12 pages, 3 figures (v2: minor modifications and clarifications). Extended abstract for FPSAC 2021
发表信息：
Séminaire Lotharingien de Combinatoire, 85B.33 (2021), Proceedings of the 33rd Conference on Formal Power Series and Algebraic Combinatorics (Ramat Gan)
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 first 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 introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root singularity in their generating function. The scaling limit involves continuous trees with branching points of order up to k+1. We derive explicit integral representations for the average profile of this k-th order multicritical continuous random tree, as well as for its history distributions measuring multi-point correlations. The latter distributions involve non-p…
We investigate models of (1+d)-D Lorentzian semi-random lattices with one random (space-like) direction and d regular (time-like) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an exact solution of a variety of new models including critical and multicritical generalized (1+1)-D Lorentzian surfaces, with fractal dimensions $d_F=k+1$, k=1,2,3,..., as well as a new model of (1+2)-D critical tetrahedral complexes, with fractal dimension $d_F=12/5$. Critical exponents and universal …
We establish the universal edge scaling limit of random partitions with the infinite-parameter distribution called the Schur measure. We explore the asymptotic behavior of the wave function, which is a building block of the corresponding kernel, based on the Schr{ö}dinger-type differential equation. We show that the wave function is in general asymptotic to the Airy function and its higher-order analogs in the edge scaling limit. We construct the corresponding higher-order Airy kernel and the Tracy-Widom distribution from the wave function in the scalins limit, and discuss its implication to t…
We consider a family of distributions on spatial random partitions that provide a coupling between different models of interest: the ideal Bose gas; the zero-range process; particle clustering; and spatial permutations. These distributions are invariant for a "chain of Chinese restaurants" stochastic process. We obtain results for the distribution of the size of the largest component.
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-edge 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):…
Random skew plane partitions of large size distributed according to an appropriately scaled Schur process develop limit shapes. In the present work we consider the limit of large random skew plane partitions where the inner boundary approaches a piecewise linear curve with non-lattice slopes, describing the limit shape and the local fluctuations in various regions. This analysis is fairly similar to that in [OR2], but we do find some new behavior. For instance, the boundary of the limit shape is now a single smooth (not algebraic) curve, whereas the boundary in [OR2] is singular. We also obser…
Using techniques from integrable systems, we obtain a number of exact results for random partitions. In particular, we prove a simple formula for correlation functions of what we call the Schur measure on partitions (which is a far reaching generalization of the Plancherel measure, see math.CO/9905032) and also show that these correlations functions are tau-functions for the Toda lattice hierarchy. Also we give a new proof of the formula due to Bloch and the author, see alg-geom/9712009, for the so called n-point functions of the uniform measure on partitions and comment on the local structure…
These are (not updated) notes from the lectures I gave at the NATO ASI ``Symmetric Functions 2001'' at the Isaac Newton Institute in Cambridge (June 25 -- July 6, 2001). Their goal is an informal introduction to asymptotic combinatorics related to partitions.
We investigate the limiting distribution of the fluctuations of the maximal summand in a random partition of a large integer with respect to a multiplicative statistics. We show that for a big family of Gibbs measures on partitions (so called generalized Bose--Einstein statistics) this distribution is the well-known Gumbel distribution which usually appears in the context of indepedent random variables. In particular, it means that the (properly rescaled) maximal energy of an individual particle in the grand canonical ensemble of the $d$-dimensional quantum ideal gas has the Gumbel distributio…
We study random skew 3D partitions weighted by $q^{\textup{vol}}$ and, specifically, the $q\to 1$ asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel asymptotics for correlations in the bulk of the disordered region, Airy kernel asymptotics near a general point of the frozen boundary, and a Pearcey kernel asymptotics near a cusp of the frozen boundary.
We consider a random process with discrete time formed by singular values of products of truncations of Haar distributed unitary matrices. We show that this process can be understood as a scaling limit of the Schur process, which gives determinantal formulas for (dynamical) correlation functions and a contour integral representation for the correlation kernel. The relation with the Schur processes implies that the continuous limit of marginals for q-distributed plane partitions coincides with the joint law of singular values for products of truncations of Haar-distributed random unitary matric…
In this paper we define the shifted Schur process as a measure on sequences of strict partitions. This process is a generalization of the shifted Schur measure introduced in [TW] and [Mat] and is a shifted version of the Schur process introduced in [OR]. We prove that the shifted Schur process defines a Pfaffian point process. We further apply this fact to compute the bulk scaling limit of the correlation functions for a measure on strict plane partitions which is an analog of the uniform measure on ordinary plane partitions. As a byproduct, we obtain a shifted analog of the famous MacMahon's …
We explore some probabilistic applications arising in connections with $K$-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. We also introduce some distributions that are naturally related to the corner growth model. Our main tools are dual symmetric Grothendieck polynomials and normalized Schur functions.
