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Random intefaces, geodesics and the directed landscape B alint Vir ag, University of Toronto with Duncan Dauvergne, Janosch Ortmann and Mihai Nica Palac Bedlewo May 21, 2019 B alint Vir ag Directed landscape 5/21/2019 1 / 45

  1. Poisson last passage percolation notation R 4 L : R 4 ↑ = { ( x , s ; y , t ) : s < t } , ↑ → N metric composition s < t < u L ( x , s ; z , u ) = max y L ( x , s ; y , t ) + L ( y , t ; z , u ) L is a “directed metric” :( wrong sign, asymmetry :) triangle inequality, geodesics Perelman’s L -distance B´ alint Vir´ ag Directed landscape 5/21/2019 16 / 45

  2. Poisson last passage percolation notation R 4 L : R 4 ↑ = { ( x , s ; y , t ) : s < t } , ↑ → N metric composition s < t < u L ( x , s ; z , u ) = max y L ( x , s ; y , t ) + L ( y , t ; z , u ) L is a “directed metric” :( wrong sign, asymmetry :) triangle inequality, geodesics Perelman’s L -distance B´ alint Vir´ ag Directed landscape 5/21/2019 16 / 45

  3. Poisson last passage percolation notation R 4 L : R 4 ↑ = { ( x , s ; y , t ) : s < t } , ↑ → N metric composition s < t < u L ( x , s ; z , u ) = max y L ( x , s ; y , t ) + L ( y , t ; z , u ) L is a “directed metric” :( wrong sign, asymmetry :) triangle inequality, geodesics Perelman’s L -distance B´ alint Vir´ ag Directed landscape 5/21/2019 16 / 45

  4. The metric composition semigroup M Elements: a , b : R 2 → R ∪ { + ∞} . Composition: a ⋆ b ( x , z ) = sup y a ( x , y ) + b ( y , z ) . Example: Fix t . a t ( x , y ) := −� ( x , 0) − ( y , t ) � 2 Then a t ⋆ a s = a s + t . Works for any path metric on R 2 B´ alint Vir´ ag Directed landscape 5/21/2019 17 / 45

  5. The metric composition semigroup M Elements: a , b : R 2 → R ∪ { + ∞} . Composition: a ⋆ b ( x , z ) = sup y a ( x , y ) + b ( y , z ) . Example: Fix t . a t ( x , y ) := −� ( x , 0) − ( y , t ) � 2 Then a t ⋆ a s = a s + t . Works for any path metric on R 2 B´ alint Vir´ ag Directed landscape 5/21/2019 17 / 45

  6. The metric composition semigroup M Elements: a , b : R 2 → R ∪ { + ∞} . Composition: a ⋆ b ( x , z ) = sup y a ( x , y ) + b ( y , z ) . Example: Fix t . a t ( x , y ) := −� ( x , 0) − ( y , t ) � 2 Then a t ⋆ a s = a s + t . Works for any path metric on R 2 B´ alint Vir´ ag Directed landscape 5/21/2019 17 / 45

  7. The metric composition semigroup M Elements: a , b : R 2 → R ∪ { + ∞} . Composition: a ⋆ b ( x , z ) = sup y a ( x , y ) + b ( y , z ) . Example: Fix t . a t ( x , y ) := −� ( x , 0) − ( y , t ) � 2 Then a t ⋆ a s = a s + t . Works for any path metric on R 2 B´ alint Vir´ ag Directed landscape 5/21/2019 17 / 45

  8. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is a stationary independent increment process on M . Random walk on M (Levy process) In semigroups, two time parameters are needed to document increments L s , t = L ( · , s ; · , t ). Groups, no: X t − X s = ( X t − X 0 ) − ( X s − X 0 ) B´ alint Vir´ ag Directed landscape 5/21/2019 18 / 45

  9. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is a stationary independent increment process on M . Random walk on M (Levy process) In semigroups, two time parameters are needed to document increments L s , t = L ( · , s ; · , t ). Groups, no: X t − X s = ( X t − X 0 ) − ( X s − X 0 ) B´ alint Vir´ ag Directed landscape 5/21/2019 18 / 45

  10. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is a stationary independent increment process on M . Random walk on M (Levy process) In semigroups, two time parameters are needed to document increments L s , t = L ( · , s ; · , t ). Groups, no: X t − X s = ( X t − X 0 ) − ( X s − X 0 ) B´ alint Vir´ ag Directed landscape 5/21/2019 18 / 45

