New edge asymptotics of skew Young diagrams via free boundaries Dan - PowerPoint PPT Presentation

New edge asymptotics of skew Young diagrams via free boundaries Dan Betea University of Bonn joint work with J. Bouttier, P. Nejjar and M. Vuleti´ c FPSAC, Ljubljana, 2019 4.VI1.MM19

Outline This talk contains stuff on ◮ partitions and tableaux ◮ the Plancherel (mostly) and uniform measures on Young diagrams ◮ main results on skew Young diagrams ◮ the beyond and a few surprises.

Partitions • • ◦ • • ◦ • • • ◦ ◦ ◦ ◦ ◦ Figure: Partition (Young diagram) λ = (2 , 2 , 2 , 1 , 1) (Frobenius coordinates (1 , 0 | 4 , 1)) in English, French and Russian notation, with associated Maya diagram (particle-hole representation). Size | λ | = 8, length ℓ ( λ ) = 5. Figure: Skew partitions (Young diagrams) (4 , 3 , 2 , 1) / (2 , 1) (but also (5 , 4 , 3 , 2 , 1) / (5 , 2 , 1) , . . . ) and (4 , 4 , 2 , 1) / (2 , 2) (but also (6 , 4 , 4 , 2 , 1) / (6 , 2 , 2) , . . . )

Counting tableaux A standard (semi-standard) Young tableau SYT (SSYT) is a filling of a (possibly skew) Young diagram with numbers 1 , 2 , . . . strictly increasing down columns and rows (rows weakly increasing for semi-standard). 1 3 5 6 1 1 2 2 1 7 1 2 2 4 9 2 2 3 3 4 1 3 7 3 2 5 2 2 8 4 6 3 dim λ := number of SYTs of shape λ, � dim λ := number of SSYTs of shape λ with entries from 1 . . . n and similarly for dim λ/µ , � dim λ/µ .

Two natural measures on partitions ◮ On partitions of n ( | λ | := � λ i = n ): Plancherel vs. uniform Prob ( λ ) = (dim λ ) 2 1 vs. Prob ( λ ) = n ! # { partitions of n } ◮ On all partitions: poissonized Plancherel vs. (grand canonical) uniform Prob ( λ ) = e − ǫ 2 ǫ 2 | λ | (dim λ ) 2 Prob ( λ ) = u | λ | � (1 − u i ) vs. ( | λ | !) 2 i ≥ 1 with ǫ > 0 , 1 > u > 0 parameters.

Ulam’s problem and Hammersley last passage percolation I PPP ( ǫ 2 ) in the unit square.

Ulam’s problem and Hammersley last passage percolation II Quantity of interest: L = longest up-right path from (0 , 0) to (1 , 1) (= 4 here).

Ulam’s problem and Hammersley last passage percolation III 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 9 4 7 2 5 8 6 1 10 3 L is the length (any) of the longest increasing subsequence in a random permutation of S N with N ∼ Poisson ( ǫ 2 ).

The poissonized Plancherel measure By the Robinson–Schensted–Knuth correspondence and Schensted’s theorem, L = λ 1 in distribution where λ has the poissonized Plancherel measure: Prob ( λ ) = e − ǫ 2 ǫ 2 | λ | (dim λ ) 2 ( | λ | !) 2 = e − ǫ 2 s λ ( pl ǫ ) s λ ( pl ǫ ) ( s is a Schur function, pl ǫ the Plancherel specialization sending p 1 → ǫ, p i → 0 , i ≥ 2) Interest: what happens to λ 1 as ǫ → ∞ ? (large PPP, large random permutation, ...)

Limit shape A Plancherel-random representation (partition!) of S 2304 ( Prob ( λ ) = (dim λ ) 2 / n !, n = 2304), at IHP. The limit shape should be obvious (VerKer, LogShe 1977).

Limit shapes: Plancherel vs uniform Random Plancherel (left) and uniform (right) partitions of N = 10000. The scale is √ √ different: N for Plancherel, N log N for uniform.

The Baik–Deift–Johansson theorem and Tracy–Widom Theorem (BaiDeiJoh 1999) If λ is distributed as poissonized Plancherel, we have: � λ 1 − 2 ǫ � ǫ →∞ Prob lim ≤ s = F GUE ( s ) := det(1 − Ai 2 ) L 2 ( s , ∞ ) ǫ 1 / 3 with � ∞ Ai 2 ( x , y ) := Ai ( x + s ) Ai ( y + s ) ds 0 and Ai the Airy function (solution of y ′′ = xy decaying at ∞ ). F GUE is the Tracy–Widom GUE distribution. It is by (original) construction the extreme distribution of the largest eigenvalue of a random hermitian matrix with iid standard Gaussian entries as the size of the matrix goes to infinity.

