd, if x = 0 and n is even, 1 if 0 < |x| ≤ ⌊n

2⌋,

if ⌊ n

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) = nδ(0,0)(x, y) then the stable sandpile for the 1D ASM is s∞(x, 0) =            1 if x = 0 and n is odd, if x = 0 and n is even, 1 if 0 < |x| ≤ ⌊n

2⌋,

if ⌊ n

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) = nδ(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 = 107. Colors correspond to heights of sandpile.

2D ASM

Let s(x, y) = nδ(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 between 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 of sandpile.

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 have

• ne grain of sand.

Toppling rule +1 −2

• +1

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 unstable and topples distributing sand to its neighbors:

• s(v) → s(v) − d · deg(v)

s(u) → s(u) + 1 if u ∼ v. Uniform ASM with dissipation +1 +1 −4d

• +1

+1

Main Results

Let s(v) = nδ(0,0)(v) and topple until stable using the uniform toppling rule. Dn,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 −1Dn,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) = nδ(0,0)(v) and topple until stable using the uniform toppling rule. Dn,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 −1Dn,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 dn = 1 + tn. If tn ≍

1 log(n) then the boundary of

√tn log(n)Dn,d converges to a circle. If tn ≍

1 n1−α with 0 < α < 1, then the boundary of

√tn log(n)Dn,d is between circles of radii c1 and c2 with c1

c2 →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 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

• f sand is added. All unstable sites topple. The distribution
• f 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

• f sand is added. All unstable sites topple. The distribution
• f 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

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

• f sand is added. All unstable sites topple. The distribution
• f 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).

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.

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 the asymptotic death probability.

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 the asymptotic death probability. Level curves of 4 n and 4(d − 1) n in the death probability bound the Leaky-ASM with n chips started at the origin.

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

Killed random walk

Let X1, X2, . . . be i.i.d random variables with P{Xj = (1, 0)} = 1 4d , P{Xj = (−1, 0)} = 1 4d , P{Xj = (0, 1)} = 1 4d , P{Xj = (0, −1)} = 1 4d , P{Xj = (0, 0)} = 1 − 4 4d = 1 − 1 d . The killed random walk (KRW) started at v ∈ Z2 is the sequence S1, S2, . . . where Sn = v +

n

• i=1

KiXi and Ki =

• 1

if the walker is alive at step i else.

Connection to sandpiles

Let Pd(v) = P(walker dies at v) be the death probability. Definition The odometer function u(v) = total sand emitted from v.

Connection to sandpiles

Let Pd(v) = P(walker dies at v) be the death probability. Definition The odometer function u(v) = total sand emitted from v. Start with initial sandpile s(v) = nδ0,0(v) and topple until reaching the stable sandpile s∞(v).

Connection to sandpiles

Let Pd(v) = P(walker dies at v) be the death probability. Definition The odometer function u(v) = total sand emitted from v. Start with initial sandpile s(v) = nδ0,0(v) and topple until reaching the stable sandpile s∞(v). Proposition (A.-Mkrtchyan 2020) For the operator T = 1 d ∆ − d − 1 d

• I

we have T(u(v) − Pd(v)) = d − 1 dn s∞(v).

Key lemma

“Invert” T = 1 d ∆ − d − 1 d

• I

and use inequality 0 ≤ s∞(v) < 4d to obtain the key lemma: Lemma (A.-Mkrtchyan 2020)

1

If Pd(v) < 4(d − 1) n , then u(v) = 0, i.e. v ∈ Dn,d.

2

If Pd(v) ≥ 4d n , then u(v) ≥ 4d, i.e. v ∈ Dn,d. Dn,d is the set of sites which topple.

Key lemma

“Invert” T = 1 d ∆ − d − 1 d

• I

and use inequality 0 ≤ s∞(v) < 4d to obtain the key lemma: Lemma (A.-Mkrtchyan 2020)

1

If Pd(v) < 4(d − 1) n , then u(v) = 0, i.e. v ∈ Dn,d.

2

If Pd(v) ≥ 4d n , then u(v) ≥ 4d, i.e. v ∈ Dn,d. Dn,d is the set of sites which topple. Consequence Asymptotics of Pd(v) give the boundary of the limit shape.