  11. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is a stationary independent increment process on M . Random walk on M (Levy process) In semigroups, two time parameters are needed to document increments L s , t = L ( · , s ; · , t ). Groups, no: X t − X s = ( X t − X 0 ) − ( X s − X 0 ) B´ alint Vir´ ag Directed landscape 5/21/2019 18 / 45

  12. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is random walk on M What is Brownian motion on M ? What is a Gaussian on M ? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

  13. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is random walk on M What is Brownian motion on M ? What is a Gaussian on M ? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

  14. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is random walk on M What is Brownian motion on M ? What is a Gaussian on M ? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

  15. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is random walk on M What is Brownian motion on M ? What is a Gaussian on M ? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

  16. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is random walk on M What is Brownian motion on M ? What is a Gaussian on M ? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

  17. Poisson last passage, algebraically M = ( { a : R 2 → R ∪ { + ∞}} , ⋆ ) . Poisson LP is random walk on M What is Brownian motion on M ? What is a Gaussian on M ? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

  18. Last passage across functions Definition For a sequence of functions f k and n ≤ m , s < t define the last passage value as � f [( s , m ) → ( t , n )] = sup f k ( v ) − f k ( u ) π k ∈ Z : π − 1 ( k ) o =( u , v ) over nonincreasing π : [ s , t ] → Z with π ( s ) = m , π ( t ) = n . k → for k nonintersecting paths. B´ alint Vir´ ag Directed landscape 5/21/2019 20 / 45

  19. Last passage across functions Definition For a sequence of functions f k and n ≤ m , s < t define the last passage value as � f [( s , m ) → ( t , n )] = sup f k ( v ) − f k ( u ) π k ∈ Z : π − 1 ( k ) o =( u , v ) over nonincreasing π : [ s , t ] → Z with π ( s ) = m , π ( t ) = n . k → for k nonintersecting paths. B´ alint Vir´ ag Directed landscape 5/21/2019 20 / 45

  20. Brownian last passage π : [ s , t ] → Z nonincreasing, π ( s ) = 4 , π ( t ) = 1 f are BMs some symmetry lost, some gained B´ alint Vir´ ag Directed landscape 5/21/2019 21 / 45

  21. RSK: the melon Wf of f Fix n , f k : R + → R , k ∈ { 1 , . . . , n } , Define the melon Wf by k Wf 1 ( y ) + . . . + Wf k ( y ) = f [(0 , n ) → ( y , 1)] One of two parts of a continuous RSK Key property: for all x < y f [( x , n ) → ( y , 1)] = Wf [( x , n ) → ( y , 1)] . B´ alint Vir´ ag Directed landscape 5/21/2019 22 / 45

  22. O’Connell-Yor, 2002 W applied to B 1 , . . . , B n . WB has the law of n Brownian motions conditioned not to intersect. Proof, essentially: RSK is a bijection Pre-limit Airy sheet: ( x , y ) �→ WB [( x , n ) → ( y , 1)] . B´ alint Vir´ ag Directed landscape 5/21/2019 23 / 45

  23. O’Connell-Yor, 2002 W applied to B 1 , . . . , B n . WB has the law of n Brownian motions conditioned not to intersect. Proof, essentially: RSK is a bijection Pre-limit Airy sheet: ( x , y ) �→ WB [( x , n ) → ( y , 1)] . B´ alint Vir´ ag Directed landscape 5/21/2019 23 / 45

  24. O’Connell-Yor, 2002 W applied to B 1 , . . . , B n . WB has the law of n Brownian motions conditioned not to intersect. Proof, essentially: RSK is a bijection Pre-limit Airy sheet: ( x , y ) �→ WB [( x , n ) → ( y , 1)] . B´ alint Vir´ ag Directed landscape 5/21/2019 23 / 45

  25. The Brownian melon The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top. B´ alint Vir´ ag Directed landscape 5/21/2019 24 / 45

  26. The Brownian melon The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top. B´ alint Vir´ ag Directed landscape 5/21/2019 24 / 45

  27. The Brownian melon The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top. B´ alint Vir´ ag Directed landscape 5/21/2019 24 / 45

  28. The Brownian melon The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top. B´ alint Vir´ ag Directed landscape 5/21/2019 24 / 45

  29. The Brownian melon The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top. B´ alint Vir´ ag Directed landscape 5/21/2019 24 / 45

  30. The Airy line ensemble A · ( t ) + t 2 / 4 is stationary A k (0) ∼ − (3 π k / 2) 2 / 3 B´ alint Vir´ ag Directed landscape 5/21/2019 25 / 45

  31. The Airy line ensemble A · ( t ) + t 2 / 4 is stationary A k (0) ∼ − (3 π k / 2) 2 / 3 B´ alint Vir´ ag Directed landscape 5/21/2019 25 / 45

