Dynamical properties of a tagged particle in TASEP with the step-type initial condition

1. Dynamical properties of a tagged particle in TASEP withthe step-type initial condition 10/31/2006 Luminy conference on random matrices Takashi Imamura, Institute of industrial science, University of Tokyo Joint work with Tomohiro Sasamoto (Chiba University)

2. Totally asymmetric simple exclusion process(TASEP) 1 discrete time version total number of particles hopping rate of the i th particle exclusion effect Step-type initial condition

3. Motivation: Dynamics of a tagged particle 2 Distance which the th particle moves during Multi-time distribution function 1. dynamical (multi-time) correlation many body effect of exclusion (non Gaussian distribution) inhomogeneity of 2. universality of the largest e.v. in multi-matrix models (Airy process, RM+external sources・・・)

4. Result 1 3 Multi-time distribution function . Growth process of Young diagram characterized by the Schur process. Okounkov and Reshetikhin(’03) Borodin and Rains(‘06)

5. Derivation of result 1 4 TASEP last passage problem in 01 matrix Growth of Young diagram (Schur process) Result 1 c.f. current fluctuation in TASEP : Johansson (’00) oriented digital boiling model: Gravner, Tracy and Widom (’01,’02)

6. From TASEP to 01 matrix 5 matrix Set of 01 matrices TASEP Particle hops to the right site Particle stays at the same site Particle cannot hop (exclusion effect) ??

7. Last passage problem 6 #of stay for the M th particle from t=0 through N+M-1 Proposition “Length” of maximum down/left path down/left path e.g. start × pick up 1s once per each row × goal

8. Dual RSK correspondence 7 Stanley,Enumerative Combinatorics vol.2. Young diagram 01matrix Pair of tableaux P,Q Length of first column

9. Growth of Young diagram 8 Schur process : Schur function O’Connell (‘03) Borodin and Olshanski(‘06) this type of growth problem

10. Multi-time distribution (result 1) 9 Fredholm determinant

11. Result 2 Scaling limit 10 heavy particles normal particle with hopping rate (total number M-n) heavy particle with hopping rate (total number n)

12. 2-1 Scaled average position 11

13. 2-2 Fluctuation 12 1: some special kernel 2: Airy process (Limiting process of the largest eigenvalue in GUE Dyson’s Brownian motion model) 3: Limiting distribution of the largest eigenvalue of Hermitian multi matrix model with rank n external source 4:Largest eigenvalue distribution of GUE Dyson’s Brownian motion model.

14. 2-2 Fluctuation 13 1: some special kernel cf. “critical regime” in Gravner, Tracy and Widom (‘01) 2: Airy process m=1:GUE Tracy-Widom distribution (‘94) GUE Dyson’s BM model H: N×N Hermitian matrix largest eigenvalue

15. 14 3:Random matrix with rank n external source m=1: Baik, Ben Arous, and Peche(‘05) Baik(‘06) :Painleve expression rank n external source 4:n×n GUE Dyson’s BM model

16. 15 Summary • Position fluctuation of a tagged particle in TASEP with step-type initial condition. Schur process • TASEP Scaling limit last passage problem Largest eigenvalue process in multi-matrix model Dual RSK Growth process of Young diagram