Plancherel Measure Research Articles

On the asymptotic support of Plancherel measures for homogeneous spaces

On the asymptotic support of Plancherel measures for homogeneous spaces

Matter coupled to 3d quantum gravity: one-loop unitarity

Abstract We expect quantum field theories for matter to acquire intricate corrections due to their coupling to quantum fluctuations of the gravitational field. This can be precisely worked out in 3d quantum gravity: after integrating out quantum gravity, matter fields are effectively described as noncommutative quantum field theories, with quantum-deformed Lorentz symmetries. An open question remains: Are such theories unitary or not? On the one hand, since these are effective field theories obtained after integrating out high energy degrees of freedom, we may expect the loss of unitarity. On the other hand, as rigorously defined field theories built with Lorentz symmetries and standing on their own, we naturally expect the conservation of unitarity. In an effort to settle this issue, we explicitly check unitarity for a scalar field at one-loop level in both Euclidean and Lorentzian space-time signatures. We find that unitarity requires adding an extra-term to the propagator of the noncommutative theory, corresponding to a massless mode and given by a representation with vanishing Plancherel measure, thus usually ignored in spinfoam path integrals for quantum gravity. This indicates that the inclusion of matter in spinfoam models, and more generally in quantum gravity, might be more subtle than previously thought.

Spectral representation in Klein space: simplifying celestial leaf amplitudes

In this paper, we explore the spectral representation in Klein space, which is the split (2, 2) signature flat spacetime. The Klein space can be foliated into Lorentzian AdS3/ℤ slices, and its identity resolution has continuous and discrete parts. We calculate the identity resolution and the Plancherel measure in these slices. Using the foliation of Klein space into the slices, the identity resolution, and the Plancherel measure in each slice, we compute the spectral representation of the massive bulk-to-bulk propagator in Klein space. It can be expressed as the sum of the product of two massive (or tachyonic) conformal primary wavefunctions, with both continuous and discrete parts, and sharing a common boundary coordinate. An interesting point in Klein space is that, since the identity resolution has discrete and continuous parts, a new type of conformal primary wavefunction naturally arises for the massive (or tachyonic) case. For the conformal primary wavefunctions, both the discrete and continuous parts involve integrating over the common boundary coordinate and the real (or imaginary) mass. The conformal dimension is summed in the discrete part, whereas it is integrated in the continuous part. The spectral representation in Klein space is a computational tool to derive conformal block expansions for celestial amplitudes in Klein space and its building blocks, called celestial leaf amplitudes, by integrating the particle interaction vertex over a single slice of foliation.

Elliptic Analogue of the Vershik–Kerov Limit Shape

We review the limit shape problem for the Plancherel measure and its generalizations found in supersymmetric gauge theory instanton count. We focus on the measure, interpolating between the Plancherel measure and the uniform measure, a \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U(1)$$\\end{document} case of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}=2^{*}$$\\end{document} gauge theory. We give the formula for its limit shape in terms of elliptic functions, generalizing the trigonometric “arcsin” law of Vershik–Kerov and Logan–Schepp.

A p-adic analog of Hasse--Davenport product relation involving ϵ-factors

Abstract In this paper we prove some generalizations of the classical Hasse–Davenport product relation for certain arithmetic factors defined on a p-adic field F, among them one finds the ϵ-factors appearing in Tate’s thesis. We then show that these generalizations are equivalent to some representation theoretic identities relating the determinant of ramified local coefficients matrices defined for coverings of SL 2 ⁢ ( F ) {\mathrm{SL}_{2}(F)} to Plancherel measures and γ-factors.

Core size of a random partition for the Plancherel measure

Core size of a random partition for the Plancherel measure

Asymptotics of Some Plancherel Averages Via Polynomiality Results

Consider Young diagrams of n boxes distributed according to the Plancherel measure. So those diagrams could be the output of the RSK algorithm, when applied to random permutations of the set {1,ldots ,n}. Here we are interested in asymptotics, as nrightarrow infty , of expectations of certain functions of random Young diagrams, such as the number of bumping steps of the RSK algorithm that leads to that diagram, the side length of its Durfee square, or the logarithm of its probability. We can express these functions in terms of hook lengths or contents of the boxes of the diagram, which opens the door for application of known polynomiality results for Plancherel averages. We thus obtain representations of expectations as binomial convolutions, that can be further analyzed with the help of Rice’s integral or Poisson generating functions. Among our results is a very explicit expression for the constant appearing in the almost equipartition property of the Plancherel measure.

An Explicit Minorant for the Amenability Constant of the Fourier Algebra

Abstract We show that if a locally compact group $G$ is non-abelian, then the amenability constant of its Fourier algebra is $\geq 3/2$, extending a result of [9] who proved that this holds for finite non-abelian groups. Our lower bound, which is known to be best possible, improves on results by previous authors and answers a question raised by [16]. To do this, we study a minorant for the amenability constant, related to the anti-diagonal in $G\times G$, which was implicitly used in Runde’s work but hitherto not studied in depth. Our main novelty is an explicit formula for this minorant when $G$ is a countable virtually abelian group, in terms of the Plancherel measure for $G$. As further applications, we characterize those non-abelian groups where the minorant attains its minimal value and present some examples to support the conjecture that the minorant always coincides with the amenability constant.

