Around the Plancherel Measure on Integer Partitions (An Introduction to Schur Processes Without Schur Functions)

Around the Plancherel measure on integer partitions (an introduction to Schur processes without Schur functions)

J´er´emieBouttier

A subject which I learned with Dan Betea, C´edricBoutillier, Guillaume Chapuy, Sylvie Corteel, Sanjay Ramassamy and Mirjana Vuleti´c

Institut de Physique Th´eorique, CEA Saclay Laboratoire de Physique, ENS de Lyon

Al´ea2019, 20-21 mars

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20-21 March 2019 1 / 28 Part I

20 March 2019

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 2 / 28 This course was roughly based on Chapters 1 and 2 of Dan Romik’s beautiful book The surprising mathematics of longest increasing subsequences (available online).

In the second part (Chapter 2) I somewhat diverged from the book by following my own favorite approach (developed mostly by Okounkov), based on fermions and saddle point computations for asymptotics.

This is the material I would like to present here: fermions because of physics, saddle point computations because, well, we are in Al´ea!

What these lectures are about In these lectures I present a very condensed version of some material which form the second part of a M2 course I gave in Lyon.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 3 / 28 In the second part (Chapter 2) I somewhat diverged from the book by following my own favorite approach (developed mostly by Okounkov), based on fermions and saddle point computations for asymptotics.

This is the material I would like to present here: fermions because of physics, saddle point computations because, well, we are in Al´ea!

What these lectures are about In these lectures I present a very condensed version of some material which form the second part of a M2 course I gave in Lyon.

This course was roughly based on Chapters 1 and 2 of Dan Romik’s beautiful book The surprising mathematics of longest increasing subsequences (available online).

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 3 / 28 This is the material I would like to present here: fermions because of physics, saddle point computations because, well, we are in Aléa!

What these lectures are about In these lectures I present a very condensed version of some material which form the second part of a M2 course I gave in Lyon.

This course was roughly based on Chapters 1 and 2 of Dan Romik’s beautiful book The surprising mathematics of longest increasing subsequences (available online).

In the second part (Chapter 2) I somewhat diverged from the book by following my own favorite approach (developed mostly by Okounkov), based on fermions and saddle point computations for asymptotics.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 3 / 28 What these lectures are about In these lectures I present a very condensed version of some material which form the second part of a M2 course I gave in Lyon.

This course was roughly based on Chapters 1 and 2 of Dan Romik’s beautiful book The surprising mathematics of longest increasing subsequences (available online).

In the second part (Chapter 2) I somewhat diverged from the book by following my own favorite approach (developed mostly by Okounkov), based on fermions and saddle point computations for asymptotics.

This is the material I would like to present here: fermions because of physics, saddle point computations because, well, we are in Al´ea!

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 3 / 28 It may be represented by a Young diagram, e.g. for λ = (4, 2, 2, 1):

8 4 9 3 6 1 2 5 7

Integer partitions and Young diagrams/tableaux An (integer) partition λ is a ﬁnite nonincreasing sequence of positive integers called parts:

λ1 ≥ λ2 ≥ · · · ≥ λ` > 0. P Its size is |λ| := λi and its length is `(λ) := ` (by convention λn = 0 for n > `).

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 4 / 28 8 4 9 3 6 1 2 5 7

Integer partitions and Young diagrams/tableaux An (integer) partition λ is a ﬁnite nonincreasing sequence of positive integers called parts:

λ1 ≥ λ2 ≥ · · · ≥ λ` > 0. P Its size is |λ| := λi and its length is `(λ) := ` (by convention λn = 0 for n > `). It may be represented by a Young diagram, e.g. for λ = (4, 2, 2, 1):

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 4 / 28 Integer partitions and Young diagrams/tableaux An (integer) partition λ is a finite nonincreasing sequence of positive integers called parts:

λ1 ≥ λ2 ≥ · · · ≥ λ` > 0. P Its size is |λ| := λi and its length is `(λ) := ` (by convention λn = 0 for n > `). It may be represented by a Young diagram, e.g. for λ = (4, 2, 2, 1):

8 4 9 3 6 1 2 5 7

A standard Young tableau (SYT) of shape λ is a filling of the Young diagram of λ by the integers 1,..., |λ| that is increasing along rows and columns. We denote by dλ the number of SYTs of shape λ. JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 4 / 28 It is a probability measure because of the “well-known” identity

X 2 n! = dλ λ`n which has (at least) two classical proofs: representation theory: n! is the dimension of the regular representation of the symmetric group Sn, and dλ is the dimension of its irreducible representation indexed by λ, bijection: the Robinson-Schensted correspondence is a bijection between Sn and the set of triples (λ, P, Q), where λ ` n and P, Q are two SYTs of shape λ.

Plancherel measure The Plancherel measure on partitions of size n is the probability measure such that ( d2 λ if λ ` n, Prob(λ) = n! 0 otherwise. Here λ ` n is a shorthand symbol to say that λ is partition of size n.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 5 / 28 representation theory: n! is the dimension of the regular representation of the symmetric group Sn, and dλ is the dimension of its irreducible representation indexed by λ, bijection: the Robinson-Schensted correspondence is a bijection between Sn and the set of triples (λ, P, Q), where λ ` n and P, Q are two SYTs of shape λ.

X 2 n! = dλ λ`n which has (at least) two classical proofs:

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 5 / 28 bijection: the Robinson-Schensted correspondence is a bijection between Sn and the set of triples (λ, P, Q), where λ ` n and P, Q are two SYTs of shape λ.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 5 / 28 Plancherel measure The Plancherel measure on partitions of size n is the probability measure such that ( d2 λ if λ ` n, Prob(λ) = n! 0 otherwise. Here λ ` n is a shorthand symbol to say that λ is partition of size n. It is a probability measure because of the “well-known” identity

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 5 / 28 Example: for σ = (3, 1, 6, 7, 2, 5, 4), we have L(σ) = 3.

