Tropical series, sandpiles and symplectic area

Nikita Kalinin (joint with Mikhail Shkolnikov)

University of Geneva

May 13, 2015

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 1 / 30 1 Tropical geometry

2 Sandpile models

3 Scaling limit

4 Harmonic functions

5 Symplectic area

6 Proofs

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 2 / 30 Tropical geometry

Tropical geometry: tropical analytic curves

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 3 / 30 A tropical curve is the set of non-smooth points of a tropical polynomial.

2 2 Motivation: let logt (x, y) = (logt (|x|), logt (|y|)), logt : C → R . P aij i j Consider a family of polynomials Ft (x, y) = (i,j)∈A t x y . 2 Define the corresponding curves Ct = {(x, y)|Ft (x, y) = 0} ⊂ C . Proposition

Then C = limt→0 logt Ct is the tropical curve defined by h(x, y) = min (aij + ix + jx). (i,j)∈A

Tropical geometry Tropical curves

Definition 2 Let A be a finite subset of Z .A tropical polynomial is a function 2 h(x, y) = min (aij + ix + jy), aij ∈ R. Note that h : R → R. (i,j)∈A

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 4 / 30 2 2 Motivation: let logt (x, y) = (logt (|x|), logt (|y|)), logt : C → R . P aij i j Consider a family of polynomials Ft (x, y) = (i,j)∈A t x y . 2 Define the corresponding curves Ct = {(x, y)|Ft (x, y) = 0} ⊂ C . Proposition

Then C = limt→0 logt Ct is the tropical curve defined by h(x, y) = min (aij + ix + jx). (i,j)∈A

Tropical geometry Tropical curves

Definition 2 Let A be a finite subset of Z .A tropical polynomial is a function 2 h(x, y) = min (aij + ix + jy), aij ∈ R. Note that h : R → R. (i,j)∈A A tropical curve is the set of non-smooth points of a tropical polynomial.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 4 / 30 P aij i j Consider a family of polynomials Ft (x, y) = (i,j)∈A t x y . 2 Define the corresponding curves Ct = {(x, y)|Ft (x, y) = 0} ⊂ C . Proposition

Then C = limt→0 logt Ct is the tropical curve defined by h(x, y) = min (aij + ix + jx). (i,j)∈A

Tropical geometry Tropical curves

Definition 2 Let A be a finite subset of Z .A tropical polynomial is a function 2 h(x, y) = min (aij + ix + jy), aij ∈ R. Note that h : R → R. (i,j)∈A A tropical curve is the set of non-smooth points of a tropical polynomial.

2 2 Motivation: let logt (x, y) = (logt (|x|), logt (|y|)), logt : C → R .

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 4 / 30 Proposition

Then C = limt→0 logt Ct is the tropical curve defined by h(x, y) = min (aij + ix + jx). (i,j)∈A

Tropical geometry Tropical curves

Definition 2 Let A be a finite subset of Z .A tropical polynomial is a function 2 h(x, y) = min (aij + ix + jy), aij ∈ R. Note that h : R → R. (i,j)∈A A tropical curve is the set of non-smooth points of a tropical polynomial.

2 2 Motivation: let logt (x, y) = (logt (|x|), logt (|y|)), logt : C → R . P aij i j Consider a family of polynomials Ft (x, y) = (i,j)∈A t x y . 2 Define the corresponding curves Ct = {(x, y)|Ft (x, y) = 0} ⊂ C .

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 4 / 30 Tropical geometry Tropical curves

Definition 2 Let A be a finite subset of Z .A tropical polynomial is a function 2 h(x, y) = min (aij + ix + jy), aij ∈ R. Note that h : R → R. (i,j)∈A A tropical curve is the set of non-smooth points of a tropical polynomial.

2 2 Motivation: let logt (x, y) = (logt (|x|), logt (|y|)), logt : C → R . P aij i j Consider a family of polynomials Ft (x, y) = (i,j)∈A t x y . 2 Define the corresponding curves Ct = {(x, y)|Ft (x, y) = 0} ⊂ C . Proposition

Then C = limt→0 logt Ct is the tropical curve defined by h(x, y) = min (aij + ix + jx). (i,j)∈A

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 4 / 30 A tropical analytic curve on Ω is the corner locus (i.e. set of non-smooth points) of a tropical series on Ω.