In this paper, we introduce polynomial time algorithms that generate random 3-noncrossing partitions and 2-regular, 3-noncrossing partitions with uniform probability. A 3-noncrossing partition does not contain any three mutually crossing arcs in its canonical representation and is 2-regular if the latter does not contain arcs of the form $(i,i+1)$. Using a bijection of Chen {\it et al.} \cite{Chen,Reidys:08tan}, we interpret 3-noncrossing partitions and 2-regular, 3-noncrossing partitions as restricted generalized vacillating tableaux. Furthermore, we interpret the tableaux as sampling paths o…
We use a coin flipping model for the random partition and Chebyshev's inequality to prove the lower bound $\lim \frac{\log p(n)}{\sqrt{n}} \ge C$ for the number of partitions $p(n)$ of $n$, where $C$ is an explicit constant.
Let $λ$ be a partition of the positive integer $n$ chosen umiformly at random among all such partitions. Let $L_n=L_n(λ)$ and $M_n=M_n(λ)$ be the largest part size and its multiplicity, respectively. For large $n$, we focus on a comparison between the partition statistics $L_n$ and $L_n M_n$. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of $L_n M_n -L_n$ grows as fast as $\frac{1}{2}\log{n}$ We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.
We consider a certain sequence of random walks. The state space of the n-th random walk is the set of all strict partitions of n (that is, partitions without equal parts). We prove that, as n goes to infinity, these random walks converge to a continuous-time Markov process. The state space of this process is the infinite-dimensional simplex consisting of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The main result about the limit process is the expression of its the pre-generator as a formal second order differential operator in a polynomial a…
We study the asymptotic behaviour of the trace (the sum of the diagonal parts) of a plane partition of the positive integer n, assuming that this parfition is chosen uniformly at random from the set of all such partitions.
We investigate combinatorial properties of a family of probability distributions on finite abelian p-groups. This family includes several well-known distributions as specializations. These specializations have been studied in the context of Cohen-Lenstra heuristics and cokernels of families of random p-adic matrices.
We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions of all of the parts together and that of the largest part as $n$ tends to infinity while $m$ is fixed or tends to infinity. In particular, if $m$ goes to infinity not too fast, the largest part satisfies the central limit theorem. The second measure is very general. It includes the Dirichlet distribution and the uniform distribution as special cases. We deri…
Fix $t \geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. We show that this converges weakly to a gamma distribution after dividing by $\sqrt{n}$. As a consequence, we find that the size of the $t$-core is of the order of $\sqrt{n}$ in expectation. We then apply this result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in the limit. Finally…
This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the two-parameter family of Poisson-Dirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions o…
We show that throughout the satisfiable phase the normalised number of satisfying assignments of a random $2$-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to `true' under a uniformly random satisfying assignment.
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an integer parameter N. Then we emulate the theory of random matrices in a combinatorial framework: for any parameter N, we introduce a family of linear forms on the partition algebras which allows us to define a notion of weak convergence similar to the convergence in moments in random matrices theory. A renormalization of the partition algebras allows us to c…
We prove a lower bound expansion on the probability that a random $\pm 1$ matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form $e_i \pm e_j$, which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form $e_i \pm e_j \pm e_k \pm e_\ell$, which corresponds to the partition 1111. The fifth…
This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is $O(n (\ln n)^3)$ in approximate-size sampling and $O(n^{4/3})$ in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to the size (there exist polynomial-time samplers of another type, which draw plane partitions…
In this paper we study random partitions of 1,...n, where every cluster of size j can be in any of w\_j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w\_j, the time-reversed process is the discrete Marcus-Lushnikov …
We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a polynomial expression for their drift. We also recover Rietsch's parametriza-tion of totally nonnegative unitriangular Toeplitz matrices without using quantum cohomology of flag varieties. All the homeomorphisms we define can moreover be made explicit by using the combinatorics of k-Schur functions and elementary computations based on Perron-Frobenius theorem.
In this paper, we introduce polynomial time algorithms that generate random $k$-noncrossing partitions and 2-regular, $k$-noncrossing partitions with uniform probability. A $k$-noncrossing partition does not contain any $k$ mutually crossing arcs in its canonical representation and is 2-regular if the latter does not contain arcs of the form $(i,i+1)$. Using a bijection of Chen {\it et al.} \cite{Chen,Reidys:08tan}, we interpret $k$-noncrossing partitions and 2-regular, $k$-noncrossing partitions as restricted generalized vacillating tableaux. Furthermore, we interpret the tableaux as sampling…
Let $P_r(n)$ be the set of partitions of n with non negative rth differences. Let $λ$ be a partition chosen uniformly at random among the set $P_r(n)$. Let $d(λ)$ be a positive rth difference chosen uniformly at random in $λ$. The aim of this work is to show that for every $m\ge 1$, the probability that $d(λ)\ge m$ approaches $m^{-1/r}$ as $n\to\infty$. To prove this result we use bijective, asymptotic/analytic, and probabilistic combinatorics.