There is a more general statement (Greene’s theorem) but we will not discuss it here.

The Longest Increasing Subsequence problem consists in understanding (n) the asymptotic behaviour as n → ∞ of Ln := L(σn) = λ1 , where σn (n) denotes a uniform random permutation in Sn, and λ the random partition to which it maps via the RS correspondence, and whose law is the Plancherel measure.

Connection with Longest Increasing Subsequences A property of the Robinson-Schensted correspondence is that if σ 7→ (λ, P, Q), then the ﬁrst part of λ satisﬁes

λ1 = L(σ)

where L(σ) is the length of a Longest Increasing Subsequence (LIS) of σ.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 6 / 28 There is a more general statement (Greene’s theorem) but we will not discuss it here.

Connection with Longest Increasing Subsequences A property of the Robinson-Schensted correspondence is that if σ 7→ (λ, P, Q), then the ﬁrst part of λ satisﬁes

λ1 = L(σ)

where L(σ) is the length of a Longest Increasing Subsequence (LIS) of σ.

Example: for σ = (3, 1, 6, 7, 2, 5, 4), we have L(σ) = 3.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 6 / 28 The Longest Increasing Subsequence problem consists in understanding (n) the asymptotic behaviour as n → ∞ of Ln := L(σn) = λ1 , where σn (n) denotes a uniform random permutation in Sn, and λ the random partition to which it maps via the RS correspondence, and whose law is the Plancherel measure.

Connection with Longest Increasing Subsequences A property of the Robinson-Schensted correspondence is that if σ 7→ (λ, P, Q), then the ﬁrst part of λ satisﬁes

λ1 = L(σ)

where L(σ) is the length of a Longest Increasing Subsequence (LIS) of σ.

Example: for σ = (3, 1, 6, 7, 2, 5, 4), we have L(σ) = 3.

There is a more general statement (Greene’s theorem) but we will not discuss it here.

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 6 / 28 Connection with Longest Increasing Subsequences A property of the Robinson-Schensted correspondence is that if σ 7→ (λ, P, Q), then the first part of λ satisfies

λ1 = L(σ)

where L(σ) is the length of a Longest Increasing Subsequence (LIS) of σ.

Example: for σ = (3, 1, 6, 7, 2, 5, 4), we have L(σ) = 3.

There is a more general statement (Greene’s theorem) but we will not discuss it here.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 6 / 28 It was then popularized by Hammersley (1972) who introduced a nice graphical method (closely related with the RSK correspondence) and √ proved that Ln/ n converges in probability to a constant c ∈ [π/2, e]. Vershik-Kerov and Logan-Shepp (1977) proved independently that c = 2, as a consequence of a more general limit shape theorem for the Plancherel measure on partitions. (See Chapter 1 of Romik’s book.) Baik-Deift-Johansson (1999) proved the most precise result √ Ln − 2 n P ≤ s = FGUE (s), n → ∞ n1/6

where FGUE is the Tracy-Widom GUE distribution. (See Chapter 2.) The unusual exponent n1/6 was previously conjectured by Odlyzko-Rains and Kim based on numerical evidence and bounds.

Some partial history of the LIS problem The problem was formulated by Ulam (1961) who suggested investigating it using Monte Carlo simulations and observed that L √ n should be of order n.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 7 / 28 Vershik-Kerov and Logan-Shepp (1977) proved independently that c = 2, as a consequence of a more general limit shape theorem for the Plancherel measure on partitions. (See Chapter 1 of Romik’s book.) Baik-Deift-Johansson (1999) proved the most precise result √ Ln − 2 n P ≤ s = FGUE (s), n → ∞ n1/6

where FGUE is the Tracy-Widom GUE distribution. (See Chapter 2.) The unusual exponent n1/6 was previously conjectured by Odlyzko-Rains and Kim based on numerical evidence and bounds.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 7 / 28 Baik-Deift-Johansson (1999) proved the most precise result √ Ln − 2 n P ≤ s = FGUE (s), n → ∞ n1/6

where FGUE is the Tracy-Widom GUE distribution. (See Chapter 2.) The unusual exponent n1/6 was previously conjectured by Odlyzko-Rains and Kim based on numerical evidence and bounds.

Some partial history of the LIS problem The problem was formulated by Ulam (1961) who suggested investigating it using Monte Carlo simulations and observed that L √ n should be of order n. It was then popularized by Hammersley (1972) who introduced a nice graphical method (closely related with the RSK correspondence) and √ proved that Ln/ n converges in probability to a constant c ∈ [π/2, e]. Vershik-Kerov and Logan-Shepp (1977) proved independently that c = 2, as a consequence of a more general limit shape theorem for the Plancherel measure on partitions. (See Chapter 1 of Romik’s book.)

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 7 / 28 Limit shape

A Plancherel random partition of size 10000 (courtesy of D. Betea)

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 8 / 28 Baik-Deift-Johansson (1999) proved the most precise result √ Ln − 2 n P ≤ s = FGUE (s), n → ∞ n1/6

where FGUE is the Tracy-Widom GUE distribution. (See Chapter 2.) The unusual exponent n1/6 was previously conjectured by Odlyzko-Rains and Kim based on numerical evidence and bounds.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 9 / 28 Some partial history of the LIS problem The problem was formulated by Ulam (1961) who suggested investigating it using Monte Carlo simulations and observed that L √ n should be of order n. It was then popularized by Hammersley (1972) who introduced a nice graphical method (closely related with the RSK correspondence) and √ proved that Ln/ n converges in probability to a constant c ∈ [π/2, e]. Vershik-Kerov and Logan-Shepp (1977) proved independently that c = 2, as a consequence of a more general limit shape theorem for the Plancherel measure on partitions. (See Chapter 1 of Romik’s book.) Baik-Deift-Johansson (1999) proved the most precise result √ Ln − 2 n P ≤ s = FGUE (s), n → ∞ n1/6

where FGUE is the Tracy-Widom GUE distribution. (See Chapter 2.) The unusual exponent n1/6 was previously conjectured by Odlyzko-Rains and Kim based on numerical evidence and bounds.

