Random vicious walks and random matrices.

*(English)*Zbl 1026.60071The author considers an infinite system of nearest-neighbor walks on \(\mathbb Z\) in discrete time. At time zero, there is one walker each at the points \(0, 2, 4, 6,\dots\). At each discrete time unit \(1,2,3,\dots\), every walker makes a step with equal probability to the right or left. The only condition is that no two walkers may be at one site at the same time, i.e., the walkers are non-colliding. Related walker systems appear in the literature under the notion random turn walks or lock step walks. The model is equivalent to a certain totally asymmetric exclusion process.

The main result of the paper is the identification of the limiting distribution of the number of left steps made by the leftmost walker by time \(N\), denoted \(L_1(N)\), conditioned on the event \(E(N,k)\) that in the system a total of \(k\) steps has been made up to time \(N\). Here \(k\) is coupled with \(N\) as \(k=t^2 N^2/(1-t^2)+o(N^{4/3})\), where \(t\in(0,1)\) is a parameter. It turns out that the limiting distribution is an elementary transformation of the well-known GOE Tracy-Widom distribution, which appeared for the first time in 1994 as the (renormalized and rescaled) limiting distribution of the largest eigenvalue of the Gaussian orthogonal ensemble, GOE. More precisely, denote this distribution function by \(F_1\) (it is defined in terms of a solution of the Painlevé II equation), then the author obtains that \[ \lim_{N\to\infty} \mathbb P\Bigl(\frac{L_1(N)-\eta(t)N}{\rho(t) N^{1/3}} \leq x \Big|E(N,k)\Bigr)=F_1(x),\qquad x\in\mathbb R, \] where \(\eta(t)=2t/(1+t)\) and \(\rho(t)=(1(1-t))^{1/3}/(1+t)\). Furthermore, it is also proved that all the rescaled conditional moments of \(L_1(N)\) converge to the corresponding moments of the Tracy-Widom distribution. Extensions to the \(j\)th walker for any \(j\in \mathbb N\) are discussed and contrasted with the corresponding results for the random turn walk. The main tools are a bijection between certain path system classes and classes of semistandard Young tableaux [established by A. J. Guttmann, A. L. Owczarek and X. G. Viennot, J. Phys. A, Math. Gen. 31, 8123-8135 (1998; Zbl 0930.05098)], representations of the generating function for the first row in terms of Hankel determinants, orthogonal polynomials, a Riemann-Hilbert approach and the Deift-Zhou steepest-descent method.

The main result of the paper is the identification of the limiting distribution of the number of left steps made by the leftmost walker by time \(N\), denoted \(L_1(N)\), conditioned on the event \(E(N,k)\) that in the system a total of \(k\) steps has been made up to time \(N\). Here \(k\) is coupled with \(N\) as \(k=t^2 N^2/(1-t^2)+o(N^{4/3})\), where \(t\in(0,1)\) is a parameter. It turns out that the limiting distribution is an elementary transformation of the well-known GOE Tracy-Widom distribution, which appeared for the first time in 1994 as the (renormalized and rescaled) limiting distribution of the largest eigenvalue of the Gaussian orthogonal ensemble, GOE. More precisely, denote this distribution function by \(F_1\) (it is defined in terms of a solution of the Painlevé II equation), then the author obtains that \[ \lim_{N\to\infty} \mathbb P\Bigl(\frac{L_1(N)-\eta(t)N}{\rho(t) N^{1/3}} \leq x \Big|E(N,k)\Bigr)=F_1(x),\qquad x\in\mathbb R, \] where \(\eta(t)=2t/(1+t)\) and \(\rho(t)=(1(1-t))^{1/3}/(1+t)\). Furthermore, it is also proved that all the rescaled conditional moments of \(L_1(N)\) converge to the corresponding moments of the Tracy-Widom distribution. Extensions to the \(j\)th walker for any \(j\in \mathbb N\) are discussed and contrasted with the corresponding results for the random turn walk. The main tools are a bijection between certain path system classes and classes of semistandard Young tableaux [established by A. J. Guttmann, A. L. Owczarek and X. G. Viennot, J. Phys. A, Math. Gen. 31, 8123-8135 (1998; Zbl 0930.05098)], representations of the generating function for the first row in terms of Hankel determinants, orthogonal polynomials, a Riemann-Hilbert approach and the Deift-Zhou steepest-descent method.

Reviewer: Wolfgang König (Berlin)

##### Keywords:

vicious walkers; lock step walk; totally asymmetric exclusion process; Tracy-Widom distribution; Gaussian orthogonal ensemble##### References:

[1] | Baik, J Amer Math Soc 12 pp 1119– (1999) |

[2] | Baik, Geom Funct Anal |

[3] | ; Algebraic aspects of increasing subsequences. LANL e-print math. CO/9905083, http://xxx.lanl.gov/find/math. |

[4] | ; The asymptotics of monotone subsequences of involutions. LANL e-print math. CO/9905084, http://xxx.lanl.gov/find/math. |

[5] | ; Symmetrized random permutations. LANL e-print math. CO/9910019, http://xxx.lanl.gov/find/math. |

[6] | ; ; On asymptotics of Plancherel measures for symmetric groups. LANL e-print math. CO/9905032, http://xxx.lanl.gov/find/math. |

[7] | Brak, J Phys A 32 pp 2921– (1999) |

[8] | Brak, J Phys A 32 pp 3497– (1999) |

[9] | Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3. New York University, Courant Institute of Mathematical Sciences, New York, 1999. |

[10] | Deift, J Approx Theory 95 pp 388– (1998) |

[11] | Deift, Comm Pure Appl Math 52 pp 1491– (1999) |

[12] | Deift, Comm Pure Appl Math 52 pp 1335– (1999) |

[13] | Fisher, J Statist Phys 34 pp 667– (1984) |

[14] | Fokas, Comm Math Phys 142 pp 313– (1991) |

[15] | Random walks and random permutations. LANL e-print math. CO/9907037, http://xxx.lanl.gov/find/math. |

[16] | Forrester, J Phys A 22 pp l607– (1989) |

[17] | Forrester, J Phys A 23 pp 1259– (1990) |

[18] | Forrester, J Phys A 24 pp 203– (1991) |

[19] | Guttmann, J Phys A 31 pp 8123– (1998) |

[20] | Johansson, Ann of Math (2) 145 pp 519– (1997) |

[21] | Johansson, Math Res Lett 5 pp 63– (1998) · Zbl 0923.60008 · doi:10.4310/MRL.1998.v5.n1.a6 |

[22] | Discrete orthogonal polynomial ensembles and the Plancherel measure. LANL e-print math. CO/9906120, http://xxx.lanl.gov/find/math. |

[23] | Knuth, Pacific J Math 34 pp 709– (1970) · Zbl 0199.31901 · doi:10.2140/pjm.1970.34.709 |

[24] | Random matrices. Second edition. Academic, Boston, 1991. · Zbl 0780.60014 |

[25] | Random matrices and random permutations. LANL e-print math. CO/9903176, http://xxx.lanl.gov/find/math. |

[26] | Representations of finite and compact groups. Graduate Studies in Mathematics, 10. American Mathematical Society, Providence, R.I., 1996. |

[27] | Enumerative combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics, 62. Cambridge University, Cambridge, 1999. |

[28] | Orthogonal polynomials. Fourth edition. American Mathematical Society, Colloquium Publications, 23. American Mathematical Society, Providence, R.I., 1975. |

[29] | Tracy, Comm Math Phys 159 pp 151– (1994) |

[30] | Tracy, Comm Math Phys 177 pp 727– (1996) |

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