## The Annals of Probability

### Nonintersecting Brownian motions on the unit circle

#### Abstract

We consider an ensemble of $n$ nonintersecting Brownian particles on the unit circle with diffusion parameter $n^{-1/2}$, which are conditioned to begin at the same point and to return to that point after time $T$, but otherwise not to intersect. There is a critical value of $T$ which separates the subcritical case, in which it is vanishingly unlikely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper, we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as $n\to\infty$, and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode processes in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painlevé II equation of size 2 $\times$ 2. The proofs are based on the determinantal structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral.

#### Article information

Source
Ann. Probab., Volume 44, Number 2 (2016), 1134-1211.

Dates
Revised: December 2014
First available in Project Euclid: 14 March 2016

https://projecteuclid.org/euclid.aop/1457960393

Digital Object Identifier
doi:10.1214/14-AOP998

Mathematical Reviews number (MathSciNet)
MR3474469

Zentralblatt MATH identifier
1342.60138

#### Citation

Liechty, Karl; Wang, Dong. Nonintersecting Brownian motions on the unit circle. Ann. Probab. 44 (2016), no. 2, 1134--1211. doi:10.1214/14-AOP998. https://projecteuclid.org/euclid.aop/1457960393

#### References

• Adler, M., Ferrari, P. L. and van Moerbeke, P. (2013). Nonintersecting random walks in the neighborhood of a symmetric tacnode. Ann. Probab. 41 2599–2647.
• Adler, M., Johansson, K. and van Moerbeke, P. (2014). Double Aztec diamonds and the tacnode process. Adv. Math. 252 518–571.
• Adler, M., Orantin, N. and van Moerbeke, P. (2010). Universality for the Pearcey process. Phys. D 239 924–941.
• Adler, M. and van Moerbeke, P. (2005). PDEs for the joint distributions of the Dyson, Airy and sine processes. Ann. Probab. 33 1326–1361.
• Adler, M., van Moerbeke, P. and Wang, D. (2013). Random matrix minor processes related to percolation theory. Random Matrices Theory Appl. 2 1350008, 72.
• Aptekarev, A. I., Bleher, P. M. and Kuijlaars, A. B. J. (2005). Large $n$ limit of Gaussian random matrices with external source. II. Comm. Math. Phys. 259 367–389.
• Baik, J., Liechty, K. and Schehr, G. (2012). On the joint distribution of the maximum and its position of the $\mathrm{Airy}_{2}$ process minus a parabola. J. Math. Phys. 53 083303, 13.
• Baik, J. and Suidan, T. M. (2007). Random matrix central limit theorems for nonintersecting random walks. Ann. Probab. 35 1807–1834.
• Baik, J., Kriecherbauer, T., McLaughlin, K. T.-R. and Miller, P. D. (2007). Discrete Orthogonal Polynomials: Asymptotics and Applications. Annals of Mathematics Studies 164. Princeton Univ. Press, Princeton, NJ.
• Bleher, P. and Kuijlaars, A. B. J. (2004). Large $n$ limit of Gaussian random matrices with external source. I. Comm. Math. Phys. 252 43–76.
• Bleher, P. M. and Kuijlaars, A. B. J. (2007). Large $n$ limit of Gaussian random matrices with external source. III. Double scaling limit. Comm. Math. Phys. 270 481–517.
• Bleher, P. and Liechty, K. (2010). Exact solution of the six-vertex model with domain wall boundary conditions: Antiferroelectric phase. Comm. Pure Appl. Math. 63 779–829.
• Bleher, P. and Liechty, K. (2011). Uniform asymptotics for discrete orthogonal polynomials with respect to varying exponential weights on a regular infinite lattice. Int. Math. Res. Not. IMRN 2 342–386.
• Bleher, P. and Liechty, K. (2014). Random Matrices and the Six-Vertex Model. CRM Monograph Series 32. Amer. Math. Soc., Providence, RI.
• Borodin, A. and Rains, E. M. (2005). Eynard–Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121 291–317.
• Byrd, P. F. and Friedman, M. D. (1971). Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. Springer, New York.
• Cardy, J. (2003). Stochastic Loewner evolution and Dyson’s circular ensembles. J. Phys. A 36 L379–L386.
• Castillo, I. P. and Dupic, T. (2014). Reunion probabilities of $N$ one-dimensional random walkers with mixed boundary conditions. J. Stat. Phys. 156 606–616.
• Claeys, T. and Kuijlaars, A. B. J. (2006). Universality of the double scaling limit in random matrix models. Comm. Pure Appl. Math. 59 1573–1603.
• Corwin, I. (2012). The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1 1130001, 76.
• Corwin, I. and Hammond, A. (2014). Brownian Gibbs property for Airy line ensembles. Invent. Math. 195 441–508.
• Deift, P. A. (1999). Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics 3. New York Univ., Courant Institute of Mathematical Sciences, New York.