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 9 / 28 These results were obtained indepently in two papers by Borodin, Okounkov and Olshanski (2000) and by Johansson (2001). But we use a different approach developed later by Okounkov et al., which may be generalized to Schur measures and Schur processes. We concentrate on the Plancherel measure for simplicity.

Topics of the lectures

We will discuss some properties of the Plancherel measure. 1 We will show that the poissonized Plancherel measure (to be deﬁned) is closely related with a determinantal point process (DPP) called the discrete Bessel process. Plan:

I Some general theory of DPPs I Connection with Plancherel measure via fermions 2 We will then investigate asymptotics, in the following regimes:

I Bulk limits: the VKLS limit shape and the discrete sine process I Edge limit: the Airy process and the Baik-Deift-Johansson theorem

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 10 / 28 Topics of the lectures

I Some general theory of DPPs I Connection with Plancherel measure via fermions 2 We will then investigate asymptotics, in the following regimes:

I Bulk limits: the VKLS limit shape and the discrete sine process I Edge limit: the Airy process and the Baik-Deift-Johansson theorem

These results were obtained indepently in two papers by Borodin, Okounkov and Olshanski (2000) and by Johansson (2001). But we use a diﬀerent approach developed later by Okounkov et al., which may be generalized to Schur measures and Schur processes. We concentrate on the Plancherel measure for simplicity.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 10 / 28 Poissonized Plancherel measure

The poissonized Plancherel measure of parameter θ is the measure

d2 Prob(λ) = λ θ|λ|e−θ. (|λ|!)2

It is a mixture of the Plancherel measures of ﬁxed size, where the size is a Poisson random variable of parameter θ. We denote by λhθi a random partition distributed according to the poissonized Plancherel measure, λ(n) denoting a Plancherel random partition of size n.

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 11 / 28 Partitions and particle configurations

0 1 To a partition λ, here (4, 2, 1), we associate a set S(λ) ⊂ Z := Z + 2 by 1 3 5 S(λ) = {λ − , λ − , λ − ,...} 1 2 2 2 3 2

7 1 −3 −7 −9 Here S(λ) = { 2 , 2 , 2 , 2 , 2 ,...}. Elements of S(λ) (“particles” •) correspond to the down-steps of the blue curve.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 12 / 28 The correlation kernel Jθ is the discrete Bessel kernel √ √ X 0 Jθ(s, t) = Js+`(2 θ)Jt+`(2 θ), s, t ∈ Z 0 `∈Z>0

where Jn is the Bessel function of order n.

By the general theory of DPPs, knowing Jθ gives all the information on the point process.

Main result of today

Theorem [Borodin-Okounkov-Olshanski 2000, Johansson 2001] The particle conﬁguration S(λhθi) associated with the poissonized Plancherel measure is a determinantal point process in the sense that, for 0 any distinct points {u1,..., un} ⊂ Z , we have

hθi P {u1,..., un} ⊂ S(λ ) = det Jθ(ui , uj ). 1≤i,j≤n

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 13 / 28 By the general theory of DPPs, knowing Jθ gives all the information on the point process.

Main result of today

hθi P {u1,..., un} ⊂ S(λ ) = det Jθ(ui , uj ). 1≤i,j≤n

The correlation kernel Jθ is the discrete Bessel kernel √ √ X 0 Jθ(s, t) = Js+`(2 θ)Jt+`(2 θ), s, t ∈ Z 0 `∈Z>0

where Jn is the Bessel function of order n.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 13 / 28 Main result of today

hθi P {u1,..., un} ⊂ S(λ ) = det Jθ(ui , uj ). 1≤i,j≤n

The correlation kernel Jθ is the discrete Bessel kernel √ √ X 0 Jθ(s, t) = Js+`(2 θ)Jt+`(2 θ), s, t ∈ Z 0 `∈Z>0

where Jn is the Bessel function of order n.

By the general theory of DPPs, knowing Jθ gives all the information on the point process.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 13 / 28 Tomorrow

Asymptotics of Jθ, using saddle point computations. Again this is diﬀerent from the original techniques of BOO/J, our approach follows Okounkov and Reshetikhin and are robust (“universality”).

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 20 March 2019 14 / 28 Part II

21 March 2019

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 15 / 28 Partitions are embedded into fermionic configurations by the mapping 1 3 5 λ 7→ S(λ) := {λ − , λ − , λ − ,...} 1 2 2 2 3 2 It is not a bijection but the mapping (λ, c) 7→ S(λ) + c, with c ∈ Z, is. The fermionic Fock space F consists of columns vectors indexed by S. The standard basis is denoted by (vS )S∈S and the dual basis (of row ∗ vectors) by (vS )S∈S . Operators on F are naively viewed as matrices with rows and columns indexed by S.

We use the shorthand notation v := v for partitions and v := v 0 λ S(λ) ∅ Z<0 corresponds to the (nonzero!) vacuum vector.

A refresher on fermions 0 1 A fermionic configuration is a subset S ⊂ Z := Z + 2 that contains finitely many positive elements, and whose complement contains finitely many negative elements. We denote by S the (countable) set of fermionic configurations.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 16 / 28 The fermionic Fock space F consists of columns vectors indexed by S. The standard basis is denoted by (vS )S∈S and the dual basis (of row ∗ vectors) by (vS )S∈S . Operators on F are naively viewed as matrices with rows and columns indexed by S.

We use the shorthand notation v := v for partitions and v := v 0 λ S(λ) ∅ Z<0 corresponds to the (nonzero!) vacuum vector.