Proposition If Ω is a convex set then a tropical series f on it can be written as f (x, y)|Ω◦ = min (aij + ix + jy)|Ω◦ for some aij ∈ R ∪ {+∞}. (i,j)∈Z2

Tropical geometry Tropical series

Definition ◦ A tropical series on Ω is a function f :Ω → R which is locally a tropical polynomial.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 5 / 30 Proposition If Ω is a convex set then a tropical series f on it can be written as f (x, y)|Ω◦ = min (aij + ix + jy)|Ω◦ for some aij ∈ R ∪ {+∞}. (i,j)∈Z2

Tropical geometry Tropical series

Definition ◦ A tropical series on Ω is a function f :Ω → R which is locally a tropical polynomial. A tropical analytic curve on Ω is the corner locus (i.e. set of non-smooth points) of a tropical series on Ω.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 5 / 30 Tropical geometry Tropical series

Definition ◦ A tropical series on Ω is a function f :Ω → R which is locally a tropical polynomial. A tropical analytic curve on Ω is the corner locus (i.e. set of non-smooth points) of a tropical series on Ω.

Proposition If Ω is a convex set then a tropical series f on it can be written as f (x, y)|Ω◦ = min (aij + ix + jy)|Ω◦ for some aij ∈ R ∪ {+∞}. (i,j)∈Z2

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 5 / 30 Tropical geometry Example

The graph of a tropical series on the circle Ω = {(x, y)|x2 + y 2 ≤ 1} and the corresponding tropical analytic curve.

f (x, y) = min(i,j)∈Z2 (aij + ix + jy) where aij is − infΩ(ix + jy).

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 6 / 30 Sandpile models

Sandpile models: emergence of tropical objects

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 7 / 30 0 1

3 4 3 4 0 4 • • •

1 3 1 4 •

Sandpile models Sandpile model

Definition A sandpile is a collection of indistinguishable sand grains distributed 2 among a finite subset Γ of Z , that is a function ϕ :Γ → N0. A vertex v is unstable if ϕ(v) ≥ 4. An unstable vertex can topple by sending one grain of sand to each of 4 neighbours.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 8 / 30 Sandpile models Sandpile model

Definition A sandpile is a collection of indistinguishable sand grains distributed 2 among a finite subset Γ of Z , that is a function ϕ :Γ → N0. A vertex v is unstable if ϕ(v) ≥ 4. An unstable vertex can topple by sending one grain of sand to each of 4 neighbours.

0 1

3 4 3 4 0 4 • • •

1 3 1 4 •

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 8 / 30 Proposition Neither the number of topplings at a given vertex depends on the order.

Sandpile models Topplings, relaxation and toppling function

Definition The process of doing topplings while it is possible is called relaxation.

Proposition The final configuration after the relaxation does not depend on the order of topplings.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 9 / 30 Sandpile models Topplings, relaxation and toppling function

Definition The process of doing topplings while it is possible is called relaxation.

Proposition The final configuration after the relaxation does not depend on the order of topplings.

Proposition Neither the number of topplings at a given vertex depends on the order.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 9 / 30 Sandpile models 28 million grains dropped at one point

2 We start from the empty grid Z , add a lot of sand to (0, 0) and relax...

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 10 / 30 Sandpile models Hidden tropical curves

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 11 / 30 We start from the maximal stable distribution of sand, i.e. ϕ ≡ 3, then we add a grain to (0, 0). What is the result of relaxation?

White: 3 grains, Green: 2 grains, Yellow: 1 grain, Red: 0 grains

Sandpile models The problem

Definition We chose a boundary ∂Γ ⊂ Γ, where we never do topplings. For example, Γ = [−N..N] × [−N..N], ∂Γ is the set (i, j) where i 2 + j2 ≥ N2.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 12 / 30 White: 3 grains, Green: 2 grains, Yellow: 1 grain, Red: 0 grains

Sandpile models The problem

Definition We chose a boundary ∂Γ ⊂ Γ, where we never do topplings. For example, Γ = [−N..N] × [−N..N], ∂Γ is the set (i, j) where i 2 + j2 ≥ N2.

We start from the maximal stable distribution of sand, i.e. ϕ ≡ 3, then we add a grain to (0, 0). What is the result of relaxation?

