We study the multipoint distribution of stationary half-space last passage percolation with exponentially weighted times. We derive both finite-size and asymptotic results for this distribution. In the latter case we observe a new one-parameter proce
ss we call half-space Airy stat. It is a one-parameter generalization of the Airy stat process of Baik-Ferrari-Peche, which is recovered far away from the diagonal. All these results extend the one-point results previously proven by the authors.
We study two families of probability measures on integer partitions, which are Schur measures with parameters tuned in such a way that the edge fluctuations are characterized by a critical exponent different from the generic $1/3$. We find that the f
irst part asymptotically follows a higher-order analogue of the Tracy-Widom GUE distribution, previously encountered by Le Doussal, Majumdar and Schehr in quantum statistical physics. We also compute limit shapes, and discuss an exact mapping between one of our families and the multicritical unitary matrix models introduced by Periwal and Shevitz.
We study probabilistic and combinatorial aspects of natural volume-and-trace weighted plane partitions and their continuous analogues. We prove asymptotic limit laws for the largest parts of these ensembles in terms of new and known hard- and soft-ed
ge distributions of random matrix theory. As a corollary we obtain an asymptotic transition between Gumbel and Tracy--Widom GUE fluctuations for the largest part of such plane partitions, with the continuous Bessel kernel providing the interpolation. We interpret our results in terms of two natural models of directed last passage percolation (LPP): a discrete $(max, +)$ infinite-geometry model with rapidly decaying geometric weights, and a continuous $(min, cdot)$ model with power weights.
In this note we study a natural measure on plane partitions giving rise to a certain discrete-time Muttalib-Borodin process (MBP): each time-slice is a discrete version of a Muttalib-Borodin ensemble (MBE). The process is determinantal with explicit
time-dependent correlation kernel. Moreover, in the $q to 1$ limit, it converges to a continuous Jacobi-like MBP with Muttalib-Borodin marginals supported on the unit interval. This continuous process is also determinantal with explicit correlation kernel. We study its hard-edge scaling limit (around 0) to obtain a discrete-time-dependent generalization of the classical continuous Bessel kernel of random matrix theory (and, in fact, of the Meijer $G$-kernel as well). We lastly discuss two related applications: random sampling from such processes, and their interpretations as models of directed last passage percolation (LPP). In doing so, we introduce a corner growth model naturally associated to Jacobi processes, a version of which is the usual corner growth of Forrester-Rains in logarithmic coordinates. The aforementioned hard edge limits for our MBPs lead to interesting asymptotics for these LPP models. In particular, a special cases of our LPP asymptotics give rise (via the random matrix Bessel kernel and following Johanssons lead) to an extremal statistics distribution interpolating between the Tracy-Widom GUE and the Gumbel distributions.
We show that the symplectic and orthogonal character analogues of Okounkovs Schur measure (on integer partitions) are determinantal, with explicit correlation kernels. We apply this to prove certain Borodin-Okounkov-Gessel-type results concerning Toe
plitz+Hankel and Fredholm determinants; a SzegH{o}-type limit theorem; an edge Baik-Deift-Johansson-type asymptotical result for certain symplectic and orthogonal analogues of the poissonized Plancherel measure; and a similar result for actual poissonized Plancherel measures supported on almost symmetric partitions.
In this paper we study stationary last passage percolation (LPP) in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new two-parameter family of distributio
ns: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik--Rains distributions for full-space geometry. We finally show that far enough away from the characteristic line, our distributions indeed converge to the Baik--Rains family. We derive our results using a related integrable model having Pfaffian structure together with careful analytic continuation and steepest descent analysis.
We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik--Deift--Johansson and Baik--Rains in resolving Ulams problem on long
est increasing subsequences of random permutations and the last passage percolation (corner growth) discre
In this note we give a(nother) combinatorial proof of an old result of Baik--Rains: that for appropriately considered independent geometric weights, the generating series for last passage percolation polymers in a $2n times n times n$ quarter square
(point-to-half-line-reflected geometry) splits as the product of two simpler generating series---that for last passage percolation polymers in a point-to-line geometry and that for last passage percolation in a point-to-point-reflected (half-space) geometry, the latter both in an $n times n times n$ triangle. As a corollary, for iid geometric random variables---of parameter $q$ off-diagonal and parameter $sqrt{q}$ on the diagonal---we see that the last passage percolation time in said quarter square obeys Tracy--Widom $mathrm{GOE}^2$ fluctuations in the large $n$ limit as both the point-to-line and the point-to-point-reflected geometries have known GOE fluctuations. This is a discrete analogue of a celebrated Baik--Rains theorem (the limit $q to 0$) and more recently of results from Bisis PhD thesis (the limit $q to 1$).
We revisit the periodic Schur process introduced by Borodin in 2007. Our contribution is threefold. First, we provide a new simpler derivation of its correlation functions via the free fermion formalism. In particular, we shall see that the process b
ecomes determinantal by passing to the grand canonical ensemble, which gives a physical explanation to Borodins shift-mixing trick. Second, we consider the edge scaling limit in the simplest nontrivial case, corresponding to a deformation of the poissonized Plancherel measure on partitions. We show that the edge behavior is described, in a certain crossover regime different from that for the bulk, by the universal finite-temperature Airy kernel, which was previously encountered by Johansson and Le Doussal et al. in other models, and whose extreme value statistics interpolates between the Tracy-Widom GUE and the Gumbel distributions. We also define and prove convergence for a stationary extension of our model. Finally, we compute the correlation functions for a variant of the periodic Schur process involving strict partitions, Schurs P and Q functions, and neutral fermions.
We show, using either Fock space techniques or Macdonald difference operators, that certain symplectic and orthogonal analogues of Okounkovs Schur measure are determinantal with kernels given by explicit double contour integrals. We give two applicat
ions: one equates certain Toeplitz+Hankel determinants of random matrix theory with appropriate Fredholm determinants and computes SzegH{o} asymptotics for the former; another finds that the simplest examples of said measures exhibit discrete sine kernel asymptotics in the bulk and Airy 2 to 1 kernel---along with a certain dual---asymptotics at the edge. We believe the edge behavior to be universal.