• Delvaux, S. (2013). The tacnode kernel: Equality of Riemann–Hilbert and Airy resolvent formulas. Available at arXiv:1211.4845v2.
• Delvaux, S., Kuijlaars, A. B. J. and Zhang, L. (2011). Critical behavior of nonintersecting Brownian motions at a tacnode. Comm. Pure Appl. Math. 64 1305–1383.
• Douglas, M. R. and Kazakov, V. A. (1993). Large $N$ phase transition in continuum QCD$_{2}$. Phys. Lett. B 319 219–230.
• Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.
• Dyson, F. J. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191–1198.
• Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1981). Higher Transcendental Functions. Vol. II. Robert E. Krieger Publishing, Melbourne, FL.
• Eynard, B. and Mehta, M. L. (1998). Matrices coupled in a chain. I. Eigenvalue correlations. J. Phys. A 31 4449–4456.
• Ferrari, P. L. and Vető, B. (2012). Non-colliding Brownian bridges and the asymmetric tacnode process. Electron. J. Probab. 17 17.
• Flaschka, H. and Newell, A. C. (1980). Monodromy- and spectrum-preserving deformations. I. Comm. Math. Phys. 76 65–116.
• Forrester, P. J. (1990). Exact solution of the lock step model of vicious walkers. J. Phys. A 23 1259–1273.
• Forrester, P. J., Majumdar, S. N. and Schehr, G. (2011). Non-intersecting Brownian walkers and Yang–Mills theory on the sphere. Nuclear Phys. B 844 500–526.
• Fulmek, M. (2004/07). Nonintersecting lattice paths on the cylinder. Sém. Lothar. Combin. 52 16 pp. (electronic).
• Gessel, I. and Viennot, G. (1985). Binomial determinants, paths, and hook length formulae. Adv. Math. 58 300–321.
• Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th ed. Elsevier/Academic Press, Amsterdam.
• Gross, D. J. and Matytsin, A. (1995). Some properties of large-$N$ two-dimensional Yang–Mills theory. Nuclear Phys. B 437 541–584.
• Hastings, S. P. and McLeod, J. B. (1980). A boundary value problem associated with the second Painlevé transcendent and the Korteweg–de Vries equation. Arch. Ration. Mech. Anal. 73 31–51.
• Hobson, D. G. and Werner, W. (1996). Non-colliding Brownian motions on the circle. Bull. Lond. Math. Soc. 28 643–650.
• Johansson, K. (2005). The arctic circle boundary and the Airy process. Ann. Probab. 33 1–30.
• Johansson, K. (2013). Non-colliding Brownian motions and the extended tacnode process. Comm. Math. Phys. 319 231–267.
• Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd ed. Wiley, Hoboken, NJ.
• Karlin, S. and McGregor, J. (1959). Coincidence probabilities. Pacific J. Math. 9 1141–1164.
• Katori, M. and Tanemura, H. (2007). Noncolliding Brownian motion and determinantal processes. J. Stat. Phys. 129 1233–1277.
• Kemp, A. W. (1997). Characterizations of a discrete normal distribution. J. Statist. Plann. Inference 63 223–229.
• Kuijlaars, A. B. J. (2000). On the finite-gap ansatz in the continuum limit of the Toda lattice. Duke Math. J. 104 433–462.
• Kuijlaars, A. B. J. (2010). Multiple orthogonal polynomials in random matrix theory. In Proceedings of the International Congress of Mathematicians. Volume III 1417–1432. Hindustan Book Agency, New Delhi.
• Kuijlaars, A. (2014). The tacnode Riemann–Hilbert problem. Constr. Approx. 39 197–222.
• Liechty, K. (2012). Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials. J. Stat. Phys. 147 582–622.
• Liechty, K. and Wang, D. (2013). Nonintersecting Brownian motions on the unit circle: Noncritical cases. Available at arXiv:1312.7390v3.
• Lindström, B. (1973). On the vector representations of induced matroids. Bull. Lond. Math. Soc. 5 85–90.
• Okounkov, A. and Reshetikhin, N. (2003). Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16 581–603 (electronic).
• Okounkov, A. and Reshetikhin, N. (2007). Random skew plane partitions and the Pearcey process. Comm. Math. Phys. 269 571–609.
• Schehr, G., Majumdar, S. N., Comtet, A. and Randon-Furling, J. (2008). Exact distribution of the maximal height of $p$ vicious walkers. Phys. Rev. Lett. 101 150601, 4.
• Schehr, G., Majumdar, S. N., Comtet, A. and Forrester, P. J. (2013). Reunion probability of $N$ vicious walkers: Typical and large fluctuations for large $N$. J. Stat. Phys. 150 491–530.
• Soshnikov, A. (2000). Determinantal random point fields. Uspekhi Mat. Nauk 55 107–160.
• Szabłowski, P. (2001). Discrete normal distribution and its relationship with Jacobi theta functions. Statist. Probab. Lett. 52 289–299.
• Szegő, G. (1975). Orthogonal Polynomials, 4th ed. Amer. Math. Soc., Providence, RI.
• Tracy, C. A. and Widom, H. (2004). Differential equations for Dyson processes. Comm. Math. Phys. 252 7–41.
• Tracy, C. A. and Widom, H. (2006). The Pearcey process. Comm. Math. Phys. 263 381–400.
• Whittaker, E. T. and Watson, G. N. (1996). A Course of Modern Analysis. Cambridge Univ. Press, Cambridge.