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 12 / 30 Sandpile models The problem

Definition We chose a boundary ∂Γ ⊂ Γ, where we never do topplings. For example, Γ = [−N..N] × [−N..N], ∂Γ is the set (i, j) where i 2 + j2 ≥ N2.

We start from the maximal stable distribution of sand, i.e. ϕ ≡ 3, then we add a grain to (0, 0). What is the result of relaxation?

White: 3 grains, Green: 2 grains, Yellow: 1 grain, Red: 0 grains Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 12 / 30 Sandpile models Sandpile on a disk: a point in the center

Extra grain at the center The grid is 3 times finer The grid is 9 times finer

White: 3 grains, Green: 2 grains, Yellow: 1 grain, Red: 0 grains

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 13 / 30 Sandpile models The origin of this problem: works by S. Caracciolo, G. Paoletti and A. Sportiello

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 14 / 30 Sandpile models Sandpile on a disk: several points

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 15 / 30 Scaling limit

Scaling limits

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 16 / 30 Definition

Let CN be the set of vertices whereϕ ˜N 6= 3.

Theorem 2 If Ω is not R and ∂Ω does not contain a line with irrational slope, then 1 Ω the sequence of sets N CN converges to a tropical analytic curve Cp¯ ⊂ Ω, passing through the points p1, p2,..., pn.

Scaling limit The existence of a scaling limit

2 Consider a convex closed set Ω ⊂ R and a collection of distinct points p¯ = {p1, ..., pn} in the interior of Ω. For each integer N > 0 we define a 2 graph ΓN as the subgraph of Z bounded by N · Ω. Define the state ϕN as result of adding extra grains of sand at the vertices [Npi ] to the maximal stable state. Letϕ ˜N be the result of relaxation for the state ϕN .

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 17 / 30 Scaling limit The existence of a scaling limit

2 Consider a convex closed set Ω ⊂ R and a collection of distinct points p¯ = {p1, ..., pn} in the interior of Ω. For each integer N > 0 we define a 2 graph ΓN as the subgraph of Z bounded by N · Ω. Define the state ϕN as result of adding extra grains of sand at the vertices [Npi ] to the maximal stable state. Letϕ ˜N be the result of relaxation for the state ϕN . Definition

Let CN be the set of vertices whereϕ ˜N 6= 3.

Theorem 2 If Ω is not R and ∂Ω does not contain a line with irrational slope, then 1 Ω the sequence of sets N CN converges to a tropical analytic curve Cp¯ ⊂ Ω, passing through the points p1, p2,..., pn.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 17 / 30 Theorem Ω The function FN converges to a tropical series Gp¯ vanishing at ∂Ω. Ω Ω Moreover, Cp¯ ⊂ Ω is defined by Gp¯ .

Scaling limit Toppling function

Let hN be the toppling function of the relaxation process ϕN → ϕ˜N , i.e. 2 the value of hN at v ∈ Z is the number of topplings at v during the 1 relaxation. Define a function FN :Ω → R by FN (x) = N hN ([Nx]).

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 18 / 30 Scaling limit Toppling function

Let hN be the toppling function of the relaxation process ϕN → ϕ˜N , i.e. 2 the value of hN at v ∈ Z is the number of topplings at v during the 1 relaxation. Define a function FN :Ω → R by FN (x) = N hN ([Nx]). Theorem Ω The function FN converges to a tropical series Gp¯ vanishing at ∂Ω. Ω Ω Moreover, Cp¯ ⊂ Ω is defined by Gp¯ .

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 18 / 30 Harmonic functions

Harmonic functions

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 19 / 30 Proof. Indeed, count the number of incoming and outgoing grains.

That is, h is harmonic at (x, y). In fact, h(x, y) is a “piece-wise” linear function.

Harmonic functions Explanation: look at the number of topplings

Consider the number h(x, y) of topplings at a point (x, y) during the relaxation. Proposition If the number of sand grains at (x, y) after relaxation is the same as before, then h(x − 1, y) + h(x + 1, y) + h(x, y − 1) + h(x, y + 1) − 4h(x, y) = 0.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 20 / 30 Harmonic functions Explanation: look at the number of topplings

Consider the number h(x, y) of topplings at a point (x, y) during the relaxation. Proposition If the number of sand grains at (x, y) after relaxation is the same as before, then h(x − 1, y) + h(x + 1, y) + h(x, y − 1) + h(x, y + 1) − 4h(x, y) = 0.