The Poisson geometry of Plancherel formulas for triangular groups

In this paper we establish the existence of canonical coordinates for generic co-adjoint orbits on triangular groups. These orbits correspond to a set of full Plancherel measure on the associated dual groups. This generalizes a well-known coordinatization of co-adjoint orbits of a minimal (non-generic) type originally discovered by Flaschka. The latter had strong connections to the classical Toda lattice and its associated Poisson geometry. Our results develop connections with the Full Kostant–Toda lattice and its Poisson geometry. This leads to novel insights relating the details of Plancherel theorems for Borel Lie groups to the invariant theory for Borels and their subgroups. We also discuss some implications for the quantum integrability of the Full Kostant Toda lattice.

Limit shape for infinite rank limit of tensor power decomposition for Lie algebras of series *

We consider the Plancherel measure on irreducible components of tensor powers of the spinor representation of . The irreducible representations correspond to the generalized Young diagrams. With respect to this measure the probability of an irreducible representation is the product of its multiplicity and dimension, divided by the total dimension of the tensor product. We study the limit shape of the generalized Young diagram when the tensor power N and the rank n of the algebra tend to infinity with fixed. We derive an explicit formula for the limit shape and prove convergence to it in probability. We prove central limit theorem for global fluctuations around the limit shape.

Orthogonal root numbers of tempered parameters

We show that an orthogonal root number of a tempered L-parameter varphi decomposes as the product of two other numbers: the orthogonal root number of the principal parameter and the value on a central involution of Langlands’s central character for varphi . The formula resolves a conjecture of Gross and Reeder and computes root numbers of Weil–Deligne representations arising in a conjectural description of the Plancherel measure.

Supergroup structure of Jackiw-Teitelboim supergravity

We develop the gauge theory formulation of mathcal{N} = 1 Jackiw-Teitelboim supergravity in terms of the underlying OSp(1|2, ℝ) supergroup, focusing on boundary dynamics and the exact structure of gravitational amplitudes. We prove that the BF description reduces to a super-Schwarzian quantum mechanics on the holographic boundary, where boundary-anchored Wilson lines map to bilocal operators in the super-Schwarzian theory. A classification of defects in terms of monodromies of OSp(1|2, ℝ) is carried out and interpreted in terms of character insertions in the bulk. From a mathematical perspective, we construct the principal series representations of OSp(1|2, ℝ) and show that whereas the corresponding Plancherel measure does not match the density of states of mathcal{N} = 1 JT supergravity, a restriction to the positive subsemigroup OSp+(1|2, ℝ) yields the correct density of states, mirroring the analogous results for bosonic JT gravity. We illustrate these results with several gravitational applications, in particular computing the late-time complexity growth in JT supergravity.

From quantum groups to Liouville and dilaton quantum gravity

We investigate the underlying quantum group symmetry of 2d Liouville and dilaton gravity models, both consolidating known results and extending them to the cases with mathcal{N} = 1 supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of Uq( mathfrak{sl} (2, ℝ)) and review its applications to Liouville gravity. We then derive the corresponding matrix element for Uq( mathfrak{osp} (1|2, ℝ)) and apply it to explain structural features of mathcal{N} = 1 Liouville supergravity. We show that this matrix element has the following properties: (1) its q → 1 limit is the classical OSp+(1|2, ℝ) Whittaker function, (2) it yields the Plancherel measure as the density of black hole states in mathcal{N} = 1 Liouville supergravity, and (3) it leads to 3j-symbols that match with the coupling of boundary vertex operators to the gravitational states as appropriate for mathcal{N} = 1 Liouville supergravity. This object should likewise be of interest in the context of integrability of supersymmetric relativistic Toda chains. We furthermore relate Liouville (super)gravity to dilaton (super)gravity with a hyperbolic sine (pre)potential. We do so by showing that the quantization of the target space Poisson structure in the (graded) Poisson sigma model description leads directly to the quantum group Uq( mathfrak{sl} (2, ℝ)) or the quantum supergroup Uq( mathfrak{osp} (1|2, ℝ)).

Infinite-dimensional groups over finite fields and Hall-Littlewood symmetric functions

The groups mentioned in the title are certain matrix groups of infinite size with elements in a finite field Fq. They are built from finite classical groups and at the same time they are similar to reductive p-adic Lie groups. In the present paper, we initiate the study of invariant measures for the coadjoint action of these infinite-dimensional groups. Of special interest for us are ergodic invariant measures, which are a substitute of orbital measures.We examine first the group GLB, a topological completion of the inductive limit group lim→GL(n,Fq). The traceable factor representations of GLB were studied by Gorin, Kerov, and Vershik (2014) [12]. We show that there exists a parallel theory for ergodic coadjoint-invariant measures, which is linked with harmonic functions on the “HL-deformed Young graph” YHL(t). Here the deformation means that the edges of the Young graph Y are endowed with certain formal multiplicities depending on the Hall-Littlewood (HL) parameter t specialized to q−1.This fact serves as a prelude to our main results, which concern topological completions of two inductive limit groups built from finite unitary groups, of even and odd dimension. We show that in this case, coadjoint-invariant measures are linked to some new branching graphs. They are still related to the HL symmetric functions, but in a nonstandard way, and the HL parameter t takes now the negative value −q−1. As an application, we find several families of unitarily invariant measures, including analogues of the Plancherel measure.

