Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures

We introduce a large class of random Young diagrams which can be regarded as a natural one-parameter deformation of some classical Young diagram ensembles; a deformation which is related to Jack polynomials and Jack characters. We show that each such a random Young diagram converges asymptotically to some limit shape and that the fluctuations around the limit are asymptotically Gaussian.

Let $$p(n)$$ be the number of all integer partitions of the positive integer $$n$$ , and let $$\lambda$$ be a partition selected uniformly at random from among all such $$p(n)$$ partitions. It is well known that each partition $$\lambda$$ has a unique graphical representation composed of $$n$$ non-overlapping cells in the plane, called a Young diagram. As a second step of our sampling experiment, we select a cell $$c$$ uniformly at random from among the $$n$$ cells of the Young diagram of the partition $$\lambda$$ . For large $$n$$ , we study the asymptotic behavior of the hook length $$Z_n=Z_n(\lambda,c)$$ of the cell $$c$$ of a random partition $$\lambda$$ . This two-step sampling procedure suggests a product probability measure, which assigns the probability $$1/np(n)$$ to each pair $$(\lambda,c)$$ . With respect to this probability measure, we show that the random variable $$\pi Z_n/\sqrt{6n}$$ converges weakly, as $$n\to\infty$$ , to a random variable whose probability density function equals $$6y/(\pi^2(e^y-1))$$ if $$0<y<\infty$$ , and zero elsewhere. Our method of proof is based on Hayman’s saddle point approach for admissible power series.

The asymptotic behavior of the lengths of the first rows and columns in the random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in n and prove a central limit theorem for their lengths in the case of distinct Thoma parameters. We also prove a more precise statement relating the growth of rows and columns of Young diagrams to a simple independent random sampling model.

We introduce and study a family of Markov processes on partitions. The processes preserve the so-called z-measures on partitions previously studied in connection with harmonic analysis on the infinite symmetric group. We show that the dynamical correlation functions of these processes have determinantal structure and we explicitly compute their correlation kernels. We also compute the scaling limits of the kernels in two different regimes. The limit kernels describe the asymptotic behavior of large rows and columns of the corresponding random Young diagrams, and the behavior of the Young diagrams near the diagonal.Our results show that recently discovered analogy between random partitions arising in representation theory and spectra of random matrices extends to the associated time– dependent models.

We prove the equivalence of ensembles or a realization of the local equilibrium for Bernoulli measures on \({\mathbb{Z}}\) conditioned on two conserved quantities under the situation that one of them is spatially inhomogeneous. For the proof, we extend the classical local limit theorem for a sum of Bernoulli independent sequences to those multiplied by linearly growing weights. The motivation comes from the study of random Young diagrams and their evolutional models, which were originally suggested by Herbert Spohn. We discuss the relation between our result and the so-called Vershik curve which appears in a scaling limit for height functions of two-dimensional Young diagrams. We also discuss a related random dynamics.

Using the original method from [4], [5] we prove the validity of the local large deviations principle for the shape of a random Young diagram with different constraints on the multiplicity of the rows of equal length.

We continue our analysis of: (i) "Quantum tomography", i.e., learning a quantum state, i.e., the quantum generalization of learning a discrete probability distribution; (ii) The distribution of Young diagrams output by the RSK algorithm on random words. Regarding (ii), we introduce two powerful new tools: first, a precise upper bound on the expected length of the longest union of k disjoint increasing subsequences in a random length-n word with letter distribution α1 ≥ α2 ≥ … ≥ αd. Our bound has the correct main term and second-order term, and holds for all n, not just in the large-n limit. Second, a new majorization property of the RSK algorithm that allows one to analyze the Young diagram formed by the lower rows λk, λk+1, … of its output. These tools allow us to prove several new theorems concerning the distribution of random Young diagrams in the nonasymptotic regime, giving concrete error bounds that are optimal, or nearly so, in all parameters. As one example, we give a fundamentally new proof of the celebrated fact that the expected length of the longest increasing sequence in a random length-n permutation is bounded by 2√n. This is the k = 1, αi ≡ 1/d, d → ∞ special case of a much more general result we prove: the expected length of the kth Young diagram row produced by an α-random word is αk n ± 2√αkd n.

In the quantum state tomography problem, one wishes to estimate an unknown d-dimensional mixed quantum state ρ, given few copies. We show that O(d/) copies suffice to obtain an estimate ρ ̂ that satisfies ‖ρ̂−ρ‖2F ≤ (with high probability). An immediate consequence is that O(rank(ρ)·d/2) ≤ O(d2/2) copies suffice to obtain an -accurate estimate in the standard trace distance. This improves on the best known prior result of O(d3/2) copies for full tomography, and even on the best known prior result of O(d2 log(d/)/2) copies for spectrum estimation. Our result is the first to show that nontrivial tomography can be obtained using a number of copies that is just linear in the dimension. Next, we generalize these results to show that one can perform efficient principal component analysis on ρ. Our main result is that O(kd/2) copies suffice to output a rank-k approximation ρ ̂ whose trace distance error is at most more than that of the best rank-k approximator to ρ. This subsumes our above trace distance tomography result and generalizes it to the case when ρ is not guaranteed to be of low rank. A key part of the proof is the analogous generalization of our spectrum-learning results: we show that the largest k eigenvalues of ρ can be estimated to trace-distance error using O(k2/2) copies. In turn, this result relies on a new coupling theorem concerning the Robinson–Schensted–Knuth algorithm that should be of independent combinatorial interest. 1

