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Piotr Śniady

Publications and source records attributed to Piotr Śniady.

At least 19 recordsLinked to original sources
arXiv2026

Jeu de taquin forests and the inverse infinite RSK correspondence

The Plancherel-random infinite Young tableau arises from applying the infinite Robinson-Schensted-Knuth (RSK) correspondence to a sequence of i.i.d. random variables distributed uniformly on the unit interval [0,1]. Building on the isomorphism of dynamical systems established in prior work, we provide an explicit geometric characterization of the inverse map. Our approach makes use of a previously unexplored structure, the jeu de taquin forest on the infinite tableau, where edges connect boxes according to local comparison rules. We prove that each tree in this forest almost surely extends toward infinity with a well-defined asymptotic direction, establishing a canonical bijection between trees and values in the i.i.d. input sequence. The ordering is recovered through tree lifetimes under iterated jeu de taquin transformations.

math.PR↗
arXiv2026

Determinant and Pfaffian formulas for particle annihilation

We consider systems of particles on a line in which colliding particles annihilate each other and vanish. Computing exact annihilation probabilities is difficult because every collision reduces the particle count, while determinantal methods require a fixed count throughout. The ghost particle method, introduced in a companion paper for coalescence, removes the obstacle: destroyed particles continue walking as invisible ghosts, so the number of trajectories never changes. Applied to annihilation, the method yields an exact determinantal formula for the probability of any prescribed outcome - the number of annihilations, the survivor positions, and the positions of the ghosts. For complete annihilation, where no particle survives, the determinant collapses to a Pfaffian, an algebraic relative of the determinant built from pairwise quantities: although the particles interact, the extinction probability is determined by pairwise annihilation probabilities alone. This gives a combinatorial explanation of the Pfaffian structure of annihilating systems, previously derived through differential equations for specific dynamics. The annihilation formula also yields results about coalescence: the event that prescribed pairs of particles have merged can be reinterpreted as complete annihilation, producing a Pfaffian coalescence formula. All formulas are exact for any finite initial configuration and apply to discrete lattice paths, birth-death chains, and continuous diffusions including Brownian motion.

math.PR↗
arXiv2026

Exact determinant formulas for coalescing particle systems

When particles on a line collide, they may coalesce into one. Such systems arise in the voter model, where boundaries between opinion clusters perform coalescing random walks, and in reaction-diffusion theory, where diffusing particles merge on contact. Computing exact coalescence probabilities has been difficult because collisions reduce the particle count, while classical determinantal methods require a fixed number of particles throughout. We introduce ghost particles: when two particles collide, one survivor continues as usual and one invisible ghost is created alongside it, preserving the total count. This restores the square matrix structure needed for a determinantal formula. We prove that the probability of any specified coalescence pattern - which initial particles merge into which survivors - is given by a determinant whose entries are transition probabilities. Integrating out ghost positions yields a closed-form formula for the surviving particles alone: the coalescence determinant. The only assumptions are the Markov property and nearest-neighbor transitions, so the results apply wherever the classical non-colliding theory does: discrete lattice paths, birth-death chains, and continuous diffusions including Brownian motion.

math.PR↗
arXiv2026

Coalescing random walks via the coalescence determinant

When identical particles on a line collide, they merge and continue as one. Exact determinantal formulas have long been available for particles conditioned never to collide, but collisions change the number of particles, and exact distributions for the survivors have been obtained only in specific settings and by ad hoc methods. Building on the coalescence determinant introduced in a companion paper, we study the wall-particle system: when every site is initially occupied, this is the joint system of survivors and the boundaries between their basins of attraction. Its finite-dimensional distributions are determinants of block matrices built from transition probabilities and their cumulative sums; a finite block matrix suffices even when the initial configuration is infinite. As applications, we recover the Rayleigh spacing density and the joint distribution of consecutive gaps - which are negatively correlated - by new methods, and give a new derivation of the determinantal formula for the joint CDF of finitely many coalescing particles starting from fixed positions. All formulas hold for arbitrary nearest-neighbor random walks and their Brownian scaling limits, with no specific transition kernels required.