We find exact and asymptotic formulas for the average values of several statistics on set partitions: of Carlitz's $q$-Stirling distributions, of the numbers of crossings in linear and circular representations of set partitions, of the numbers of overlappings and embracings, and of the numbers of occurrences of a 2-pattern.
The material gives a new combinatorial proof of the multiplicative property of the S-transform. In particular, several properties of the coefficients of its inverse are connected to non-crossing linked partitions and planar trees.
Cohen-Lenstra heuristics for Jacobians of random graphs give rise to random partitions. We connect these random partitions to the Hall-Littlewood polynomials of symmetric function theory, and use this connection to give combinatorial proofs of properties of these random partitions. In addition, we use Markov chains to give an algorithm for generating these partitions.
An integer partition of $n$ is a decreasing sequence of positive integers that add up to $[n]$. Back in $1979$ Macdonald posed a question about the limit value of the probability that two partitions chosen uniformly at random, and independently of each other, are comparable in terms of the dominance order. In $1982$ Wilf conjectured that the uniformly random partition is a size-ordered degree sequence of a simple graph with the limit probability $0$. In $1997$ we showed that in both, seemingly unrelated, cases the limit probabilities are indeed zero, but our method left open the problem of con…
We propose an aproach for asymptotic analysis of plane partition statistics related to counts of parts whose sizes exceed a certain suitably chosen level. In our study, we use the concept of conjugate trace of a plane partition of the positive integer $n$, introduced by Stanley in 1973. We derive generating functions and determine the asymptotic behavior of counts of large parts using a general scheme based on the saddle point method. In this way, we are able to prove a Poisson limit theorem for the number of parts of a random and uniformly chosen plane partition of $n$, whose sizes are greate…
We suggest an hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with ``beta=2 ensembles'' arising in the random matrix theory. We show that all such results are, essentially, degenerations of one general situation arising from so-called generalized regular representations of the infinite symmetric group.
Let $B_t(n)$ be the number of set partitions of a set of size~$t$ into at most $n$ parts and let $B'_t(n)$ be the number of set partitions of $\{1,\ldots, t\}$ into at most $n$ parts such that no part contains both $1$ and~$t$ or both $i$ and $i+1$ for any $i \in \{1,\ldots,t-1\}$. We give two new combinatorial interpretations of the numbers $B_t(n)$ and $B'_t(n)$ using sequences of random-to-top shuffles, %that leave a deck of cards invariant, and sequences of box moves on the Young diagrams of partitions. Using these ideas we obtain a very short proof of a generalization of a result of Phata…
Let g be a random element of a finite classical group G, and let λ_{z-1}(g) denote the partition corresponding to the polynomial z-1 in the rational canonical form of g. As the rank of G tends to infinity, λ_{z-1}(g) tends to a partition distributed according to a Cohen-Lenstra type measure on partitions. We give sharp upper and lower bounds on the total variation distance between the random partition λ_{z-1}(g) and the Cohen-Lenstra type measure.
Let $X_1,\dots, X_n$ be independent integers distributed uniformly on $\{1,\dots, M\}$, $M=M(n)\to\infty$ however slow. A partition $S$ of $[n]$ into $ν$ non-empty subsets $S_1,\dots, S_ν$ is called perfect, if all $ν$ values $\sum_{j\in S_{\a}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $ν$. For $ν=2$, Borgs et al. proved, among other results, that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $κ:=\lim \tfrac{n}{\log M}>\tfrac{1}{\log 2}$, and that w.h.p. no perfect partition exists if $κ<\tfrac{1}{\log 2…
We study $k$-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer $k=1,2,\ldots$. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for $k>1$, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeable $k$-divisible partitions that are consistent under random deletio…
We study the asymptotic behavior of the maximal multiplicity $M_n=M_n(σ)$ of the blocks in a set partition of $[n]=\{1,2,...,n\}$, assuming that $σ$ is chosen uniformly at random from the set of all such partitions. Let $W=W(n)$ be the unique positive root of the equation $We^W=n$ and let $f_n$ be the fractional part of $W(n)$. Furthermore, let $R_n=W^{\lfloor W\rfloor}/\lfloor W\rfloor !$ and let $\vartheta_n=\min{\{f_n,1-f_n\}}$. We show that, over a subsequence $\{n_k\}_{k\ge 1}$, $(M_{n_k}-R_{n_k})/\sqrt{R_{n_k}}$ converges weakly, as $k\to\infty$, to $\max{\{Z_1,Z_2-u\}}$, where $Z_1$ and…
