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Mathematical Physics Seminar

On the extension of the FKG inequality to n functions

Siddhartha Sahi - Princeton University

Location:  Hill Center Room 705
Date & time: Thursday, 04 November 2021 at 1:20PM - 2:20PM

The 1971 Fortuin-Kasteleyn-Ginibre (FKG) correlation inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics, combinatorics, statistics, probability, and other fields of mathematics.

In 2008 the speaker conjectured an extended version of this inequality for all n>2 monotone functions on a distributive lattice. This reveals an intriguing connection with the representation theory of the symmetric group.

We give a proof of the conjecture for two special cases: for monotone functions on the unit square in R^k whose (upper) level sets are k-dimensional rectangles, and, more significantly, for {\it arbitrary} monotone functions on the unit square R^2. The general case for R^k, k>2 remains open.

This is joint work with Elliott Lieb.

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