  32. The Airy line ensemble Theorem Let WB be a Brownian n-melon. Define the rescaled melon by i ( t ) = n 1 / 6 ( WB i (1 + tn − 1 / 3 ) − 2 √ n − tn 1 / 6 ) . A n Then A n converges in law to a random sequence functions A , the Airy line ensemble. Proof: formulas + tightness. (Prahofer-Spohn, Adler-Van Moerbeke, Corwin-Hammond). B´ alint Vir´ ag Directed landscape 5/21/2019 26 / 45

  33. The Airy line ensemble Theorem Let WB be a Brownian n-melon. Define the rescaled melon by i ( t ) = n 1 / 6 ( WB i (1 + tn − 1 / 3 ) − 2 √ n − tn 1 / 6 ) . A n Then A n converges in law to a random sequence functions A , the Airy line ensemble. Proof: formulas + tightness. (Prahofer-Spohn, Adler-Van Moerbeke, Corwin-Hammond). B´ alint Vir´ ag Directed landscape 5/21/2019 26 / 45

  34. Defintion of the Airy sheet S A random continuous function so that S has the same law as S ( · + z , · + z ). S can be coupled with an Airy line ensemble so that S (0 , · ) = A 1 ( · ) and for all ( x , y , z ) ∈ Q + × Q 2 a.s. S ( x , z ) − S ( x , y ) = � A [( − k / 2 x , k ) → ( z , 1)] − � A [( − k / 2 x , k ) → ( y , 1)] for all large enough k . B´ alint Vir´ ag Directed landscape 5/21/2019 27 / 45

  35. Defintion of the Airy sheet S A random continuous function so that S has the same law as S ( · + z , · + z ). S can be coupled with an Airy line ensemble so that S (0 , · ) = A 1 ( · ) and for all ( x , y , z ) ∈ Q + × Q 2 a.s. S ( x , z ) − S ( x , y ) = � A [( − k / 2 x , k ) → ( z , 1)] − � A [( − k / 2 x , k ) → ( y , 1)] for all large enough k . B´ alint Vir´ ag Directed landscape 5/21/2019 27 / 45

  36. Defintion of the Airy sheet S A random continuous function so that S has the same law as S ( · + z , · + z ). S can be coupled with an Airy line ensemble so that S (0 , · ) = A 1 ( · ) and for all ( x , y , z ) ∈ Q + × Q 2 a.s. S ( x , z ) − S ( x , y ) = � A [( − k / 2 x , k ) → ( z , 1)] − � A [( − k / 2 x , k ) → ( y , 1)] for all large enough k . B´ alint Vir´ ag Directed landscape 5/21/2019 27 / 45

  37. Defintion of the Airy sheet S A random continuous function so that S has the same law as S ( · + z , · + z ). S can be coupled with an Airy line ensemble so that S (0 , · ) = A 1 ( · ) and for all ( x , y , z ) ∈ Q + × Q 2 a.s. S ( x , z ) − S ( x , y ) = � A [( − k / 2 x , k ) → ( z , 1)] − � A [( − k / 2 x , k ) → ( y , 1)] for all large enough k . B´ alint Vir´ ag Directed landscape 5/21/2019 27 / 45

  38. Hello, I am the Airy sheet B´ alint Vir´ ag Directed landscape 5/21/2019 28 / 45

  39. Hello, in a different dress B´ alint Vir´ ag Directed landscape 5/21/2019 29 / 45

  40. Airy sheet Theorem For every n, there exists a coupling so that B [(2 x / n 1 / 3 , n ) → (1 + 2 y / n 1 / 3 , 1)] = 2 √ n + ( y − x ) n 1 / 6 + n − 1 / 6 ( S + o n )( x , y ) , on every compact K ⊂ R 2 there exists a > 1 with Ea sup K | o n | 3 / 2 → 1 . B´ alint Vir´ ag Directed landscape 5/21/2019 30 / 45

  41. Properties of the Airy sheet S ( x , y ) + ( x − y ) 2 has Tracy-Widom law. y �→ S ( x , x + y ) are Airy-2 processes. x , y -swap symmetry Skew symmetry: S ( x , y ) + ( x − y ) 2 is shift-invariant in R 2 ! Quadrangle inequality Local sum structure S ( x , y ) is the CDF of the random shock measure ! B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