Proof. Indeed, count the number of incoming and outgoing grains.

That is, h is harmonic at (x, y). In fact, h(x, y) is a “piece-wise” linear function.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 20 / 30 Hence, coloured vertices constitute a tropical curve!

Harmonic functions Conclusions

Recap: The function h of topplings is (almost everywhere) a piece-wise linear (Theorem). On its linear parts h is harmonic, therefore we have 3 grains there. Finally, we have 0,1,2 grains in a neighbourhoods of non-linearity locus.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 21 / 30 Harmonic functions Conclusions

Recap: The function h of topplings is (almost everywhere) a piece-wise linear (Theorem). On its linear parts h is harmonic, therefore we have 3 grains there. Finally, we have 0,1,2 grains in a neighbourhoods of non-linearity locus.

Hence, coloured vertices constitute a tropical curve! Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 21 / 30 Symplectic area

Tropical symplectic area

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 22 / 30 Remark It is sort of Least Action Principle: sandpiles are “lazy”.

Symplectic area Limiting curve minimizes a functional

Denote by V (Ω, p¯) the set of tropical series F on Ω such that F |Ω ≥ 0 and F is not smooth at each of the points pi . Theorem The functional Z F 7→ Fdxdy Ω defined on the space V (Ω, p¯) has the unique minimum which coincides Ω with the function Gp¯ .

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 23 / 30 Symplectic area Limiting curve minimizes a functional

Denote by V (Ω, p¯) the set of tropical series F on Ω such that F |Ω ≥ 0 and F is not smooth at each of the points pi . Theorem The functional Z F 7→ Fdxdy Ω defined on the space V (Ω, p¯) has the unique minimum which coincides Ω with the function Gp¯ .

Remark It is sort of Least Action Principle: sandpiles are “lazy”.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 23 / 30 pq Let v = (p, q) be a primitive integer vector. Let Ct be the lift of the ∗ 2 interval [0, v] to the torus (C ) under Logt , i.e. pq p q Ct = {(z , z )|1 ≤ |z| ≤ t}. Then Z 2 2 d log(z1) ∧ d log(¯z1) = −4iπ p log t. pq Ct

Theorem (N. K.,M. S. (arXiv:1502.06284)) Ω If Ω is a polygon with sides of rational slopes then Cp¯ minimizes the tropical symplectic area.

Symplectic area Sandpile on a polygon with rational slopes: symplectic area

Definition 2 The tropical symplectic area of an edge e (of a direction (p, q) ∈ Z and multiplicity m(e)) of a tropical curve is given by EuclideanLength(e) · m(e) · pp2 + q2.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 24 / 30 Theorem (N. K.,M. S. (arXiv:1502.06284)) Ω If Ω is a polygon with sides of rational slopes then Cp¯ minimizes the tropical symplectic area.

Symplectic area Sandpile on a polygon with rational slopes: symplectic area

Definition 2 The tropical symplectic area of an edge e (of a direction (p, q) ∈ Z and multiplicity m(e)) of a tropical curve is given by EuclideanLength(e) · m(e) · pp2 + q2.

pq Let v = (p, q) be a primitive integer vector. Let Ct be the lift of the ∗ 2 interval [0, v] to the torus (C ) under Logt , i.e. pq p q Ct = {(z , z )|1 ≤ |z| ≤ t}. Then Z 2 2 d log(z1) ∧ d log(¯z1) = −4iπ p log t. pq Ct

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 24 / 30 Symplectic area Sandpile on a polygon with rational slopes: symplectic area

Definition 2 The tropical symplectic area of an edge e (of a direction (p, q) ∈ Z and multiplicity m(e)) of a tropical curve is given by EuclideanLength(e) · m(e) · pp2 + q2.

pq Let v = (p, q) be a primitive integer vector. Let Ct be the lift of the ∗ 2 interval [0, v] to the torus (C ) under Logt , i.e. pq p q Ct = {(z , z )|1 ≤ |z| ≤ t}. Then Z 2 2 d log(z1) ∧ d log(¯z1) = −4iπ p log t. pq Ct

Theorem (N. K.,M. S. (arXiv:1502.06284)) Ω If Ω is a polygon with sides of rational slopes then Cp¯ minimizes the tropical symplectic area.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 24 / 30 Symplectic area Symplectic area for tropical analytic curves

Remark Symplectic area of a tropical analytic curve in general is infinite.