Partitions are embedded into fermionic conﬁgurations by the mapping 1 3 5 λ 7→ S(λ) := {λ − , λ − , λ − ,...} 1 2 2 2 3 2 It is not a bijection but the mapping (λ, c) 7→ S(λ) + c, with c ∈ Z, is.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 16 / 28 We use the shorthand notation v := v for partitions and v := v 0 λ S(λ) ∅ Z<0 corresponds to the (nonzero!) vacuum vector.

Partitions are embedded into fermionic conﬁgurations by the mapping 1 3 5 λ 7→ S(λ) := {λ − , λ − , λ − ,...} 1 2 2 2 3 2 It is not a bijection but the mapping (λ, c) 7→ S(λ) + c, with c ∈ Z, is. The fermionic Fock space F consists of columns vectors indexed by S. The standard basis is denoted by (vS )S∈S and the dual basis (of row ∗ vectors) by (vS )S∈S . Operators on F are naively viewed as matrices with rows and columns indexed by S.

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 16 / 28 A refresher on fermions 0 1 A fermionic configuration is a subset S ⊂ Z := Z + 2 that contains finitely many positive elements, and whose complement contains finitely many negative elements. We denote by S the (countable) set of fermionic configurations.

We use the shorthand notation v := v for partitions and v := v 0 λ S(λ) ∅ Z<0 corresponds to the (nonzero!) vacuum vector. J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 16 / 28 These operators satisfy the canonical anticommutation relations (CAR)

ψk ψ` + ψ`ψk = 0 ∗ ∗ ∗ ∗ ψk ψ` + ψ` ψk = 0 ∗ ∗ ψk ψ` + ψ` ψk = δk,`.

∗ The diagonal operator Nk := ψk ψk “measures” whether there is a particle at position k.

A refresher on fermions We deﬁned the fermionic creation/annihilation operators through their action on the standard basis: ( 0 if k ∈ S, ψk vS := 0 #(S∩Z ) (−1) >k vS∪{k} if k ∈/ S, ( ∗ 0 if k ∈/ S, ψ vS := 0 k #(S∩Z ) (−1) >k vS\{k} if k ∈ S.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 17 / 28 ∗ The diagonal operator Nk := ψk ψk “measures” whether there is a particle at position k.

These operators satisfy the canonical anticommutation relations (CAR)

ψk ψ` + ψ`ψk = 0 ∗ ∗ ∗ ∗ ψk ψ` + ψ` ψk = 0 ∗ ∗ ψk ψ` + ψ` ψk = δk,`.

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 17 / 28 A refresher on fermions We defined the fermionic creation/annihilation operators through their action on the standard basis: ( 0 if k ∈ S, ψk vS := 0 #(S∩Z ) (−1) >k vS∪{k} if k ∈/ S, ( ∗ 0 if k ∈/ S, ψ vS := 0 k #(S∩Z ) (−1) >k vS\{k} if k ∈ S.

These operators satisfy the canonical anticommutation relations (CAR)

ψk ψ` + ψ`ψk = 0 ∗ ∗ ∗ ∗ ψk ψ` + ψ` ψk = 0 ∗ ∗ ψk ψ` + ψ` ψk = δk,`.

∗ The diagonal operator Nk := ψk ψk “measures” whether there is a particle at position k. J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 17 / 28 In terms of partitions their action read ∗ ∗ X ∗ X vλα = vµ, αvλ = vµ µ:λ%µ µ:λ%µ where % means “adding a box”.

By iterating we get ∗ ∗ n X ∗ n X v∅ (α ) = dλvλ, α v∅ = dλvλ λ`n λ`n or, equivalently, |λ| |λ| ∗ X dλx X dλx v ∗exα = v ∗, exαv = v . ∅ |λ|! λ ∅ |λ|! λ λ λ

A refresher on fermions We deﬁned the “box” creation/annihilation operators by ∗ X ∗ X ∗ α := ψk ψk+1, α := ψk+1ψk . k∈Z0 k∈Z0

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 18 / 28 By iterating we get ∗ ∗ n X ∗ n X v∅ (α ) = dλvλ, α v∅ = dλvλ λ`n λ`n or, equivalently, |λ| |λ| ∗ X dλx X dλx v ∗exα = v ∗, exαv = v . ∅ |λ|! λ ∅ |λ|! λ λ λ

A refresher on fermions We deﬁned the “box” creation/annihilation operators by ∗ X ∗ X ∗ α := ψk ψk+1, α := ψk+1ψk . k∈Z0 k∈Z0 In terms of partitions their action read ∗ ∗ X ∗ X vλα = vµ, αvλ = vµ µ:λ%µ µ:λ%µ where % means “adding a box”.

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 18 / 28 A refresher on fermions We defined the “box” creation/annihilation operators by ∗ X ∗ X ∗ α := ψk ψk+1, α := ψk+1ψk . k∈Z0 k∈Z0 In terms of partitions their action read ∗ ∗ X ∗ X vλα = vµ, αvλ = vµ µ:λ%µ µ:λ%µ where % means “adding a box”.

By iterating we get ∗ ∗ n X ∗ n X v∅ (α ) = dλvλ, α v∅ = dλvλ λ`n λ`n or, equivalently, |λ| |λ| ∗ X dλx X dλx v ∗exα = v ∗, exαv = v . ∅ |λ|! λ ∅ |λ|! λ λ λ

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 18 / 28 It remains to identify the rhs as a determinant with Bessel kernel entries. There are two main steps, which both exploit the CAR algebra structure: eliminate the α’s to rewrite ∗ ∗ ∗ X ρ(U) = v∅ ψbu1 ψbu1 ··· ψbun ψbun v∅, with ψbu := J`(2x)ψu−` `∈Z apply Wick’s lemma to get ∗ ∗ ρ(U) = det v∅ ψbui ψbu v∅ = det Jθ(ui , uj ). 1≤i,j≤n j 1≤i,j≤n