  42. Properties of the Airy sheet S ( x , y ) + ( x − y ) 2 has Tracy-Widom law. y �→ S ( x , x + y ) are Airy-2 processes. x , y -swap symmetry Skew symmetry: S ( x , y ) + ( x − y ) 2 is shift-invariant in R 2 ! Quadrangle inequality Local sum structure S ( x , y ) is the CDF of the random shock measure ! B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

  43. Properties of the Airy sheet S ( x , y ) + ( x − y ) 2 has Tracy-Widom law. y �→ S ( x , x + y ) are Airy-2 processes. x , y -swap symmetry Skew symmetry: S ( x , y ) + ( x − y ) 2 is shift-invariant in R 2 ! Quadrangle inequality Local sum structure S ( x , y ) is the CDF of the random shock measure ! B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

  44. Properties of the Airy sheet S ( x , y ) + ( x − y ) 2 has Tracy-Widom law. y �→ S ( x , x + y ) are Airy-2 processes. x , y -swap symmetry Skew symmetry: S ( x , y ) + ( x − y ) 2 is shift-invariant in R 2 ! Quadrangle inequality Local sum structure S ( x , y ) is the CDF of the random shock measure ! B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

  45. Properties of the Airy sheet S ( x , y ) + ( x − y ) 2 has Tracy-Widom law. y �→ S ( x , x + y ) are Airy-2 processes. x , y -swap symmetry Skew symmetry: S ( x , y ) + ( x − y ) 2 is shift-invariant in R 2 ! Quadrangle inequality Local sum structure S ( x , y ) is the CDF of the random shock measure ! B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

  46. Properties of the Airy sheet S ( x , y ) + ( x − y ) 2 has Tracy-Widom law. y �→ S ( x , x + y ) are Airy-2 processes. x , y -swap symmetry Skew symmetry: S ( x , y ) + ( x − y ) 2 is shift-invariant in R 2 ! Quadrangle inequality Local sum structure S ( x , y ) is the CDF of the random shock measure ! B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

  47. Properties of the Airy sheet S ( x , y ) + ( x − y ) 2 has Tracy-Widom law. y �→ S ( x , x + y ) are Airy-2 processes. x , y -swap symmetry Skew symmetry: S ( x , y ) + ( x − y ) 2 is shift-invariant in R 2 ! Quadrangle inequality Local sum structure S ( x , y ) is the CDF of the random shock measure ! B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

  48. The 1-2-3 scaling Airy sheet of scale s is S s ( x , y ) = s S ( x / s 2 , y / s 2 ) . Metric composition. r 3 = s 3 + t 3 . S r ( x , z ) = max y ∈ R S s ( x , y ) + S t ( y , z ) . Is the Airy sheet uniquely defined by this property? B´ alint Vir´ ag Directed landscape 5/21/2019 32 / 45

  49. The 1-2-3 scaling Airy sheet of scale s is S s ( x , y ) = s S ( x / s 2 , y / s 2 ) . Metric composition. r 3 = s 3 + t 3 . S r ( x , z ) = max y ∈ R S s ( x , y ) + S t ( y , z ) . Is the Airy sheet uniquely defined by this property? B´ alint Vir´ ag Directed landscape 5/21/2019 33 / 45

  50. The directed landscape The directed landscape is a stationary independent increment process with respect to metric composition. The increments are Airy sheets. L ( x , t ; y , s ) continuous, no technical issues. Increment L ( · , t ; · , t + s 3 ): Airy sheet of scale s Increments are independent over disjoint time-intervals. B´ alint Vir´ ag Directed landscape 5/21/2019 34 / 45

  51. The directed landscape The directed landscape is a stationary independent increment process with respect to metric composition. The increments are Airy sheets. L ( x , t ; y , s ) continuous, no technical issues. Increment L ( · , t ; · , t + s 3 ): Airy sheet of scale s Increments are independent over disjoint time-intervals. B´ alint Vir´ ag Directed landscape 5/21/2019 34 / 45

  52. The directed landscape The directed landscape is a stationary independent increment process with respect to metric composition. The increments are Airy sheets. L ( x , t ; y , s ) continuous, no technical issues. Increment L ( · , t ; · , t + s 3 ): Airy sheet of scale s Increments are independent over disjoint time-intervals. B´ alint Vir´ ag Directed landscape 5/21/2019 34 / 45

  53. The directed landscape The directed landscape is a stationary independent increment process with respect to metric composition. The increments are Airy sheets. L ( x , t ; y , s ) continuous, no technical issues. Increment L ( · , t ; · , t + s 3 ): Airy sheet of scale s Increments are independent over disjoint time-intervals. B´ alint Vir´ ag Directed landscape 5/21/2019 34 / 45

