About

Jinho Baik is a faculty member in the Department of Mathematics at the University of Michigan, holding the position of Professor and Director of Graduate Admissions. He earned his Ph.D. from the Courant Institute in 1999. His research interests encompass analysis, probability, and mathematical physics, with a focus on using analytic tools to study probabilistic models. His work includes the investigation of eigenvalues of large random matrices, heights of randomly growing interfaces such as melting, wetting, and burning, as well as the locations of vicious random walkers in models like random tiling. He also studies the longest increasing subsequence of random permutations, which involves maximization in random environments. These models are often explicitly solvable and demonstrate remarkable universal behavior. Although these problems are probabilistic, he employs analytic techniques such as complex analysis, asymptotic analysis, functional analysis, potential theory, and combinatorics to analyze them. Additionally, he has an interest in integrable differential equations and explores various applications of these models to statistics and engineering.

Research topics

• Mathematics
• Combinatorics
• Pure mathematics
• Mathematical analysis
• Statistical physics

Selected publications

• Pinched-up periodic KPZ fixed point

  arXiv (Cornell University) · 2024-03-03

  preprintOpen access1st authorCorresponding

  The periodic KPZ fixed point is the conjectural universal limit of the KPZ universality class models on a ring when both the period and time critically tend to infinity. For the case of the periodic narrow wedge initial condition, we consider the conditional distribution when the periodic KPZ fixed point is unusually large at a particular position and time. We prove a conditional limit theorem up to the ``pinch-up" time. When the period is large enough, the result is the same as that for the KPZ fixed point on the line obtained by Liu and Wang in 2022. We identify the regimes in which the result changes and find probabilistic descriptions of the limits.

  PublisherOA PDFDOI

• Differential Equations for the KPZ and Periodic KPZ Fixed Points

  Communications in Mathematical Physics · 2023-04-13 · 7 citations

  articleOpen access1st author
  PublisherOA PDFDOI

• KPZ limit theorems

  EMS Press eBooks · 2023-12-15

  book-chapterOpen access1st authorCorresponding

  One-dimensional interacting particle systems, 1+1 random growth models, and two-dimensional directed polymers define 2D height fields. The KPZ universality conjecture posits that an appropriately scaled height function converges to a model-independent universal random field for a large class of models. We survey limit theorems for a few models and discuss changes that arise in different domains. In particular, we present recent results on periodic domains. We also comment on integrable probability models, integrable differential equations, and universality.

  PublisherOA PDFDOI

• Limiting one-point distribution of periodic TASEP

  Annales de l Institut Henri Poincaré Probabilités et Statistiques · 2022-02-01 · 6 citations

  preprintOpen access1st authorCorresponding

  Il est attendu que la limite du temps de relaxation pour la distribution à un point du processus d’exclusion simple totalement asymétrique et périodique en espace est la distribution à un point de la classe universelle KPZ en domaine périodique. Contrairement au cas de la ligne infinie, la distribution à un point dépend de façon non-triviale des paramètres d’échelle en temps. Nous étudions plusieurs propriétés de cette distribution dans le cas des lois initiales à pas périodiques et plates. Nous montrons que la distribution change de la loi de Tracy–Widom dans la limite en temps petits à la loi Gaussienne en temps grands, et nous obtenons aussi une estimée de queue pour tous les temps. De plus, nous établissons une relation avec les équations différentielles intégrables telles que les équations KP, les systèmes couplés mKdV et équations de la chaleur non-linéaires, et l’équation KdV.

  PublisherOA PDFDOI

• KPZ limit theorems

  arXiv (Cornell University) · 2022-06-28 · 4 citations

  preprintOpen access1st authorCorresponding

  One-dimensional interacting particle systems, 1+1 random growth models, and two-dimensional directed polymers define 2d height fields. The KPZ universality conjecture posits that an appropriately scaled height function converges to a model-independent universal random field for a large class of models. We survey limit theorems for a few models and discuss changes that arise in different domains. In particular, we present recent results on periodic domains. We also comment on integrable probability models, integrable differential equations, and universality.