math.PR↗
arXiv2026

Pfaffian structure of basin walls for coalescing particles

Coalescing particles on a line merge when they meet. As they do, their basins of attraction merge and the walls between basins disappear. If every site is initially occupied, these walls at any positive time form a Pfaffian point process: all correlation functions are determined by pairwise quantities arranged in antisymmetric matrices. Tribe, Zaboronski, Garrod, and Poplavskyi established this structure using analytic methods for time-homogeneous dynamics; our combinatorial approach works for any skip-free process (one where particles cannot change order without first meeting). We show that the Pfaffian structure lives naturally at the wall level: we prove an exact Pfaffian empty-interval formula for the walls and compute the cumulants of the wall indicators (higher-order analogs of the variance) as signed sums of probabilities that independent particles started at the interval endpoint positions reorder in specific ways. A structural property of these sums, indecomposability - every nonzero term couples all wall positions together - implies a central limit theorem for the wall count. A checkerboard duality identifies the walls of one process with the surviving particles of the dual process. This covers totally asymmetric dynamics and position-dependent transition rules, and for Brownian motion recovers the known Pfaffian point process.

math.PR↗
arXiv2025

Fluctuations of Schensted row insertion

We investigate asymptotic probabilistic phenomena arising from the application of the Schensted row insertion algorithm, a key component of the Robinson-Schensted-Knuth (RSK) correspondence, to random inputs. Our analysis centers on a random tableau $T$ with a given shape $λ$, which may itself be random or deterministic. We examine the stochastic properties of the position of the new box created when inserting a deterministic entry into $T$. Specifically, we focus on the fluctuations of this position around its expected value as the size of the Young diagram $λ$ approaches infinity. Our findings reveal that these fluctuations are asymptotically Gaussian, with the mean and variance expressed in terms of Kerov's transition measure of the diagram $λ$. An important application of this analysis is the RSK algorithm applied to a finite, long sequence of independent, identically distributed random variables. While there remains a gap in the reasoning for this case, we present an explicit conjecture regarding its behavior.

math.PR↗
arXiv2025

Modulus of continuity of Kerov transition measure for continual Young diagrams

The transition measure is a foundational concept introduced by Sergey Kerov to represent the shape of a Young diagram as a centered probability measure on the real line. Over a period of decades the transition measure turned out to be an invaluable tool for many problems of the asymptotic representation theory of the symmetric groups. Kerov also showed how to expand this notion for a wider class of continual diagrams so that the transition measure provides a homeomorphism between a subclass of continual diagrams (having a specific support) and a class of centered probability measures with a support contained in a specific interval. We quantify the modulus of continuity of this homeomorphism. More specifically, we study the dependence of the cumulative distribution function of Kerov transition measure on the profile of a diagram at the locations where the profile is not too steep.

math.PR↗
arXiv2024

Cumulants of threshold for Schensted row insertion into random tableaux

Schensted row insertion is a fundamental component of the Robinson-Schensted-Knuth (RSK) algorithm, a powerful tool in combinatorics and representation theory. This study examines the insertion of a deterministic number into a random tableau of a specified shape, focusing on the relationship between the value of the inserted number and the position of the new box created by the Schensted row insertion. Specifically, for a given tableau and a point on its boundary, we consider the threshold that separates values which, if inserted, would result in the new box being created above the point from those that would result in a new box below. We analyze a random tableau of fixed shape and study the corresponding random threshold value. Explicit combinatorial formulas for the cumulants of this random variable are provided, expressed in terms of Kerov's transition measure of the diagram. These combinatorial formulas involve summing over non-crossing alternating trees. As a first application of these results, we demonstrate that for random Young tableaux of prescribed large shape, the rightmost entry in the first row converges in distribution to an explicit Gaussian distribution.

math.CO↗
arXiv2023

Bijection between trees in Stanley character formula and factorizations of a cycle

Stanley and Féray gave a formula for the irreducible character of the symmetric group related to a multi-rectangular Young diagram. This formula shows that the character is a polynomial in the multi-rectangular coordinates and gives an explicit combinatorial interpretation for its coefficients in terms of counting certain decorated maps (i.e., graphs drawn on surfaces). In the current paper we concentrate on the coefficients of the top-degree monomials in the Stanley character polynomial, which corresponds to counting certain decorated plane trees. We give an explicit bijection between such trees and minimal factorizations of a cycle.