  54. Geodesics Length of a path is k � | π | L = inf inf L ( π ( t i − 1 ) , t i − 1 ; π ( t i ) , t i ) k ∈ N t = t 0 < ··· < t k = s i =1 Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0 , 0) and (0 , 1) is γ (i.e. ( γ ( t ) , t )). Almost all geodesics are unique! Point pairs with non-unique geodesics exist. B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

  55. Geodesics Length of a path is k � | π | L = inf inf L ( π ( t i − 1 ) , t i − 1 ; π ( t i ) , t i ) k ∈ N t = t 0 < ··· < t k = s i =1 Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0 , 0) and (0 , 1) is γ (i.e. ( γ ( t ) , t )). Almost all geodesics are unique! Point pairs with non-unique geodesics exist. B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

  56. Geodesics Length of a path is k � | π | L = inf inf L ( π ( t i − 1 ) , t i − 1 ; π ( t i ) , t i ) k ∈ N t = t 0 < ··· < t k = s i =1 Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0 , 0) and (0 , 1) is γ (i.e. ( γ ( t ) , t )). Almost all geodesics are unique! Point pairs with non-unique geodesics exist. B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

  57. Geodesics Length of a path is k � | π | L = inf inf L ( π ( t i − 1 ) , t i − 1 ; π ( t i ) , t i ) k ∈ N t = t 0 < ··· < t k = s i =1 Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0 , 0) and (0 , 1) is γ (i.e. ( γ ( t ) , t )). Almost all geodesics are unique! Point pairs with non-unique geodesics exist. B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

  58. Geodesics Length of a path is k � | π | L = inf inf L ( π ( t i − 1 ) , t i − 1 ; π ( t i ) , t i ) k ∈ N t = t 0 < ··· < t k = s i =1 Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0 , 0) and (0 , 1) is γ (i.e. ( γ ( t ) , t )). Almost all geodesics are unique! Point pairs with non-unique geodesics exist. B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

  59. Geodesic trees B´ alint Vir´ ag Directed landscape 5/21/2019 36 / 45

  60. Airy sheet Theorem For every n, there exists a coupling so that B [(2 x / n 1 / 3 , n ) → (1 + 2 y / n 1 / 3 , 1)] = 2 √ n + ( y − x ) n 1 / 6 + n − 1 / 6 ( S + o n )( x , y ) , on every compact K ⊂ R 2 there exists a > 1 with Ea sup K | o n | 3 / 2 → 1 . B´ alint Vir´ ag Directed landscape 5/21/2019 37 / 45

  61. The directed landscape as a limit Let ( x , s ) n = ( s + 2 x / n 1 / 3 , −⌊ sn ⌋ ), translation between locations. Theorem There exists a coupling of Brownian last passage percolation and the directed landcape L so that B n [( x , s ) n → ( y , t ) n ] = 2( t − s ) √ n + ( y − x ) n 1 / 6 + n − 1 / 6 ( L + o n )( x , s ; y , t ) . P B´ alint Vir´ ag Directed landscape 5/21/2019 38 / 45

  62. Last passage path as a limit π n denote an optimizing path for B [(0 , n ) → (1 , 1)]. Theorem In law, as random functions in the uniform norm π n ( s ) − n (1 − s ) d → γ ( s ) . n 2 / 3 B´ alint Vir´ ag Directed landscape 5/21/2019 39 / 45

  63. The limit of TASEP h t ( y ) = sup h 0 ( x ) + L ( x , 0; y , t ) x ∈ R Like the variational formula for Burger’s equation. KPZ fixed point. (Matetski-Quastel-Remenik) B´ alint Vir´ ag Directed landscape 5/21/2019 40 / 45

  64. Previous work Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. h t ( y ) = sup h 0 ( x ) + L ( x , 0; y , t ) x ∈ R B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

  65. Previous work Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. h t ( y ) = sup h 0 ( x ) + L ( x , 0; y , t ) x ∈ R B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

  66. Previous work Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. h t ( y ) = sup h 0 ( x ) + L ( x , 0; y , t ) x ∈ R B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

  67. Previous work Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. h t ( y ) = sup h 0 ( x ) + L ( x , 0; y , t ) x ∈ R B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

  68. Previous work Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. h t ( y ) = sup h 0 ( x ) + L ( x , 0; y , t ) x ∈ R B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

  69. Previous work II Random functional wrt. metric composition: g �→ g S MQR gives marginals for fixed g , used to construct a Markov process. Here: full distribution of functional. Baik-Liu: (17,JAMS 19) Two-parameter function on the cylinder B´ alint Vir´ ag Directed landscape 5/21/2019 42 / 45

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