Jinyoung Park and Huy Tuan Pham Prove the Kahn-Kalai Conjecture

Past Member (2020–21) Jinyoung Park, a Szegö Assistant Professor at Stanford University, and Huy Tuan Pham, a Stanford Ph.D. student, proved the Kahn-Kalai Conjecture, a central problem in probabilistic combinatorics.

The conjecture concerns determining the precise point (e.g. temperature, pressure, probability, etc.) at which a “phase transition’’ occurs in a large variety of systems. The systems are studied widely in statistical mechanics and graph theory. While this point is extremely hard to compute, in 2006 Jeff Kahn and Gil Kalai, past IAS Member (1995, 2000) and frequent visitor, conjectured that it is very close to another parameter which is much easier to compute. If true, it could be possible to approximate well when phase transitions occur. This has been called “the expectation threshold conjecture.’’

Two years ago, Park—and a team including Kahn, Keith Frankston, and past IAS Visitor (2020–21) Bhargav Narayanan—proved a weaker form of the conjecture. Despite excitement and attempts by experts, proving the full conjecture using their technique evaded all. In a blog post, Kalai said that proving “the full expectation threshold conjecture looked like a difficult task”. A few weeks ago, Park, with Pham, surprised everyone, finding a direct simple argument (which settles another conjecture of French Mathematician Michel Pierre Talagrand), solving one of the major puzzles in the field. 

Avi Wigderson, Herbert H. Maass Professor in the School of Mathematics, said “Jinyoung Park was a postdoc in my group last year, working relentlessly on this conjecture and related ones. Her stamina and ingenuity are remarkable’’.

Learn more about Park’s academic journey and her motivation to explore “deep” mathematical questions in the IAS video series “Paths to Math.”