  PublisherOA PDFDOI

• Edge Distribution of Thinned Real Eigenvalues in the Real Ginibre Ensemble

  Annales Henri Poincaré · 2022-04-12 · 4 citations

  articleOpen access1st author

  Abstract This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed independently with constant likelihood. We show that the recently discovered integrable structures in [2] generalize from the real Ginibre ensemble to its thinned equivalent. Concretely, we express the aforementioned limiting distribution function as a convex combination of two simple Fredholm determinants and connect the same function to the inverse scattering theory of the Zakharov–Shabat system. As corollaries, we provide a Zakharov–Shabat evaluation of the ensemble’s real eigenvalue generating function and obtain precise control over the limiting distribution function’s tails. The latter part includes the explicit computation of the usually difficult constant factors.

  PublisherOA PDFDOI

• Differential equations for the KPZ and periodic KPZ fixed points

  arXiv (Cornell University) · 2022-08-24

  preprintOpen access1st authorCorresponding

  The KPZ fixed point is a 2d random field, conjectured to be the universal limiting fluctuation field for the height function of models in the KPZ universality class. Similarly, the periodic KPZ fixed point is a conjectured universal field for spatially periodic models. For both fields, their multi-point distributions in the space-time domain have been computed recently. We show that for the case of the narrow-wedge initial condition, these multi-point distributions can be expressed in terms of so-called integrable operators. We then consider a class of operators that include the ones arising from the KPZ and the periodic KPZ fixed points, and find that they are related to various matrix integrable differential equations such as coupled matrix mKdV equations, coupled matrix NLS equations with complex time, and matrix KP-II equations. When applied to the KPZ fixed points, our results extend previously known differential equations for one-point distributions and equal-time, multi-position distributions to multi-time, multi-position setup.

  PublisherOA PDFDOI

• Free energy of bipartite spherical Sherrington–Kirkpatrick model

  Annales de l Institut Henri Poincaré Probabilités et Statistiques · 2020-10-21 · 9 citations

  preprintOpen access1st authorCorresponding

  We consider the free energy of the bipartite spherical Sherrington--Kirkpatrick model. We find the critical temperature and prove the limiting free energy for all non-critical temperature. We also show that the law of the fluctuation of the free energy converges to the Gaussian distribution when the temperature is above the critical temperature, and to the GOE Tracy--Widom distribution when the temperature is below the critical temperature. The result is universal, and the analysis is applicable to a more general setting including the case where the disorders are non-identically distributed.

  PublisherOA PDFDOI

• Periodic TASEP with general initial conditions

  Probability Theory and Related Fields · 2020-10-27 · 22 citations

  article1st author
  PublisherDOI

• Edge distribution of thinned real eigenvalues in the real Ginibre ensemble

  arXiv (Cornell University) · 2020-08-04

  preprintOpen access1st authorCorresponding

  This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed independently with constant likelihood. We show that the recently discovered integrable structures in \cite{BB} generalize from the real Ginibre ensemble to its thinned equivalent. Concretely, we express the aforementioned limiting distribution function as a convex combination of two simple Fredholm determinants and connect the same function to the inverse scattering theory of the Zakharov-Shabat system. As corollaries, we provide a Zakharov-Shabat evaluation of the ensemble's real eigenvalue generating function and obtain precise control over the limiting distribution function's tails. The latter part includes the explicit computation of the usually difficult constant factors.

  PublisherOA PDFDOI

Recent grants

• Random Matrices, Spin Glass, and Interacting Particle Systems

  NSF · $345k · 2020–2024

• FRG: Collaborative Research: Integrable Probability

  NSF · $265k · 2017–2021

• Random Matrices and Related Topics

  NSF · $288k · 2014–2017

• Random Matrices and Applications

  NSF · $117k · 2005–2008

• Random Matrices and Applications

  NSF · $302k · 2011–2014

Frequent coauthors

• Percy Deift

  Courant Institute of Mathematical Sciences

  52 shared

• Eric M. Rains

  43 shared

• Toufic Suidan

  University of Michigan–Ann Arbor

  22 shared

• Zhipeng Liu

  University of Kansas

  16 shared

• Sandrine Péché

  University of Chicago

  15 shared

• Ji Oon Lee

  Korea Advanced Institute of Science and Technology

  12 shared

• K. T-R McLaughlin

  12 shared

• Patrik L. Ferrari

  University of Bonn

  12 shared

Labs

• U-M LSA MathematicsPI