arXiv subjects

Alexander Moll

Publications and source records attributed to Alexander Moll.

6 recordsLinked to original sources

arXiv2023

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

This paper establishes universal formulas describing the global asymptotics of two distinct discrete versions of $\beta$-ensembles in the high, low and fixed temperature regimes. Our results affirmatively answer a question posed by the second author and \'Sniady. We first introduce a special class of Jack measures on Young diagrams of arbitrary size, called the ``Jack--Thoma measures'', and prove the LLN and CLT in the three aforementioned limit regimes. In each case, we provide explicit formulas for polynomial observables of the limit shape and Gaussian fluctuations around the limit shape. These formulas have surprising positivity properties and are expressed as sums of weighted lattice paths. Second, we show that the previous formulas are universal: they also describe the limit shape and Gaussian fluctuations for the model of random Young diagrams of a fixed size derived from Jack characters with the approximate factorization property. Finally, in stark contrast with continuous $\beta$-ensembles, we show that the limit shapes at high and low temperatures of our random Young diagrams are one-sided infinite staircase shapes. For the Jack--Plancherel measure, we describe this shape explicitly by relating its local minima with the zeroes of Bessel functions.

math.PR↗

arXiv2022

Spectral Theory of the Nazarov-Sklyanin Lax Operator

In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${\mathcal L} \colon F[w] \rightarrow F[w]$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1) establish a cyclic decomposition $F[w] \cong \bigoplus_{\lambda} Z(j_{\lambda}, {\mathcal L})$ into finite-dimensional ${\mathcal L}$-cyclic subspaces in which Jack polynomials $j_{\lambda}$ may be taken as cyclic vectors and (2) prove that the restriction of ${\mathcal L}$ to each $Z(j_{\lambda}, {\mathcal L})$ has simple spectrum given by the anisotropic contents $[s]$ of the addable corners $s$ of the Young diagram of $\lambda$. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated to ${\mathcal L}$, both established by Nazarov-Sklyanin. Finally, we conjecture that the ${\mathcal L}$-eigenfunctions $\psi_{\lambda}^s {\in F[w]}$ {with eigenvalue $[s]$ and constant term} $\psi_{\lambda}^s|_{w=0} = j_{\lambda}$ are polynomials in the rescaled power sum basis $V_{\mu} w^l$ of $F[w]$ with integer coefficients.

math.CO↗

arXiv2020

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

In this paper we determine two asymptotic results for Jack measures on partitions, a model defined by two specializations of Jack polynomials proposed by Borodin-Olshanski in [European J. Combin. 26.6 (2005): 795-834]. Assuming these two specializations are the same, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to diverging, fixed, and vanishing values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call "ribbon paths", show for general Jack measures that certain joint cumulants 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-Dol\k{e}ga.

math.PR↗

arXiv2019

Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation

In this paper we describe the spectrum of the quantum periodic Benjamin-Ono equation in terms of the multi-phase solutions of the underlying classical system (the periodic multi-solitons). To do so, we show that the semi-classical quantization of this system given by Abanov-Wiegmann is exact and equivalent to the geometric quantization by Nazarov-Sklyanin. First, for the Liouville integrable subsystems defined from the multi-phase solutions, we use a result of G\'{e}rard-Kappeler to prove that if one neglects the infinitely-many transverse directions in phase space, the regular Bohr-Sommerfeld conditions on the actions are equivalent to the condition that the singularities of the Dobrokhotov-Krichever multi-phase spectral curves define an anisotropic partition (Young diagram). Next, we locate the renormalization of the classical dispersion coefficient by Abanov-Wiegmann in the realization of Jack functions as quantum periodic Benjamin-Ono stationary states. Finally, we show that the classical energies of Bohr-Sommerfeld multi-phase solutions in the renormalized theory give the exact quantum spectrum found by Nazarov-Sklyanin without any Maslov index correction.

math-ph↗

arXiv2019

Finite Gap Conditions and Small Dispersion Asymptotics for the Classical Periodic Benjamin-Ono Equation

In this paper we characterize the Nazarov-Sklyanin hierarchy for the classical periodic Benjamin-Ono equation in two complementary degenerations: for the multi-phase initial data (the periodic multi-solitons) at fixed dispersion and for bounded initial data in the limit of small dispersion. First, we express this hierarchy in terms of a piecewise-linear function of an auxiliary real variable which we call a dispersive action profile and whose regions of slope $\pm 1$ we call gaps and bands, respectively. Our expression uses Kerov's theory of profiles and Kre\u{\i}n's spectral shift functions. Next, for multi-phase initial data, we identify Baker-Akhiezer functions in Dobrokhotov-Krichever and Nazarov-Sklyanin and prove that multi-phase dispersive action profiles have finitely-many gaps determined by the singularities of their Dobrokhotov-Krichever spectral curves. Finally, for bounded initial data independent of the coefficient of dispersion, we show that in the small dispersion limit, the dispersive action profile concentrates weakly on a convex profile which encodes the conserved quantities of the dispersionless equation. To establish the weak limit, we reformulate Szeg\H{o}'s first theorem for Toeplitz operators using spectral shift functions. To illustrate our results, we identify the dispersive action profile of sinusoidal initial data with a profile found by Nekrasov-Pestun-Shatashvili and its small dispersion limit with the convex profile found by Vershik-Kerov and Logan-Shepp.

math-ph↗

arXiv2017

Random Partitions and the Quantum Benjamin-Ono Hierarchy

We derive exact and asymptotic results for random partitions from general results in the semi-classical analysis of coherent states applied to the classical periodic Benjamin-Ono equation at critical regularity $s= -1/2$. We find classical $dF_{\star |v} (c| \overline{\varepsilon})$ and quantum $d\hat{F}^{η_{NS}}( c | \hbar, \overline{\varepsilon})|_Ψ$ conserved densities for this system with dispersion coefficient $\overline{\varepsilon}$ extending Nazarov-Sklyanin (2013). For quantum stationary states, this conserved density is $dF_λ(c | \varepsilon_2, \varepsilon_1)$ the Rayleigh measure of the profile of a partition $λ$ of anisotropy $(\varepsilon_2, \varepsilon_1) \in \mathbb{C}^2$ for $\hbar = - \varepsilon_1 \varepsilon_2$, $\overline{\varepsilon}= \varepsilon_1 + \varepsilon_2$ invariant under $\varepsilon_2 \longleftrightarrow \varepsilon_1$. As Jack polynomials are the quantum stationary states and Stanley's Cauchy kernel (1989) is the reproducing kernel, the random values of the quantum periodic Benjamin-Ono hierarchy in a coherent state $Υ_v ( \cdot | \hbar)$ are a "Jack measure" on partitions, a dispersive generalization of Okounkov's Schur measures (1999). By our general results for coherent states, we have concentration on a limit shape as $\hbar \rightarrow 0$, the classical conserved density at $v$, and quantum fluctuations are an explicit Gaussian field. Our results follow from an enumerative asymptotic expansion in $\hbar$ and $\overline{\varepsilon}$ of joint cumulants over new combinatorial objects we call "ribbon paths". Our results reflect the fact that at fixed $\hbar>0$ the weight defining Fock space is already a fractional Brownian motion of variance $\hbar$ and Hurst index $(-s) - \tfrac{1}{2} \dim \mathbb{T} = + \tfrac{1}{2} - \tfrac{1}{2} = 0.$

math-ph↗