The Erd˝ os–Lehner theorem and Gumbel Theorem (ErdLeh 1941) For the uniform measure Prob ( λ ) ∝ u | λ | we have: � � λ 1 < − log(1 − u ) ξ = e − e − ξ . u → 1 − Prob lim + log u | log u |

The finite temperature Plancherel measure On pairs of partitions µ ⊂ λ ⊃ µ consider the measure (Bor 06) Prob ( µ, λ ) ∝ u | µ | · ε | λ |−| µ | dim 2 ( λ/µ ) ( | λ/µ | !) 2 with u = e − β , β = inverse temperature. ◮ u = 0 yields the poissonized Plancherel measure ◮ ε = 0 yields the (grand canonical) uniform measure

What is in a part? PPP ( u 4 ǫ 2 ) PPP ( u 3 ǫ 2 ) PPP ( u 2 ǫ 2 ) PPP ( uǫ 2 ) PPP ( ǫ 2 ) With L the longest up-right path in this cylindric geometry, in distribution, Schensted’s theorem states that λ 1 = L + κ 1 where κ is a uniform partition Prob ( κ ) ∝ u | κ | independent of everything else.

The finite temperature Plancherel measure II Theorem (B/Bouttier 2019) √ ε 1 − u → ∞ and u = exp( − α M − 1 / 3 ) → 1 . Then Let M = � λ 1 − 2 M � = F α ( s ) := det(1 − Ai α ) L 2 ( s , ∞ ) M →∞ Prob lim ≤ s M 1 / 3 with � ∞ e α s Ai α ( x , y ) := 1 + e α s · Ai ( x + s ) Ai ( y + s ) ds −∞ the finite temperature Airy kernel.

A word on the finite temperature Airy kernel Ai α is Johansson’s (2007) Airy kernel in finite temperature (also appearing as the KPZ crossover kernel: SasSpo10 and AmiCorQua11, in random directed polymers BorCorFer11, cylindric OU processes LeDMajSch15): � ∞ e α s Ai α ( x , y ) = 1 + e α s Ai ( x + s ) Ai ( y + s ) ds −∞ and interpolates between the Airy kernel and a diagonal exponential kernel: α →∞ Ai α ( x , y ) = Ai 2 ( x , y ) , lim � x � 1 α − 1 2 α log(4 πα 3 ) , y α − 1 α Ai α 2 α log(4 πα 3 ) = e − x δ x , y . lim α → 0+ If F α ( s ) , F GUE ( s ), and G ( s ) are the Fredholm determinants on ( s , ∞ ) of Ai α , Ai 2 and e − x δ x , y , then (Joh 2007) � s � α − 1 = G ( s ) = e − e − s . α →∞ F α ( s ) = F GUE ( s ) , α → 0+ F α 2 α log(4 πα 3 ) lim lim It appeared in seemingly two different situations: ◮ random matrix models on the cylinder/in finite temperature (Joh, LeDMajSch, ...) ◮ the KPZ equation with wedge I.C. at finite time (SasSpo, AmiCorQua, ...)

Three limiting regimes for edge fluctuations Theorem (B/Bouttier 2019) With u = e − r → 1 as r → 0+ and ǫ → ∞ (or finite) we have: ◮ ǫ r 2 → 0+ leads to Gumbel behavior; thermal fluctuations win ◮ ǫ r 2 → ∞ leads to Tracy–Widom; quantum fluctuations win ◮ ǫ r 2 → α ∈ (0 , ∞ ) leads to finite temperature Tracy–Widom F α ; equilibrium between thermal and quantum

The stuff that’s in the FPSAC abstract Consider the following measures (oc = number of odd columns, n letters for � dim): · u | µ | · ǫ | λ/µ | dim( λ/µ ) M ր ( µ, λ ) ∝ a oc ( µ ) a oc ( λ ) , 1 2 | λ/µ | ! · u | µ | v | ν | · ǫ | λ/µ | + | λ/ν | dim( λ/µ ) dim( λ/ν ) M րց ( µ, λ, ν ) ∝ a oc ( µ ) a oc ( λ ) , 1 2 | λ/µ | ! · | λ/ν | ! · u | µ | · q | λ/µ | · � M ր ( µ, λ ) ∝ a oc ( µ ) � a oc ( λ ) dim( λ/µ ) , 1 2 M րց ( µ, λ, ν ) ∝ a oc ( µ ) a oc ( λ ) · u | µ | v | ν | · q | λ/µ | + | λ/ν | · � � dim( λ/µ ) � dim( λ/ν ) . 1 2 They all interpolate between Plancherel-type ( u = 0) and uniform ( ǫ, q = 0) measures.

