Sho Matsumoto
We introduce a generalization of the Schur measure. And we give a combinatorial interpretation to this generalized measure and prove that a scaling limit for this measure is given by the Tracy-Widom distribution.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Multicritical Scaling Limit of Shifted Schur Measure
研究移位Schur测度的多临界标度极限，确定严格划分的极限形状，并展示在多临界条件下相关函数的边缘标度极限收敛于高阶Airy核行列式。
To each partition $λ$ with distinct parts we assign the probability $Q_λ(x) P_λ(y)/Z$ where $Q_λ$ and $P_λ$ are the Schur $Q$-functions and $Z$ is a normalization constant. This measure, which we call the shifted Schur measure, is analogous to the much-studied Schur measure. For the specialization of the first $m$ coordinates of $x$ and the first $n$ coordinates of $y$ equal to $α$ ($0<α<1$) and the rest equal to zero, we derive a limit law for $λ_1$ as $m,n\ra\infty$ with $τ=m/n$ fixed. For the Schur measure the $α$-specialization limit law was derived by Johansson. Our main result implies th…
The shifted Schur measure introduced by Tracy and Widom is a measure on the set of all strict partitions, which is defined by Schur $Q$-functions. The main aim of this paper is to calculate the correlation function of this measure, which is given by a pfaffian. As an application, we prove that a limit distribution of $λ_j$'s with respect to a shifted version of the Plancherel measure for symmetric groups is identical with the corresponding distribution of the original Plancherel measure. Further we give expressions of the mean value and the variance of the size of a partition with respect to t…
For any complex number $α$ and any even-size skew-symmetric matrix $B$, we define a generalization $\pfaα(B)$ of the pfaffian $\pf(B)$ which we call the $α$-pfaffian. The $α$-pfaffian is a pfaffian analogue of the $α$-determinant. It gives the pfaffian at $α=-1$. We give some formulas for $α$-pfaffians and study the positivity. Further we define point processes determined by the $α$-pfaffian. Also we provide a linear algebraic proof of the explicit pfaffian expression for the correlation function of the shifted Schur measure.
Disordered Schur Measures
本文研究参数来自圆正交群的随机Schur测度，发现其行为类似自旋玻璃。
We obtain a new formula to relate the value of a Schur polynomial with variables $(x_1,\ldots,x_N)$ with values of Schur polynomials at $(1,\ldots,1)$. This allows to study the limit shape of perfect matchings on a square hexagon lattice with periodic weights and piecewise boundary conditions. In particular, when the edge weights satisfy certain conditions, asymptotics of the Schur function imply that the liquid region of the model in the scaling limit has multiple connected components, while the frozen boundary consists of disjoint cloud curves.
The 2D Toda lattice hierarchy for multiplicative statistics of Schur measures
研究显示，基于Schur测度的广义Fredholm行列式可构建二维Toda晶格级数的tau函数，扩展了Okounkov和Cafasso-Ruzza关于有限温Plancherel测度的结果。
Relative dimensions of isotypic components of N-th order tensor representations of the symmetric group on n letters give a Plancherel-type measure on the space of Young diagrams with n cells and at most N rows. It was conjectured by G. Olshanski that dimensions of isotypic components of tensor representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to this family of Plancherel-type measures in the limit when N/sqrt{n} converges to a constant. The main result of the paper is the proof of this conjecture.
We construct a vertex model whose partition function is a refined dual Grothendieck polynomial, where the states are interpreted as nonintersecting lattice paths. Using this, we show refined dual Grothendieck polynomials are multi-Schur functions and give a number of identities, including a Littlewood and Cauchy(-Littlewood) identity. We then refine Yeliussizov's connection between dual Grothendieck polynomials and the last passage percolation (LPP) stochastic process discussed by Johansson. By refining algebraic techniques of Johansson, we show Jacobi-Trudi formulas for skew refined dual Grot…
The main results imply that the probability P(\ZZ\in A+þ) is Schur-concave/Schur-convex in (þ_1^2,\dots,þ_k^2) provided that the indicator function of a set A in \R^k is so, respectively; here, þ=(þ_1,\dots,þ_k) in \R^k and \ZZ is a standard normal random vector in \R^k. Moreover, it is shown that the Schur-concavity/Schur-convexity is strict unless the set A is equivalent to a spherically symmetric set. Applications to testing hypotheses on multivariate means are given.
