The free boundary Schur process and applications I

We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of"free boundary states". For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions, and for plane overpartitions.

The extra factor q 2N|λ (0) | is needed, as otherwise constant sequences would all have weight 1.

Different type of boundary conditions
We have encountered instances of Schur process with various types of "boundary conditions": Determinantal and pfaffian point processes Determinantal is a subcase of pfaffian, when taking

General definition
The free boundary Schur process is a random sequence of partitions Here: u, v are nonnegative real parameters (recover empty boundary conditions by taking them zero), the ρ ± k are specializations (e.g. single variables for plane partitions), The partition function There is a general product formula for the partition function. To state it in full generality we need some notations: The free boundary Schur process 19 October 2017 21 / 47 The partition function There is a general product formula for the partition function. To state it in full generality we need some notations: The free boundary Schur process 19 October 2017 21 / 47 The partition function There is a general product formula for the partition function. To state it in full generality we need some notations:

Correlation functions
Recall that the point configuration is We will encounter pfaffian correlation kernels with the following "universal" form where F and κ are Laurent series in z and w (obtained as expansions of meromorphic functions in certain compatible annuli). Only F depends on the specializations ρ ± k , κ just depends on the boundary conditions. For u = 0, the point process S(λ) is pfaffian, and its correlation kernel takes the universal form with .
We shall expand the κ's in the annuli |z|, |w | > v , with |z| > |w | for i ≤ i and vice versa otherwise. Remarks: in [Borodin-Raines 2005], the expressions appear slightly different because the first partition is assumed to have even columns, for v = 0, the diagonal entries K 1,1 and K 2,2 vanish, and we recover the result from [Okounkov-Reshetikhin 2003] that S(λ) is determinantal with kernel . Correlation function: two free boundaries For uv > 0 (the general free boundary case), the process is not Pfaffian anymore, but a closely related process is: define the shifted point configuration and take d independant of λ with law t being an arbitrary parameter (θ 3 : Jacobi theta function). Correlation function: two free boundaries We find that S d (λ) is pfaffian, and its correlation kernel takes the universal form with Correlation function: two free boundaries For u = 0 we recover the previous case of one free boundary (exercise !).
It is actually possible to evaluate the pfaffians and express the general n-point correlation for both S(λ) and S d (λ) as a coefficient in a Laurent series in 2n variables. Behind this, there is an elliptic pfaffian identity which can be rewritten as a particular case of an identify due to Okada (2006). For u = 0, we recover Schur's pfaffian identity pf 1≤i<j≤2n An elliptic pfaffian identity . It took us some time to understand how to compute correlation functions, as "Wick's lemma" does not generalize straightforwardly.
The key idea was to introduce "extended" free boundary states, which are not eigenvalues of the charge operator, but correspond to "rotations" (Bogoliubov transformations) of the empty boundary state. This explains the need to consider a shifted point process.
With free boundaries, we also need the Littlewood "reflection" identity

Transfer matrix method: correlation functions
To compute correlation, we need to insert "observables". Let N k be the diagonal matrix such that Then, for instance, counts sequences λ such that (1, k) ∈ S(λ). Insert several N's at appropriate places and normalize for general correlation functions.
To evaluate such expressions, we represent the observables N k in terms of "fermionic" operators.

Semi-infinite wedge space
We consider an infinite-dimensional vector space with basis |S indexed by admissible sets. Denoting s 1 > s 2 > · · · the elements of S, we may identify |S := s 1 ∧ s 2 ∧ s 3 ∧ · · · as a semi-infinite wedge (exterior) product. Partitions form the subspace of charge 0.
The ψ k ψ * 's span the Lie algebra A ∞ .
Evaluation the partition function: empty bc Jérémie Bouttier (CEA/ENS de Lyon) The free boundary Schur process 19 October 2017 46 / 47 Evaluation the partition function: free bc