What is in a part? ( λ 1 = L + κ 1 via RSK) 0 0 0 0 0 0 0 0 0 0 Geom (( uv ) 4 x 3 y 2 ) 0 0 0 0 0 0 0 0 0 κ κ κ κ κ 0 κ 0 0 0 1 2 8 0 0 0 2 0 0 0 Geom ( u ( uv ) x 3 ) κ Geom ( u 2 y 3 ) 8 κ 0 0 1 2 3 2 5 0 0 0 5 0 0 κ 2 κ Geom ( u 2 ( uv ) 2 x 1 x 2 ) 0 0 0 3 0 0 2 10 0 2 10 0 2 κ 9 κ Geom (( uv ) 2 x 2 y 3 ) 0 3 2 2 1 0 2 6 0 0 2 3 1 κ 1 κ 8 2 0 2 9 4 25 1 2 4 25 1 Geom ( vy 3 ) 2 Geom ( uy 3 ) 9 Geom (( uv ) 4 x 2 y 4 ) 1 Geom ( u 8 y 2 y 4 ) 1 1 3 11 6 7 0 0 2 7 11 2 0 Geom ( v 2 ( uv ) 2 y 2 y 4 ) Geom ( v 2 y 1 y 2 ) 4 Geom ( u 6 y 2 y 4 ) 4 4 8 0 10 2 0 2 2 13 0 12 1 0 Geom ( u 2 y 2 ) Geom ( x 3 y 2 ) Geom ( v ( uv ) y 2 ) 0 17 1 2 1 4 0 1 3 3 0 8 ν 1 0 x 4 10 1 14 5 0 2 1 2 2 1 2 0 Geom ( y 3 ) Geom (( uv ) 2 x 4 y 2 ) 0 Geom ( u 4 y 2 y 4 ) x 3 0 5 15 1 3 0 3 3 0 3 11 Geom ( u 2 x 2 x 4 ) 1 Geom ( u 2 y 2 y 4 ) x 2 6 4 0 2 1 1 3 2 1 4 3 Geom ( uy 2 ) x 1 Geom ( ux 2 ) 3 7 3 1 2 3 1 2 µ λ 7 µ λ y 1 y 2 y 3 y 4 y 1 y 2 y 3 y 4 Geom ( x 2 y 4 ) Geom ( y 2 y 4 ) M րց ( µ, λ, ν ) (left) and � M ր ( µ, λ ) (right); Figure: Longest up-right path in orange of length L = 199 (left) and L = 130 (right). � xi = yi = q ; case a 1 = a 2 = 0 (for generic, multiply the parameters in the boundary triangles by a 1 and a 2 for the two different boundaries; κ is uniform with prob. ∝ ( uv ) | κ | (left) and ∝ u | κ | (right).

Main theorem: edge limits (SYT case) Theorem (B/Bouttier/Nejjar/Vuleti´ c FPSAC 2019) ǫ Fix η, α i , i = 1 , 2 positive reals. Let M := 1 − u 2 → ∞ and set u = v = exp( − η M − 1 / 3 ) , a i = u α i /η , i = 1 , 2 all going to 1 as M → ∞ . (In particular, ǫ ∼ M 2 / 3 → ∞ .) We have: � � η log M 1 / 3 λ 1 − 2 M ≤ s + 1 M →∞ M ր = F 1; α 1 ,α 2 ; η ( s ) , lim M 1 / 3 η � � 2 η log M 1 / 3 λ 1 − 2 M ≤ s + 1 M →∞ M րց = F 2; α 1 ,α 2 ; η ( s ) lim M 1 / 3 2 η with the distributions F ··· explicit Fredholm pfaffians. Remark: This theorem generalizes celebrated results of Baik–Rains (2000) on longest increasing subsequences in symmetrized permutations, as well as the classical Baik–Deift–Johansson theorem.