Question: How can we properly state the problem of minimization of tropical symplectic area on an arbitrary convex domain? Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 25 / 30 Proofs

Ideas of proofs

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 26 / 30 The toppling function h is the pointwise minimal integer valued function with the conditions: h|∂Ω = 0,

∆h < 0 at p1, p2,..., pn,

∆h ≤ 0, elsewhere

Proofs Proofs: superharmonic functions

Definition f (x+1,y)+f (x−1,y)+f (x,y+1)+f (x,y−1) Let (∆f )(x, y) = 4 − f (x, y)

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 27 / 30 Proofs Proofs: superharmonic functions

Definition f (x+1,y)+f (x−1,y)+f (x,y+1)+f (x,y−1) Let (∆f )(x, y) = 4 − f (x, y) The toppling function h is the pointwise minimal integer valued function with the conditions: h|∂Ω = 0,

∆h < 0 at p1, p2,..., pn,

∆h ≤ 0, elsewhere

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 27 / 30 R R R Ω ∆h = ∂Ω h ≤ ∂Ω l = something small, therefore the number of coloured vertices is small. now, ∆h = 0 and less than a linear function, therefore, h is linear almost everywhere.

Proofs The toppling function is linear almost everywhere

Estimate the toppling function h from above by a piecewise linear function l,

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 28 / 30 now, ∆h = 0 and less than a linear function, therefore, h is linear almost everywhere.

Proofs The toppling function is linear almost everywhere

Estimate the toppling function h from above by a piecewise linear function l, R R R Ω ∆h = ∂Ω h ≤ ∂Ω l = something small, therefore the number of coloured vertices is small.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 28 / 30 therefore, h is linear almost everywhere.

Proofs The toppling function is linear almost everywhere

Estimate the toppling function h from above by a piecewise linear function l, R R R Ω ∆h = ∂Ω h ≤ ∂Ω l = something small, therefore the number of coloured vertices is small. now, ∆h = 0 and less than a linear function,

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 28 / 30 Proofs The toppling function is linear almost everywhere

Estimate the toppling function h from above by a piecewise linear function l, R R R Ω ∆h = ∂Ω h ≤ ∂Ω l = something small, therefore the number of coloured vertices is small. now, ∆h = 0 and less than a linear function, therefore, h is linear almost everywhere.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 28 / 30 Suppose the toppling function was h(x, y) = min(aij + ix + jy). Let us make a wave from a point in a part where a + ix + jy dominates. This results as a → a + 1. So, we have better than a scaling limit, because we have a continuous dynamic on tropical series, and sand pictures are its discrete approximations.

Proofs Proofs: waves and sand dynamics

We decompose the relaxation process on waves.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 29 / 30 Let us make a wave from a point in a part where a + ix + jy dominates. This results as a → a + 1. So, we have better than a scaling limit, because we have a continuous dynamic on tropical series, and sand pictures are its discrete approximations.

Proofs Proofs: waves and sand dynamics

We decompose the relaxation process on waves.

Suppose the toppling function was h(x, y) = min(aij + ix + jy).

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 29 / 30 So, we have better than a scaling limit, because we have a continuous dynamic on tropical series, and sand pictures are its discrete approximations.

Proofs Proofs: waves and sand dynamics

We decompose the relaxation process on waves.

Suppose the toppling function was h(x, y) = min(aij + ix + jy). Let us make a wave from a point in a part where a + ix + jy dominates. This results as a → a + 1.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 29 / 30 Proofs Proofs: waves and sand dynamics

We decompose the relaxation process on waves.

Suppose the toppling function was h(x, y) = min(aij + ix + jy). Let us make a wave from a point in a part where a + ix + jy dominates. This results as a → a + 1. So, we have better than a scaling limit, because we have a continuous dynamic on tropical series, and sand pictures are its discrete approximations.

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 29 / 30 Proofs

Thank you for your attention!

Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 30 / 30