Asymptotic death probability

The spectral curve of the random walker is P(z, w) = 4d − z − z−1 − w − w−1 4(d − 1)

Asymptotic death probability

The spectral curve of the random walker is P(z, w) = 4d − z − z−1 − w − w−1 4(d − 1) Expand P−1(z, w) as a power series to compute probabilities: P−1(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∈Z

Pd(k, l)zkwl, where Pd(k, l) is the probability the walker dies at (k, l).

Contour integration gives the coefficients along a ray: Pd(r, ar) = 1 (2πi)2

• Cw
• Cz

P−1(z, w) dz zr+1 dw war+1 = 4(d − 1) 2πi

• C

G(w)erS(w)dw where G(w) = 1 w

• (4d − w − 1/w)2 − 4

S(w) = log  4d − w − 1

w −

• 4d − w − 1

w

2 − 4 2wa   .

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

Contour integration gives the coefficients along a ray: Pd(r, ar) = 1 (2πi)2

• Cw
• Cz

P−1(z, w) dz zr+1 dw war+1 = 4(d − 1) 2πi

• C

G(w)erS(w)dw where G(w) = 1 w

• (4d − w − 1/w)2 − 4

S(w) = log  4d − w − 1

w −

• 4d − w − 1

w

2 − 4 2wa   . Use the steepest descent method to compute the asymptotics.

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

Steepest descent method

Let w+ be the real critical point of S(w) with w+ > 1 Deform the contour of integration to pass through the critical point and make the change of variable w = w+ + i y

√r :

Pd(r, ar) = 4(d − 1) 2πi

• C

G(w)erS(w)dw = 4(d − 1) 2π√r G(w+)erS(w+) ∞

−∞

e− S′′(w+)y2

2

(1 + o(1))dy. = 4(d − 1)

• 2πS′′(w+)r

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

Steepest descent method

Let w+ be the real critical point of S(w) with w+ > 1 Deform the contour of integration to pass through the critical point and make the change of variable w = w+ + i y

√r :

Pd(r, ar) = 4(d − 1) 2πi

• C

G(w)erS(w)dw = 4(d − 1) 2π√r G(w+)erS(w+) ∞

−∞

e− S′′(w+)y2

2

(1 + o(1))dy. = 4(d − 1)

• 2πS′′(w+)r

G(w+)erS(w+)(1 + o(1)). Solving Pd(ro, aro) = 4(d − 1) n and Pd(ri, ari) = 4d n . gives the boundaries for the limit shape.

The limit shape for initial sandpile s0 = nδ(0,0) is parametrized by − log(n)

• 1

S(w+), a S(w+)

• for 0 ≤ a ≤ 1,

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

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

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

Limit Shapes and Amoebae

The amoeba of a polynomial P(z, w) is the image of {(z, w) ∈ C2 : P(z, w) = 0} under the map (z, w) → (log |z|, log |w|).

Figure: The boundary of the amoeba of P(z, w) = 4d−z−z−1−w−w−1

4(d−1)

and its dual curve. The red curve bounds the gaseous phase.

Definition The bounded complementary component of an amoeba is the gaseous phase.

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

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

Theorem (A.-Mkrtchyan 2020) The limit shape of the Leaky-ASM is (up to scale) the dual of the boundary of the gaseous phase in the amoeba. For P(z, w) = 4d−z−z−1−w−w−1

4(d−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.

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

Theorem (A.-Mkrtchyan 2020) The limit shape of the Leaky-ASM is (up to scale) the dual of the boundary of the gaseous phase in the amoeba. For P(z, w) = 4d−z−z−1−w−w−1

4(d−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.

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

Why do amoebae appear?

Asymptotic level curves of Pd(r, ar) = 4(d − 1) 2πi

• C

G(w)erS(w)dw correspond to the limit shape. If the model has a spectral curve P(z, w) and S(w) = − ln(zwa) for (z, w) satisfying P(z, w) = 0 then the asymptotic level curves of Pd(r, ar) are given by the boundary of the gaseous phase in the amoeba.

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

Thank you!

• I. Alevy and S. Mkrtchyan, The Limit Shape of the Leaky

Abelian Sandpile Model, arXiv e-prints , arXiv:2010.01946 (October 2020), 2010.01946.

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