A refresher on fermions Final result of yesterday The correlation function ρ(U) of the poissonized Plancherel measure admit the fermionic expression (with θ = x2)

∗ xα∗ xα hθi v∅ e Nu1 ··· Nun e v∅ ρ(U) := P {u1,..., un} ⊂ S(λ ) = 2 ex ∗ where we recall that Nu = ψuψu “indicates” if there is a particle at u.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 19 / 28 eliminate the α’s to rewrite ∗ ∗ ∗ X ρ(U) = v∅ ψbu1 ψbu1 ··· ψbun ψbun v∅, with ψbu := J`(2x)ψu−` `∈Z apply Wick’s lemma to get ∗ ∗ ρ(U) = det v∅ ψbui ψbu v∅ = det Jθ(ui , uj ). 1≤i,j≤n j 1≤i,j≤n

A refresher on fermions Final result of yesterday The correlation function ρ(U) of the poissonized Plancherel measure admit the fermionic expression (with θ = x2)

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 19 / 28 apply Wick’s lemma to get ∗ ∗ ρ(U) = det v∅ ψbui ψbu v∅ = det Jθ(ui , uj ). 1≤i,j≤n j 1≤i,j≤n

A refresher on fermions Final result of yesterday The correlation function ρ(U) of the poissonized Plancherel measure admit the fermionic expression (with θ = x2)

∗ xα∗ xα hθi v∅ e Nu1 ··· Nun e v∅ ρ(U) := P {u1,..., un} ⊂ S(λ ) = 2 ex ∗ where we recall that Nu = ψuψu “indicates” if there is a particle at u. It remains to identify the rhs as a determinant with Bessel kernel entries. There are two main steps, which both exploit the CAR algebra structure: eliminate the α’s to rewrite ∗ ∗ ∗ X ρ(U) = v∅ ψbu1 ψbu1 ··· ψbun ψbun v∅, with ψbu := J`(2x)ψu−` `∈Z

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 19 / 28 A refresher on fermions Final result of yesterday The correlation function ρ(U) of the poissonized Plancherel measure admit the fermionic expression (with θ = x2)

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 19 / 28 From the CAR we deduce the commutation relations

∗ [α , ψk ] = ψk−1, [ψk , α] = −ψk+1

P k and their duals. Equivalently, in terms of ψ(z) := 0 ψ z , k∈Z k [α∗, ψ(z)] = zψ(z), [ψ(z), α] = −z−1ψ(z).

By “exponentiating” we get

xz −xz−1 Adexα∗ (ψ(z)) = e ψ(z), Ade−xα (ψ(z)) = e ψ(z)

−1 where Adg (Y ) := ghg . Equivalently,

X x` X (−x)` Ad xα∗ (ψ ) = ψ , Ad −xα (ψ ) = ψ e k `! k−` e k `! k+` `∈Z `∈Z

Eliminating the α’s

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 20 / 28 P k Equivalently, in terms of ψ(z) := 0 ψ z , k∈Z k [α∗, ψ(z)] = zψ(z), [ψ(z), α] = −z−1ψ(z).

By “exponentiating” we get

xz −xz−1 Adexα∗ (ψ(z)) = e ψ(z), Ade−xα (ψ(z)) = e ψ(z)

−1 where Adg (Y ) := ghg . Equivalently,

X x` X (−x)` Ad xα∗ (ψ ) = ψ , Ad −xα (ψ ) = ψ e k `! k−` e k `! k+` `∈Z `∈Z

Eliminating the α’s From the CAR we deduce the commutation relations

∗ [α , ψk ] = ψk−1, [ψk , α] = −ψk+1

and their duals.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 20 / 28 By “exponentiating” we get

xz −xz−1 Adexα∗ (ψ(z)) = e ψ(z), Ade−xα (ψ(z)) = e ψ(z)

−1 where Adg (Y ) := ghg . Equivalently,

X x` X (−x)` Ad xα∗ (ψ ) = ψ , Ad −xα (ψ ) = ψ e k `! k−` e k `! k+` `∈Z `∈Z

Eliminating the α’s From the CAR we deduce the commutation relations

∗ [α , ψk ] = ψk−1, [ψk , α] = −ψk+1

P k and their duals. Equivalently, in terms of ψ(z) := 0 ψ z , k∈Z k [α∗, ψ(z)] = zψ(z), [ψ(z), α] = −z−1ψ(z).

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 20 / 28 Equivalently,

X x` X (−x)` Ad xα∗ (ψ ) = ψ , Ad −xα (ψ ) = ψ e k `! k−` e k `! k+` `∈Z `∈Z

Eliminating the α’s From the CAR we deduce the commutation relations

∗ [α , ψk ] = ψk−1, [ψk , α] = −ψk+1

P k and their duals. Equivalently, in terms of ψ(z) := 0 ψ z , k∈Z k [α∗, ψ(z)] = zψ(z), [ψ(z), α] = −z−1ψ(z).

By “exponentiating” we get

xz −xz−1 Adexα∗ (ψ(z)) = e ψ(z), Ade−xα (ψ(z)) = e ψ(z)

−1 where Adg (Y ) := ghg .

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 20 / 28 Eliminating the α’s From the CAR we deduce the commutation relations

∗ [α , ψk ] = ψk−1, [ψk , α] = −ψk+1

P k and their duals. Equivalently, in terms of ψ(z) := 0 ψ z , k∈Z k [α∗, ψ(z)] = zψ(z), [ψ(z), α] = −z−1ψ(z).