Large N limit of Wilson Loops on orientable closed surfaces in the light of Koike-Schur-Weyl duality and Spin Networks
研究在大N极限下，杨-米尔斯测度下可定向闭曲面上威尔逊环的收敛性，利用科伊基-施楚尔-韦伊尔对偶性和旋网络方法。
The aim of this note is to prove a law of large numbers for local patterns in discrete point processes. We investigate two different situations: a class of point processes on the one dimensional lattice including certain Schur measures, and a model of random plane partitions, introduced by Okounkov and Reshetikhin. The results state in both cases that the linear statistic of a function, weighted by the appearance of a fixed pattern in the random configuration and conveniently normalized, converges to the deterministic integral of that function weighted by the expectation with respect to the li…
We establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure. These expectations turn out to be the generating series of certain paths in the Cayley graph of the symmetric group. We then compute the asymptotic distribution of a random unitary matrix under the heat kernel measure on the unitary group U(N) as N tends to infinity, and prove a result of asymptotic freeness result for independent large unitary matrices, thus recovering results obtained previously by Xu and Biane. We give an interpretat…
Vertex operators, infinite wedge representations, and correlation functions of the t-Schur measure
本文通过顶点算子方法研究t-Schur测度，推导其相关函数并证明其确定性，同时揭示其与Schur和Schur-Q理论的联系。
Identity between Restricted Cauchy Sums for the $q$-Whittaker and Skew Schur Polynomials
本研究针对$q$-Whittaker与斜Schur多项式的受限柯西和，借助可积概率中的$q$-Whittaker测度、周期Schur测度及其Fredholm行列式公式，证明了二者间的恒等式，拓展了对称函数柯西恒等式的相关结论。
A Two-Color Lift of the Shifted $t$-Schur Measure
通过插入中间严格分拆，将移位 Schur 测度提升为双色测度，计算了归一化、边缘分布、马尔可夫核，证明了半群性质和体积独立性，并给出了关联核的 Pfaffian 形式。
We show, using either Fock space techniques or Macdonald difference operators, that certain symplectic and orthogonal analogues of Okounkov's Schur measure are determinantal with kernels given by explicit double contour integrals. We give two applications: one equates certain Toeplitz+Hankel determinants of random matrix theory with appropriate Fredholm determinants and computes Szegő 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 e…
Multicritical Schur measures and higher-order analogues of the Tracy-Widom distribution
本文引入多临界 Schur 测度，刻画边缘非通用涨落，其普适类由高阶 Airy 核的 Fredholm 行列式描述，推广了 Tracy-Widom GUE 分布，并建立了与多临界酉矩阵模型的精确映射。
We explore probabilistic consequences of correspondences between $q$-Whittaker measures and periodic and free boundary Schur measures established by the authors in the recent paper [arXiv:2106.11922]. The result is a comprehensive theory of solvability of stochastic models in the KPZ class where exact formulas descend from mapping to explicit determinantal and pfaffian point processes. We discover new variants of known results as determinantal formulas for the current distribution of the ASEP on the line and new results such as Fredholm pfaffian formulas for the distribution of the point-to-po…
The Symplectic Schur Process
本文定义了辛施尔过程，该过程是Okounkov-Reshetikhin对于Cartan型C的类比。通过引入辛字符和新的Down-Up施尔函数，证明了相关点过程的行列性，并给出了显式相关核。还研究了保形过程的动力学和采样方案，以及极限行为。
Universal Cusp Scaling in Random Partitions
本文研究Schur测度下随机划分的普适标度极限，扩展先前工作获得高阶Pearcey核，并推导间隙概率的耦合非线性微分方程及大间隙渐近行为。
Large Deviations for Ablowitz-Ladik lattice, and the Schur flow
本文研究 Ablowitz-Ladik 格点与 Schur 流的广义吉布斯系综，推导经验测度的大偏差原理，并证明其几乎必然收敛，极限由圆与 Jacobi beta 系综的平衡测度刻画。
Random Fibonacci Words via Clone Schur Functions
本文通过克隆Schur函数生成随机斐波那契词，研究其正性、概率性质及与多项式、正交序列等的联系，扩展了Plancherel测度的渐进行为。
Integral formulas for two-layer Schur and Whittaker processes
本文证明了刻画两层 Schur 和 Whittaker 过程多点联合分布的围道积分公式，将其表示为具有显式转移核的 Doob 变换 Markov 过程，并以此计算了 $[0,L]$ 上 KPZ 方程的增长率。
Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests
本文研究基于对称 Grothendieck 多项式的概率测度，证明其并非行列式点过程，并通过倾斜双正交系综框架将其识别为二维 Schur 过程的截面，进而推导出其极限形状结果。
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…
Last passage percolation in lower triangular domain
本文研究下三角区域中最后通过渗流的渐近分布，证明反对角线情形由GOE Tracy-Widom分布控制，主对角线情形由GSE Tracy-Widom分布控制，并在截断下三角情形中得到了GSE Tracy-Widom分布的新推广。
Recent work on recurrence in quantum walks has provided a representation of Schur functions in terms of unitary operators. We propose a generalization of Schur functions by extending this operator representation to arbitrary operators on Banach spaces. Such generalized Schur functions meet the formal structure of first return generating functions, thus we call them FR-functions. We derive general properties of FR-functions, among them a generalization of the renewal equation already known for random and quantum walks, as well as splitting properties which extend useful factorizations of Schur …
