On the Shuffling Algorithm for the Aztec Nordenstam eno@kth.se - PowerPoint PPT Presentation

On the Shuffling Algorithm for the Aztec Diamond Eric On the Shuffling Algorithm for the Aztec Nordenstam eno@kth.se Diamond Background Shuffling algorithm Eric Nordenstam Warren’s Process eno@kth.se Aztec diamond point Universit´ e Catholique de Louvain, Belgium process Asymptotics Statcomb 09 Borodin & Ferrari

The Aztec Diamond On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Aztec diamonds of orders 1, 2, 3 and 4. Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

The Aztec Diamond On the Shuffling Algorithm for the Aztec Diamond Eric Aztec diamonds of orders 1, 2, 3 and 4. Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process The diamond of order n can be tiled in 2 n ( n +1) / 2 ways. Aztec diamond point Elkies et al, ’92 process Asymptotics Borodin & Ferrari

The Aztec Diamond On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

The Aztec Diamond On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

The Aztec Diamond On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

The Aztec Diamond y = 1 x = 16 On the y = 2 x = 15 Shuffling y = 3 x = 14 Algorithm for y = 4 x = 13 the Aztec y = 5 x = 12 Diamond y = 6 x = 11 y = 7 x = 10 Eric y = 8 x = 9 Nordenstam y = 9 x = 8 eno@kth.se y = 10 x = 7 y = 11 x = 6 Background y = 12 x = 5 y = 13 x = 4 Shuffling y = 14 x = 3 algorithm y = 15 x = 2 y = 16 x = 1 Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

The Aztec Diamond y = 1 x = 16 On the y = 2 x = 15 Shuffling y = 3 x = 14 Algorithm for y = 4 x = 13 the Aztec y = 5 x = 12 Diamond y = 6 x = 11 y = 7 x = 10 Eric y = 8 x = 9 Nordenstam y = 9 x = 8 eno@kth.se y = 10 x = 7 y = 11 x = 6 Background y = 12 x = 5 y = 13 x = 4 Shuffling y = 14 x = 3 algorithm y = 15 x = 2 y = 16 x = 1 Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

GUE Minor Process On the Shuffling Let H be a large GUE matrix, i.e. a random matrix with Algorithm for the Aztec probability density Z − 1 e − Tr H 2 . Diamond Let H n = [ H i , j ] 1 ≤ i , j ≤ n be the n :th minor of H . Eric Nordenstam Let H n have eigenvalues λ n 1 , . . . , λ n eno@kth.se n Background Shuffling algorithm Warren’s Process Aztec diamond point process Theorem (Johansson&N ’06) Asymptotics Borodin & The Aztec diamond point process in a suitable rescaling Ferrari converges to the GUE minor process.

On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se 1 The shuffling algorithm Background 2 Warren’s Interlaced Brownian motions Shuffling algorithm 3 Asymptotics Warren’s Process 4 Borodin & Ferrari Aztec diamond point process Asymptotics Borodin & Ferrari

On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Three phases of the algorithm. Background 1 Delete Shuffling algorithm 2 Shuffle Warren’s 3 Create Process Aztec diamond point process Asymptotics Borodin & Ferrari

Delete On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling �→ algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Shuffle On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling �→ algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Create On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling �→ algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Particle dynamics On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

TASEP with step initial condition On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

TASEP with step initial condition On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

TASEP with step initial condition On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

TASEP with step initial condition On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

TASEP with step initial condition On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

TASEP with step initial condition On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Particle dynamics On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Particle dynamics On the Shuffling Algorithm for the Aztec Diamond x 1 1 ( t ) = x 1 1 ( t − 1) + γ 1 1 ( t ) Eric Nordenstam x j 1 ( t ) = x j 1 ( t − 1) + γ j 1 ( t ) − 1 { x j 1 ( t − 1) + γ j 1 ( t ) = x j − 1 eno@kth.se ( t − 1) + 1 } 1 x j j ( t ) = x j j ( t − 1) + γ j j ( t ) + 1 { x j j ( t − 1) + γ j j ( t ) = x j − 1 Background j − 1 ( t − 1) } Shuffling x j i ( t ) = x j i ( t − 1) + γ j i ( t ) − 1 { x j i ( t − 1) + γ j i ( t ) = x j − 1 ( t − 1) + 1 } algorithm j Warren’s + 1 { x j i ( t − 1) + γ j i ( t ) = x j − 1 j − 1 ( t − 1) } . Process Aztec diamond point Here, γ j i ( t ) are i.i.d. fair coin tosses and initial conditions x j i ( j ) = i process for j = 1 , 2 , . . . and 1 ≤ i ≤ j . At each time t , Asymptotics Borodin & i ( t ) ≤ x j − 1 x j ( t ) ≤ x j i +1 ( t ) . Ferrari i

Aztec diamond particle dynamics On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam Perform substitution eno@kth.se Background X j i ( t ) = x j i ( t − j ) (1) Shuffling algorithm For all t , Warren’s Process i ( t ) ≤ X j − 1 X j ( t ) < X j i +1 ( t ) (2) i Aztec diamond point process Asymptotics Borodin & Ferrari

Aztec diamond particle dynamics On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Aztec diamond particle dynamics On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Aztec diamond particle dynamics On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Warren’s Process On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Warren’s Process On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Warren’s Process On the Shuffling Algorithm for the Aztec Diamond Eric Nordenstam eno@kth.se Background Shuffling algorithm Warren’s Process Aztec diamond point process Asymptotics Borodin & Ferrari

Transition Density for Dyson’s BM On the Shuffling Algorithm for the Aztec Let W n = { x ∈ R n : x 1 ≤ x 2 ≤ · · · ≤ x n } . For x , x ′ ∈ W n Diamond Eric Nordenstam ( x , x ′ ) = h n ( x ′ ) eno@kth.se p n , + � ϕ t ( x ′ � h n ( x ) det i − x j ) (3) t Background Shuffling where algorithm � h n ( x ) = ( x j − x i ) (4) Warren’s Process i < j Aztec diamond point and process 1 e − x 2 / 2 t √ ϕ t ( x ) = (5) Asymptotics 2 π t Borodin & Ferrari