math.CO↗
arXiv2022

Poisson limit theorems for the Robinson-Schensted correspondence and for the multi-line Hammersley process

We consider Robinson-Schensted-Knuth algorithm applied to a random input and study the growth of the bottom rows of the corresponding Young diagrams. We prove multidimensional Poisson limit theorem for the resulting Plancherel growth process. In this way we extend the result of Aldous and Diaconis to more than just one row. This result can be interpreted as convergence of the multi-line Hammersley process to its stationary distribution which is given by a collection of independent Poisson point processes.

math.PR↗
arXiv2022

Second class particles and limit shapes of evacuation and sliding paths for random tableaux

We investigate two closely related setups. In the first one we consider a TASEP-style system of particles with specified initial and final configurations. The probability of each history of the system is assumed to be equal. We show that the rescaled trajectory of the \emph{second class particle} converges (as the size of the system tends to infinity) to a random arc of an ellipse. In the second setup we consider a uniformly random Young tableau of square shape and look for typical (in the sense of probability) sliding paths and evacuation paths in the asymptotic setting as the size of the square tends to infinity. We show that the probability distribution of such paths converges to a random meridian connecting the opposite corners of the square. We also discuss analogous results for non-square Young tableaux.

math.CO↗
arXiv2021

Symmetric group characters of almost square shape

We give closed product formulas for the irreducible characters of the symmetric groups related to rectangular `almost square' Young diagrams $p \times(p+δ)$ for a fixed value of an integer $δ$ and an arbitrary integer $p$.

math.CO↗
arXiv2021

Poisson limit of bumping routes in the Robinson-Schensted correspondence

We consider the Robinson-Schensted-Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the $y$-axis) when a specified number is inserted into a large Plancherel-distributed tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process.

math.CO↗
arXiv2020

Random strict partitions and random shifted tableaux

We study asymptotics of random shifted Young diagrams which correspond to a given sequence of reducible projective representations of the symmetric groups. We show limit results (Law of Large Numbers and Central Limit Theorem) for their shapes, provided that the representation character ratios and their cumulants converge to zero at some prescribed speed. Our class of examples includes uniformly random shifted standard tableaux with prescribed shape as well as shifted tableaux generated by some natural combinatorial algorithms (such as shifted Robinson-Schensted-Knuth correspondence) applied to a random input.

math.CO↗
arXiv2019

Linear versus spin: representation theory of the symmetric groups

We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition $ξ$ with analogous normalized linear characters evaluated on the double partition $D(ξ)$. We also relate some natural filtration on the usual (linear) Kerov-Olshanski algebra of polynomial functions on the set of Young diagrams with its spin counterpart. Finally, we give a spin counterpart to Stanley formula for the characters of the symmetric groups.

math.CO↗
arXiv2018

Structure coefficients for Jack characters: approximate factorization property

Jack characters are a generalization of the characters of the symmetric groups; a generalization that is related to Jack symmetric functions. We investigate the structure coefficients for Jack characters; they are a generalization of the connection coefficients for the symmetric groups. More specifically, we study the cumulants which measure the discrepancy between these structure coefficients and the simplistic structure coefficients related to the disjoint product. We show that Jack characters satisfy approximate factorization property: their cumulants are of very small degree and the character related to a given partition is approximately equal to the product of the characters related to its parts. This result will play a key role in the proof of Gaussianity of fluctuations for a class of random Young diagrams related to the Jack polynomials.

math.CO↗
arXiv2018

Asymptotics of Jack characters

Jack characters are a one-parameter deformation of the characters of the symmetric groups; a deformation given by the coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions. We study Jack characters from the viewpoint of the asymptotic representation theory. In particular, we give explicit formulas for their asymptotically top-degree part, in terms of bicolored oriented maps with an arbitrary face structure. We also study their multiplicative structure and their structure constants and we prove that they fulfill approximate factorization property, a convenient tool for proving Gaussianity of fluctuations of random Young diagrams.

math.CO↗