Let $\{x_α\}_{α\in \mathbb{Z}}$ and $\{y_α\}_{α\in \mathbb{Z}}$ be two independent collections of zero mean, unit variance random variables with uniformly bounded moments of all orders. Consider a nonsymmetric Toeplitz matrix $X_n = ((x_{i - j}))_{1 \le i, j \le n}$ and a Hankel matrix $Y_n = ((y_{i + j}))_{1 \le i, j \le n}$, and let $M_n = X_n \odot Y_n$ be their elementwise/Schur-Hadamard product. In this article, we show that almost surely, $n^{-1/2}M_n$, as an element of the $*$-probability space $(\mathcal{M}_n(\mathbb{C}), \frac{1}{n}\mathrm{tr})$, converges in $*$-distribution to a cir…
Non-Markovian chains with long-range dependence and their scaling limits
本文研究了具有长程依赖性的非马尔可夫链及其缩放极限，探讨了para-马尔可夫链和时间改变的马尔可夫链模型，揭示了其与分数阶方程和异常扩散的关系。
The basic $κ$-color box-ball (BBS) system is an integrable cellular automaton on one dimensional lattice whose local states take $\{0,1,\cdots,κ\}$ with $0$ regarded as an empty box. The time evolution is defined by a combinatorial rule of quantum group theoretical origin, and the complete set of conserved quantities is given by a $κ$-tuple of Young diagrams. In the randomized BBS, a probability distribution on $\{0,1,\cdots,κ\}$ to independently fill the consecutive $n$ sites in the initial state induces a highly nontrivial probability measure on the $κ$-tuple of those invariant Young diagram…
The paper deals with a 3-parameter family of probability measures on the set of partitions, called the z-measures. The z-measures first emerged in connection with the problem of harmonic analysis on the infinite symmetric group. They are a special and distinguished case of Okounkov's Schur measures. It is known that any Schur measure determines a determinantal point process on the 1-dimensional lattice. In the particular case of z-measures, the correlation kernel of this process, called the discrete hypergeometric kernel, has especially nice properties. The aim of the paper is to derive the di…
We establish a fundamental connection between the geometric RSK correspondence and GL(N,R)-Whittaker functions, analogous to the well known relationship between the RSK correspondence and Schur functions. This gives rise to a natural family of measures associated with GL(N,R)-Whittaker functions which are the analogues in this setting of the Schur measures on integer partitions. The corresponding analogue of the Cauchy-Littlewood identity can be seen as a generalisation of an integral identity for GL(N,R)-Whittaker functions due to Bump and Stade. As an application, we obtain an explicit integ…
We study large random partitions boxed into a rectangle and coming from skew Howe duality, or alternatively from dual Schur measures. As the sides of the rectangle go to infinity, we obtain: 1) limit shape results for the profiles generalizing the Vershik--Kerov--Logan--Shepp curve; and 2) universal edge asymptotic results for the first parts in the form of the Tracy--Widom distribution, as well as less-universal critical regime results introduced by Gravner, Tracy and Widom. We do this for a large class of Schur parameters going beyond the Plancherel or principal specializations previously st…
In this paper we prove that the stationary harmonic measure of an infinite set in the upper planar lattice can be represented as the proper scaling limit of the classical harmonic measure of truncations of the infinite set.
Scaling limit of the occupation measure of random walk cut points
本文研究二维和三维立方晶格上简单随机游走割点的占据测度，证明其弱拓扑下的标度极限为基于闵可夫斯基容度定义的布朗割点自然分形测度。
We prove that ergodic measures on one-sided shift spaces are uniformly scaling in the sense of Gavish. That is, given a shift ergodic measure we prove that at almost every point the scenery distributions weakly converge to a common distribution on the space of measures. Moreover, we give an explicit description of the limiting distribution in terms of a `reverse Jacobian' function associated with the corresponding measure on the space of left infinite sequences.
The main result of this paper, Theorem 1.1, gives explicit formulae for the kernels of the ergodic decomposition measures for infinite Pickrell measures on spaces of infinite complex matrices. The kernels are obtained as the scaling limits of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。