Gaussian Asymptotics of Jack Measures on Partitions From Weighted Enumeration of Ribbon Paths

Abstract In this paper, we determine two asymptotic results for Jack measures $M(v^{\textrm {out}}, v^{\textrm {in}})$, a measure on partitions defined by two specializations $v^{\textrm {out}}, v^{\textrm {in}}$ of Jack polynomials proposed by Borodin–Olshanski in [10]. Assuming $v^{\textrm {out}} = v^{\textrm {in}}$, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to vanishing, fixed, and diverging values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call “ribbon paths,” show for arbitrary $v^{\textrm {out}}, v^{\textrm {in}}$ that certain Jack measure joint cumulants ${\kappa _n}$ are weighted sums of connected ribbon paths on $n$ sites with $n-1+g$ pairings, and derive our two results from the contributions of $(n,g)=(1,0)$ and $(2,0)$, respectively. Our analysis makes use of Nazarov–Sklyanin’s spectral theory for Jack polynomials. As a consequence, we give new proofs of several results for Schur measures, Plancherel measures, and Jack–Plancherel measures. In addition, we relate our weighted sums of ribbon paths to the weighted sums of ribbon graphs of maps on non-oriented real surfaces recently introduced by Chapuy–Dołęga.

A unitary matrix model for q-deformed Plancherel growth

In this paper we construct a unitary matrix model that captures the asymptotic growth of Young diagrams under q-deformed Plancherel measure. The matrix model is a q analog of Gross-Witten-Wadia (GWW) matrix model. In the large N limit the model exhibits a third order phase transition between no-gap and gapped phases, which is a q-deformed version of the GWW phase transition. We show that the no-gap phase of this matrix model captures the asymptotic growth of Young diagrams equipped with q-deformed Plancherel measure. The no-gap solutions also satisfy a differential equation which is the q-analogue of the automodel equation. We further provide a droplet description for these growing Young diagrams. Quantising these droplets we identify the Young diagrams with coherent states in the Hilbert space. We also elaborate the connection between moments of Young diagrams and the infinite number of commuting Hamiltonians obtained from the large N droplets and explicitly compute the moments for asymptotic Young diagrams.

QRSt: A Probabilistic Robinson–Schensted Correspondence for Macdonald Polynomials

Abstract We present a probabilistic generalization of the Robinson–Schensted correspondence in which a permutation maps to several different pairs of standard Young tableaux with nonzero probability. The probabilities depend on two parameters $q$ and $t$, and the correspondence gives a new proof of the squarefree part of the Cauchy identity for Macdonald polynomials (i.e., the equality of the coefficients of $x_1 \cdots x_n y_1 \cdots y_n$ on either side, which are related to permutations and standard Young tableaux). By specializing $q$ and $t$ in various ways, one recovers the row and column insertion versions of the Robinson–Schensted correspondence, several $q$- and $t$-deformations of row and column insertion which have been introduced in recent years in connection with $q$-Whittaker and Hall–Littlewood processes, and the Plancherel measure on partitions. Our construction is based on Fomin’s growth diagrams and the recently introduced notion of a probabilistic bijection between weighted sets.

On the asymptotics of Hecke operators for reductive groups

In this paper, we study the asymptotic behavior of the traces of Hecke operators for spherical discrete automorphic representations of fixed level on general split reductive groups over \(\mathbb {Q}\). Under a condition on the analytic behavior of intertwining operators, which is known for the classical groups and the exceptional group \(G_2\), we obtain the expected asymptotics in terms of the spherical Plancherel measure and an explicit estimate for the remainder.

Quantum mechanics of Plancherel growth

Growth of Young diagrams, equipped with Plancherel measure, follows the automodel equation of Kerov. Using the technology of unitary matrix model we show that such growth process is exactly same as the growth of gap-less phase in Gross-Witten and Wadia (GWW) model. The limit shape of asymptotic Young diagrams corresponds to GWW transition point. Our analysis also offers an alternate proof of limit shape theorem of Vershik-Kerov and Logan-Shepp. Using the connection between unitary matrix model and free Fermi droplet description, we map the Young diagrams in automodel class to different shapes of two dimensional phase space droplets. Quantising these droplets we further set up a correspondence between automodel diagrams and coherent states in the Hilbert space. Thus growth of Young diagrams are mapped to evolution of coherent states in the Hilbert space. Gaussian fluctuations of large N Young diagrams are also mapped to quantum (large N) fluctuations of the coherent states.

Statistical Properties of Eigenvalues of Laplace–Beltrami Operators

We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $$\mu $$ , the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of $$\mu $$ is obtained by a function of independent random variables. Two open problems are also asked.