By “exponentiating” we get

xz −xz−1 Adexα∗ (ψ(z)) = e ψ(z), Ade−xα (ψ(z)) = e ψ(z)

−1 where Adg (Y ) := ghg . Equivalently,

X x` X (−x)` Ad xα∗ (ψ ) = ψ , Ad −xα (ψ ) = ψ e k `! k−` e k `! k+` `∈Z `∈Z

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 20 / 28 X Adexα∗ e−xα (ψk ) = J`(2x)ψk−` =: ψbk `∈Z ∗ xα ∗ xα∗ v∅ e = v∅ e v∅ = v∅

Note that

x(z−z−1) Adexα∗ e−xα (ψ(z)) = e ψ(z)

and combining everything

v ∗exα∗ ψ ψ∗ ··· ψ ψ∗ exαv ∅ u1 u1 un un ∅ ∗ ∗ ∗ ρ(U) = = v ψbu ψb ··· ψbu ψb v . ex2 ∅ 1 u1 n un ∅

Eliminating the α’s Also, we have [α∗, α] = 1 which implies, by the Baker-Campbell-Hausdorﬀ formula,

∗ 2 ∗ exα exα = ex exαexα .

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 21 / 28 X Adexα∗ e−xα (ψk ) = J`(2x)ψk−` =: ψbk `∈Z ∗ xα ∗ xα∗ v∅ e = v∅ e v∅ = v∅ and combining everything

v ∗exα∗ ψ ψ∗ ··· ψ ψ∗ exαv ∅ u1 u1 un un ∅ ∗ ∗ ∗ ρ(U) = = v ψbu ψb ··· ψbu ψb v . ex2 ∅ 1 u1 n un ∅

Eliminating the α’s Also, we have [α∗, α] = 1 which implies, by the Baker-Campbell-Hausdorﬀ formula,

∗ 2 ∗ exα exα = ex exαexα .

Note that

x(z−z−1) Adexα∗ e−xα (ψ(z)) = e ψ(z)

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 21 / 28 ∗ xα ∗ xα∗ v∅ e = v∅ e v∅ = v∅ and combining everything

v ∗exα∗ ψ ψ∗ ··· ψ ψ∗ exαv ∅ u1 u1 un un ∅ ∗ ∗ ∗ ρ(U) = = v ψbu ψb ··· ψbu ψb v . ex2 ∅ 1 u1 n un ∅

Eliminating the α’s Also, we have [α∗, α] = 1 which implies, by the Baker-Campbell-Hausdorﬀ formula,

∗ 2 ∗ exα exα = ex exαexα .

Note that

x(z−z−1) Adexα∗ e−xα (ψ(z)) = e ψ(z) X Adexα∗ e−xα (ψk ) = J`(2x)ψk−` =: ψbk `∈Z

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 21 / 28 and combining everything

v ∗exα∗ ψ ψ∗ ··· ψ ψ∗ exαv ∅ u1 u1 un un ∅ ∗ ∗ ∗ ρ(U) = = v ψbu ψb ··· ψbu ψb v . ex2 ∅ 1 u1 n un ∅

Eliminating the α’s Also, we have [α∗, α] = 1 which implies, by the Baker-Campbell-Hausdorﬀ formula,

∗ 2 ∗ exα exα = ex exαexα .

Note that

x(z−z−1) Adexα∗ e−xα (ψ(z)) = e ψ(z) X Adexα∗ e−xα (ψk ) = J`(2x)ψk−` =: ψbk `∈Z ∗ xα ∗ xα∗ v∅ e = v∅ e v∅ = v∅

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 21 / 28 Eliminating the α’s Also, we have [α∗, α] = 1 which implies, by the Baker-Campbell-Hausdorff formula,

∗ 2 ∗ exα exα = ex exαexα .

Note that

x(z−z−1) Adexα∗ e−xα (ψ(z)) = e ψ(z) X Adexα∗ e−xα (ψk ) = J`(2x)ψk−` =: ψbk `∈Z ∗ xα ∗ xα∗ v∅ e = v∅ e v∅ = v∅ and combining everything

v ∗exα∗ ψ ψ∗ ··· ψ ψ∗ exαv ∅ u1 u1 un un ∅ ∗ ∗ ∗ ρ(U) = = v ψbu ψb ··· ψbu ψb v . ex2 ∅ 1 u1 n un ∅

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 21 / 28 Wick’s lemma (see Gaudin 1960 for a simple proof using CAR)

Let ϕ1, ϕ3, . . . , ϕ2n−1 denote linear combinations of the ψk ’s and ∗ ∗ ∗ ∗ ϕ2, ϕ4, . . . , ϕ2n denote linear combinations of the ψk ’s. Then we have

∗ ∗ ∗ hϕ1ϕ2ϕ3ϕ4 ··· ϕ2n−1ϕ2ni = det Ci,j 1≤i,j≤n ( hϕ ϕ∗ i if i ≤ j where C = 2i−1 2j (“time-ordered correlator”). i,j ∗ −hϕ2j ϕ2i−1i if i > j

Example For n = 2 we have

∗ ∗ ∗ ∗ hϕ1ϕ2i hϕ1ϕ4i ∗ ∗ ∗ ∗ hϕ1ϕ2ϕ3ϕ4i = ∗ ∗ = hϕ1ϕ2i · hϕ3ϕ4i + hϕ1ϕ4i · hϕ2ϕ3i. −hϕ2ϕ3i hϕ3ϕ4i

Wick’s lemma (fermionic version) ∗ Let hOi := v∅ Ov∅ denote the vacuum expectation value of an operator O.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 22 / 28 Example For n = 2 we have

∗ ∗ ∗ ∗ hϕ1ϕ2i hϕ1ϕ4i ∗ ∗ ∗ ∗ hϕ1ϕ2ϕ3ϕ4i = ∗ ∗ = hϕ1ϕ2i · hϕ3ϕ4i + hϕ1ϕ4i · hϕ2ϕ3i. −hϕ2ϕ3i hϕ3ϕ4i

Wick’s lemma (fermionic version) ∗ Let hOi := v∅ Ov∅ denote the vacuum expectation value of an operator O. Wick’s lemma (see Gaudin 1960 for a simple proof using CAR)

