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Tropical Series, Sandpiles and Symplectic Area

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Tropical Series, Sandpiles and Symplectic Area 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 (jxj); logt (jyj)), logt : C ! R : P aij i j Consider a family of polynomials Ft (x; y) = (i;j)2A t x y : 2 Define the corresponding curves Ct = f(x; y)jFt (x; y) = 0g ⊂ C . Proposition Then C = limt!0 logt Ct is the tropical curve defined by h(x; y) = min (aij + ix + jx). (i;j)2A 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 2 R. Note that h : R ! R. (i;j)2A Nikita Kalinin (University of Geneva) Tropical series, sandpiles and symplectic area 4 / 30 2 2 Motivation: let logt (x; y) = (logt (jxj); logt (jyj)), logt : C ! R : P aij i j Consider a family of polynomials Ft (x; y) = (i;j)2A t x y : 2 Define the corresponding curves Ct = f(x; y)jFt (x; y) = 0g ⊂ C . Proposition Then C = limt!0 logt Ct is the tropical curve defined by h(x; y) = min (aij + ix + jx). (i;j)2A 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 2 R. Note that h : R ! R. (i;j)2A 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)2A t x y : 2 Define the corresponding curves Ct = f(x; y)jFt (x; y) = 0g ⊂ C . Proposition Then C = limt!0 logt Ct is the tropical curve defined by h(x; y) = min (aij + ix + jx). (i;j)2A 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 2 R. Note that h : R ! R. (i;j)2A A tropical curve is the set of non-smooth points of a tropical polynomial. 2 2 Motivation: let logt (x; y) = (logt (jxj); logt (jyj)), 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)2A 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 2 R. Note that h : R ! R. (i;j)2A A tropical curve is the set of non-smooth points of a tropical polynomial. 2 2 Motivation: let logt (x; y) = (logt (jxj); logt (jyj)), logt : C ! R : P aij i j Consider a family of polynomials Ft (x; y) = (i;j)2A t x y : 2 Define the corresponding curves Ct = f(x; y)jFt (x; y) = 0g ⊂ 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 2 R. Note that h : R ! R. (i;j)2A A tropical curve is the set of non-smooth points of a tropical polynomial. 2 2 Motivation: let logt (x; y) = (logt (jxj); logt (jyj)), logt : C ! R : P aij i j Consider a family of polynomials Ft (x; y) = (i;j)2A t x y : 2 Define the corresponding curves Ct = f(x; y)jFt (x; y) = 0g ⊂ C . Proposition Then C = limt!0 logt Ct is the tropical curve defined by h(x; y) = min (aij + ix + jx). (i;j)2A 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)jΩ◦ = min (aij + ix + jy)jΩ◦ for some aij 2 R [ f+1g. (i;j)2Z2 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)jΩ◦ = min (aij + ix + jy)jΩ◦ for some aij 2 R [ f+1g. (i;j)2Z2 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)jΩ◦ = min (aij + ix + jy)jΩ◦ for some aij 2 R [ f+1g. (i;j)2Z2 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 = f(x; y)jx2 + y 2 ≤ 1g and the corresponding tropical analytic curve. f (x; y) = min(i;j)2Z2 (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 ' :Γ ! N0. A vertex v is unstable if '(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 ' :Γ ! N0. A vertex v is unstable if '(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. ' ≡ 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. ' ≡ 3, then we add a grain to (0; 0).
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