Concentration phenomena in statistical ensembles of Young diagrams have been investigated as static models first for the Plancherel ensemble by Vershik–Kerov and Logan–Shepp in 1970s and later in some other group-theoretical setting by Biane. On the other hand, a dynamical model of concentration for Young diagrams, which is not directly connected with group representations, was shown by Funak–{Sasada in the framework of hydrodynamic limit. The aim of this paper is to present a new dynamical model of concentration for Young diagrams featuring the group-theoretical sense. Starting from an initial state yielding concentration and a microscopic dynamics keeping the Plancherel measure invariant, we derive an evolution of the profiles of Young diagrams under a diffusive scaling limit. The resulting evolution along macroscopic time is described in terms of the notions of Voiculescu's free probability theory such as free compression and free convolution of Kerov transition measures.

Using the local method, we prove the large deviation principle for one model distribution, derive an explicit expression for the rate function, and outline ways to further apply the method presented.

Using the original method we prove the validness of the local large deviations principle for the shape of the random Young diagram with different constrains on the multiplicity of the rows of equal length.

Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure M n. That is, the weight M n(λ) of a diagram λ equals dim2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group EquationSource$$ \mathfrak{S}_n $$ indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan-Shepp 1977, Vershik-Kerov 1977). In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov’s unpublished work notes, 1999Key wordsrandom Young diagramsPlancherel measurecentral limit theoremgeneralized Gaussian processes

Пусть $p(n)$ - количество всех целочисленных разбиений положительного целого числа $n$, и пусть $\lambda $ - разбиение, выбранное случайно и равновероятно из всех таких $p(n)$ разбиений. Известно, что каждое разбиение $\lambda $ имеет единственное графическое представление, состоящее из $n$ неперекрывающихся ячеек на плоскости, называемое диаграммой Юнга. В качестве второго шага нашего выборочного эксперимента мы выбираем из $n$ ячеек диаграммы Юнга разбиения $\lambda $ случайно и равновероятно ячейку $c$. Для больших значений $n$ мы изучаем асимптотическое поведение длины крюка $Z_n=Z_n(\lambda ,c)$ ячейки $c$ случайного разбиения $\lambda $. Эта двухэтапная выборочная процедура порождает вероятностную меру, которая приписывает вероятность $1/np(n)$ каждой паре $(\lambda ,c)$. Показано, что относительно этой вероятностной меры случайная величина $\pi Z_n/\sqrt {6n}$ слабо сходится при $n\to \infty $ к случайной величине, плотность функции распределения которой равна $6y/(\pi ^2(e^y-1))$, если $0&lt;y&lt;\infty $, и нулю в остальных случаях. Доказательство основано на подходе Хеймана к исследованию седловой точки для допустимых степенных рядов.

Asymptotic representation theory of general linear groups $$\hbox {GL}(n,F_\mathfrak {q})$$ over a finite field leads to studying probability measures $$\rho $$ on the group $$\mathbb {U}$$ of all infinite uni-uppertriangular matrices over $$F_\mathfrak {q}$$ , with the condition that $$\rho $$ is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points (in other words, ergodic measures $$\rho $$ ) was conjectured by Kerov in connection with nonnegative specializations of Hall–Littlewood symmetric functions. Vershik and Kerov also conjectured the following Law of Large Numbers. Consider an infinite random matrix drawn from an ergodic measure coming from the Kerov’s conjectural classification and its $$n \times n$$ submatrix formed by the first rows and columns. The sizes of Jordan blocks of the submatrix can be interpreted as a (random) partition of $$n$$ , or, equivalently, as a (random) Young diagram $$\lambda (n)$$ with $$n$$ boxes. Then, as $$n\rightarrow \infty $$ , the rows and columns of $$\lambda (n)$$ have almost sure limiting frequencies corresponding to parameters of this ergodic measure. Our main result is the proof of this Law of Large Numbers. We achieve it by analyzing a new randomized Robinson–Schensted–Knuth (RSK) insertion algorithm which samples random Young diagrams $$\lambda (n)$$ coming from ergodic measures. The probability weights of these Young diagrams are expressed in terms of Hall–Littlewood symmetric functions. Our insertion algorithm is a modified and extended version of a recent construction by Borodin and Petrov (2013). On the other hand, our randomized RSK insertion generalizes a version of the RSK insertion introduced by Vershik and Kerov (SIAM J. Algebr. Discret. Math. 7(1):116–124, 1986) in connection with asymptotic representation theory of symmetric groups (which is governed by nonnegative specializations of Schur symmetric functions).

Consider the standard Poisson process in the first quadrant of the Euclidean plane, and for any point (u,v) of this quadrant take the Young diagram obtained by applying the Robinson-Schensted correspondence to the intersection of the Poisson point configuration with the rectangle with vertices (0,0), (u,0), (u,v), (0,v). It is known that the distribution of the random Young diagram thus obtained is the poissonized Plancherel measure with parameter uv. We show that for (u,v) moving along any southeast-directed curve in the quadrant, these Young diagrams form a Markov chain with continuous time. We also describe these chains in terms of jump rates. Our main result is the computation of the dynamical correlation functions of such Markov chains and their bulk and edge scaling limits.