Let ϕ1, ϕ3, . . . , ϕ2n−1 denote linear combinations of the ψk ’s and ∗ ∗ ∗ ∗ ϕ2, ϕ4, . . . , ϕ2n denote linear combinations of the ψk ’s. Then we have

∗ ∗ ∗ hϕ1ϕ2ϕ3ϕ4 ··· ϕ2n−1ϕ2ni = det Ci,j 1≤i,j≤n ( hϕ ϕ∗ i if i ≤ j where C = 2i−1 2j (“time-ordered correlator”). i,j ∗ −hϕ2j ϕ2i−1i if i > j

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 22 / 28 Wick’s lemma (fermionic version) ∗ Let hOi := v∅ Ov∅ denote the vacuum expectation value of an operator O. Wick’s lemma (see Gaudin 1960 for a simple proof using CAR)

Let ϕ1, ϕ3, . . . , ϕ2n−1 denote linear combinations of the ψk ’s and ∗ ∗ ∗ ∗ ϕ2, ϕ4, . . . , ϕ2n denote linear combinations of the ψk ’s. Then we have

∗ ∗ ∗ hϕ1ϕ2ϕ3ϕ4 ··· ϕ2n−1ϕ2ni = det Ci,j 1≤i,j≤n ( hϕ ϕ∗ i if i ≤ j where C = 2i−1 2j (“time-ordered correlator”). i,j ∗ −hϕ2j ϕ2i−1i if i > j

Example For n = 2 we have

∗ ∗ ∗ ∗ hϕ1ϕ2i hϕ1ϕ4i ∗ ∗ ∗ ∗ hϕ1ϕ2ϕ3ϕ4i = ∗ ∗ = hϕ1ϕ2i · hϕ3ϕ4i + hϕ1ϕ4i · hϕ2ϕ3i. −hϕ2ϕ3i hϕ3ϕ4i

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 22 / 28 The final step is to observe that

∗ 1 hψk ψ` i = δk,` k<0

and therefore

∗ X hψbs ψbt i = Js+u(2x)Jt+u(2x) =: Jθ(s, t). 0 u∈Z>0

Applying Wick’s lemma

We deduce that

∗ ∗ ∗ ρ(U) = hψbu1 ψbu ··· ψbun ψbu i = det hψbui ψbu i 1 n 1≤i,j≤n j

∗ ∗ (“time-ordering” does not matter here as −ψbuψbv = ψbv ψbu for u 6= v).

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 23 / 28 Applying Wick’s lemma

We deduce that

∗ ∗ ∗ ρ(U) = hψbu1 ψbu ··· ψbun ψbu i = det hψbui ψbu i 1 n 1≤i,j≤n j

∗ ∗ (“time-ordering” does not matter here as −ψbuψbv = ψbv ψbu for u 6= v).

The ﬁnal step is to observe that

∗ 1 hψk ψ` i = δk,` k<0

and therefore

∗ X hψbs ψbt i = Js+u(2x)Jt+u(2x) =: Jθ(s, t). 0 u∈Z>0

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 23 / 28 Main result of “yesterday”

hθi P {u1,..., un} ⊂ S(λ ) = det Jθ(ui , uj ). 1≤i,j≤n

The correlation kernel Jθ is the discrete Bessel kernel √ √ X 0 Jθ(s, t) = Js+`(2 θ)Jt+`(2 θ), s, t ∈ Z 0 `∈Z>0

where Jn is the Bessel function of order n.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 24 / 28 We will use contour integral representations:

1 x(z−z−1) dz Jn(2x) = e n+1 2iπ ˛|z|=r z 1 ex(z−z−1) dz · dw Jθ(s, t) = · . 2 x(w−w −1) s+ 1 −t+ 1 (2iπ) ‹|z|>|w|>0 e (z − w)z 2 w 2

“Today”: asymptotics

Basically all we need to do is to understand the asymptotics of Jθ.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 25 / 28 “Today”: asymptotics

Basically all we need to do is to understand the asymptotics of Jθ. We will use contour integral representations:

1 x(z−z−1) dz Jn(2x) = e n+1 2iπ ˛|z|=r z 1 ex(z−z−1) dz · dw Jθ(s, t) = · . 2 x(w−w −1) s+ 1 −t+ 1 (2iπ) ‹|z|>|w|>0 e (z − w)z 2 w 2

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 25 / 28 Theorem 1 [BOO/J] 2 Fix A ∈ R and consider the asymptotic regime θ = x → ∞ with s, t ∼ Ax and s − t fixed. Then we have

( χ π if s = t, Jθ(s, t) → Ksin(s − t; χ) := sin χ(s−t) π(s−t) if s 6= t,

where  arccos(A/2) if |A| ≤ 2,  χ := 0 if A > 2, π if A < −2.

We deduce immediately that, if u1,..., un are such that ui ∼ Ax and ui − uj remains ﬁxed for all i, j, then hθi P {u1,..., un} ⊂ S(λ ) → det Ksin(ui − uj ; χ). 1≤i,j≤n

Bulk limit: discrete sine kernel

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 26 / 28 We deduce immediately that, if u1,..., un are such that ui ∼ Ax and ui − uj remains fixed for all i, j, then hθi P {u1,..., un} ⊂ S(λ ) → det Ksin(ui − uj ; χ). 1≤i,j≤n

Bulk limit: discrete sine kernel Theorem 1 [BOO/J] 2 Fix A ∈ R and consider the asymptotic regime θ = x → ∞ with s, t ∼ Ax and s − t ﬁxed. Then we have

( χ π if s = t, Jθ(s, t) → Ksin(s − t; χ) := sin χ(s−t) π(s−t) if s 6= t,

where  arccos(A/2) if |A| ≤ 2,  χ := 0 if A > 2, π if A < −2.

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 26 / 28 Bulk limit: discrete sine kernel Theorem 1 [BOO/J] 2 Fix A ∈ R and consider the asymptotic regime θ = x → ∞ with s, t ∼ Ax and s − t fixed. Then we have

( χ π if s = t, Jθ(s, t) → Ksin(s − t; χ) := sin χ(s−t) π(s−t) if s 6= t,

where  arccos(A/2) if |A| ≤ 2,  χ := 0 if A > 2, π if A < −2.

We deduce immediately that, if u1,..., un are such that ui ∼ Ax and ui − uj remains ﬁxed for all i, j, then hθi P {u1,..., un} ⊂ S(λ ) → det Ksin(ui − uj ; χ). 1≤i,j≤n

JérémieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 26 / 28 We do not quite recover their theorem: here we do a first moment calculation, we should also do second moment to prove concentration, and depoissonize.

Connection with Vershik-Kerov-Logan-Shepp

In particular, for s = t we obtain the one-point function (particle density).

1.0 3.0

2.5 0.8

2.0 0.6 1.5 0.4 1.0

0.2 0.5

-3 -2 -1 1 2 3 -3 -2 -1 1 2 3

It is consistent with the VKLS limit shape.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 27 / 28 Connection with Vershik-Kerov-Logan-Shepp

In particular, for s = t we obtain the one-point function (particle density).

1.0 3.0

2.5 0.8

2.0 0.6 1.5 0.4 1.0

0.2 0.5

-3 -2 -1 1 2 3 -3 -2 -1 1 2 3

It is consistent with the VKLS limit shape.

We do not quite recover their theorem: here we do a ﬁrst moment calculation, we should also do second moment to prove concentration, and depoissonize.

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 27 / 28 Theorem 2 [BOO/J] Fix σ, τ ∈ R and consider the asymptotic regime θ = x2 → ∞ with

s = 2x + σx1/3 + o(x1/3), t = 2x + τx1/3 + o(x1/3).

Then we have ∞ 1/3 x Jθ(s, t) → A(σ, τ) := Ai(σ + υ) Ai(τ + υ)dυ ˆ0

ζ3 1 3 −yζ where Ai is the Airy function given by Ai(y) = 2iπ <(ζ)=1 e dζ. ´ For the LIS problem we are interested in the gap probability

hθi P(λ1 < t) = det(I − Jθ){t,t+1,...} → det(I − A)L2(τ,∞) = F2(τ).

The Baik-Deift-Johansson theorem follows by a depoissonization argument!

Edge limit (A = 2): Airy kernel

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 28 / 28 For the LIS problem we are interested in the gap probability

hθi P(λ1 < t) = det(I − Jθ){t,t+1,...} → det(I − A)L2(τ,∞) = F2(τ).

The Baik-Deift-Johansson theorem follows by a depoissonization argument!

Edge limit (A = 2): Airy kernel Theorem 2 [BOO/J] Fix σ, τ ∈ R and consider the asymptotic regime θ = x2 → ∞ with

s = 2x + σx1/3 + o(x1/3), t = 2x + τx1/3 + o(x1/3).

Then we have ∞ 1/3 x Jθ(s, t) → A(σ, τ) := Ai(σ + υ) Ai(τ + υ)dυ ˆ0

ζ3 1 3 −yζ where Ai is the Airy function given by Ai(y) = 2iπ <(ζ)=1 e dζ. ´

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 28 / 28 → det(I − A)L2(τ,∞) = F2(τ).

The Baik-Deift-Johansson theorem follows by a depoissonization argument!

Edge limit (A = 2): Airy kernel Theorem 2 [BOO/J] Fix σ, τ ∈ R and consider the asymptotic regime θ = x2 → ∞ with

s = 2x + σx1/3 + o(x1/3), t = 2x + τx1/3 + o(x1/3).

Then we have ∞ 1/3 x Jθ(s, t) → A(σ, τ) := Ai(σ + υ) Ai(τ + υ)dυ ˆ0

ζ3 1 3 −yζ where Ai is the Airy function given by Ai(y) = 2iπ <(ζ)=1 e dζ. ´ For the LIS problem we are interested in the gap probability

hθi P(λ1 < t) = det(I − Jθ){t,t+1,...}

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 28 / 28 The Baik-Deift-Johansson theorem follows by a depoissonization argument!

Edge limit (A = 2): Airy kernel Theorem 2 [BOO/J] Fix σ, τ ∈ R and consider the asymptotic regime θ = x2 → ∞ with

s = 2x + σx1/3 + o(x1/3), t = 2x + τx1/3 + o(x1/3).

Then we have ∞ 1/3 x Jθ(s, t) → A(σ, τ) := Ai(σ + υ) Ai(τ + υ)dυ ˆ0

ζ3 1 3 −yζ where Ai is the Airy function given by Ai(y) = 2iπ <(ζ)=1 e dζ. ´ For the LIS problem we are interested in the gap probability

hθi P(λ1 < t) = det(I − Jθ){t,t+1,...} → det(I − A)L2(τ,∞) = F2(τ).

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 28 / 28 Edge limit (A = 2): Airy kernel Theorem 2 [BOO/J] Fix σ, τ ∈ R and consider the asymptotic regime θ = x2 → ∞ with

s = 2x + σx1/3 + o(x1/3), t = 2x + τx1/3 + o(x1/3).

Then we have ∞ 1/3 x Jθ(s, t) → A(σ, τ) := Ai(σ + υ) Ai(τ + υ)dυ ˆ0

ζ3 1 3 −yζ where Ai is the Airy function given by Ai(y) = 2iπ <(ζ)=1 e dζ. ´ For the LIS problem we are interested in the gap probability

hθi P(λ1 < t) = det(I − Jθ){t,t+1,...} → det(I − A)L2(τ,∞) = F2(τ).

The Baik-Deift-Johansson theorem follows by a depoissonization argument!

J´er´emieBouttier (CEA/ENS de Lyon) Around the Plancherel measure on partitions 21 March 2019 28 / 28