Bessel kernel determinants and integrable equations

We derive differential equations for multiplicative statistics of the Bessel determinantal point process depending on two parameters. In particular, we prove that such statistics are solutions to an integrable nonlinear partial differential equation describing isospectral deformations of a Sturm--Liouville equation. We also derive identities relating solutions to the integrable partial differential equation and to the Sturm--Liouville equation which imply an analogue for Painlev\'e V of Amir--Corwin--Quastel ``integro-differential Painlev\'e II equation''. This equation reduces, in a degenerate limit, to the system of coupled Painlev\'e V equations derived by Charlier and Doeraene for the generating function of the Bessel process, and to the Painlev\'e V equation derived by Tracy and Widom for the gap probability of the Bessel process. Finally, we study an initial value problem for the integrable partial differential equation. The approach is based on Its--Izergin--Korepin--Slavnov theory of integrable operators and their associated Riemann--Hilbert problems.Comment: 22 page

Jacobi beta ensemble and b-Hurwitz numbers

We express correlators of the Jacobi beta ensemble in terms of (a special case of) b-Hurwitz numbers, a deformation of Hurwitz numbers recently introduced by Chapuy and Do{\l}\k{e}ga. The proof relies on a generalized Selberg integral. The Laguerre limit is also considered. All the relevant b-Hurwitz numbers are interpreted (following Bonzom, Chapuy, and Do{\l}\k{e}ga) in terms of colored monotone Hurwitz maps.Comment: 15 pages, 1 figur

Tau functions: theory and applications to matrix models and enumerative geometry

In this thesis we study partition functions given by matrix integrals from the point of view of isomon- odromic deformations, or more generally of Riemann\u2013Hilbert problems depending on parameters

On the spectral problem of the quantum KdV hierarchy

The spectral problem for the quantum dispersionless Korteweg-de Vries (KdV) hierarchy, aka the quantum Hopf hierarchy, is solved by Dubrovin. In this article, following Dubrovin, we study Buryak-Rossi's quantum KdV hierarchy. In particular, we prove a symmetry property and a non-degeneracy property for the quantum KdV Hamiltonians. On the basis of this we construct a complete set of common eigenvectors. The analysis underlying this spectral problem implies certain vanishing identities for combinations of characters of the symmetric group. We also comment on the geometry of the spectral curves of the quantum KdV hierarchy and we give a representation of the quantum dispersionless KdV Hamiltonians in terms of multiplication operators in the class algebra of the symmetric group.Comment: 21 page

Integrable equations associated with the finite-temperature deformation of the discrete Bessel point process

We study the finite-temperature deformation of the discrete Bessel point process. We show that its largest particle distribution satisfies a reduction of the 2D Toda equation, as well as a discrete version of the integro-differential Painlev\'e II equation of Amir-Corwin-Quastel, and we compute initial conditions for the Poissonization parameter equal to 0. As proved by Betea and Bouttier, in a suitable continuum limit the last particle distribution converges to that of the finite-temperature Airy point process. We show that the reduction of the 2D Toda equation reduces to the Korteweg-de Vries equation, as well as the discrete integro-differential Painlev\'e II equation reduces to its continuous version. Our approach is based on the discrete analogue of Its-Izergin-Korepin-Slavnov theory of integrable operators developed by Borodin and Deift.Comment: 28 page

Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

We express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulae for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.Comment: 27 page

Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

We consider the Laguerre partition function, and derive explicit generating func-tions for connected correlators with arbitrary integer powers oftraces in terms of products ofHahn polynomials. It was recently proven in [22] that correlators have a topological expansionin terms of weakly or strictly monotone Hurwitz numbers, that can be explicitly computed fromour formul\ue6. As a second result we identify the Laguerre partition function with only positivecouplings and a special value of the parameter\u3b1= 121/2 with the modified GUE partitionfunction, which has recently been introduced in [28] as a generating function for Hodge inte-grals. This identification provides a direct and new link between monotone Hurwitz numbersand Hodge integrals

Matrix models for stationary Gromov-Witten invariants of the Riemann sphere

Inspired by recent formul\ae\ of Dubrovin, Yang, and Zagier, we interpret the tau function enumerating stationary Gromov-Witten invariants of $\mathbb{P}^1$ as an isomonodromic tau function associated with a difference equation. As a byproduct we obtain an analogue of the Kontsevich matrix model for this tau function. A connection with the Charlier ensemble is also considered.Comment: v3: new appendix

Stroke management during the coronavirus disease 2019 (COVID-19) pandemic: experience from three regions of the north east of Italy (Veneto, Friuli-Venezia-Giulia, Trentino-Alto-Adige)

Background: Efficiency of care chain response and hospital reactivity were and are challenged for stroke acute care management during the pandemic period of coronavirus disease 2019 (COVID-19) in North-Eastern Italy (Veneto, Friuli-Venezia-Giulia, Trentino-Alto-Adige), counting 7,193,880 inhabitants (ISTAT), with consequences in acute treatment for patients with ischemic stroke. Methods: We conducted a retrospective data collection of patients admitted to stroke units eventually treated with thrombolysis and thrombectomy, ranging from January to May 2020 from the beginning to the end of the main first pandemic period of COVID-19 in Italy. The primary endpoint was the number of patients arriving to these stroke units, and secondary endpoints were the number of thrombolysis and/or thrombectomy. Chi-square analysis was used on all patients; furthermore, patients were divided into two cohorts (pre-lockdown and lockdown periods) and the Kruskal-Wallis test was used to test differences on admission and reperfusive therapies. Results: In total, 2536 patients were included in 22 centers. There was a significant decrease of admissions in April compared to January. Furthermore, we observed a significant decrease of thrombectomy during the lockdown period, while thrombolysis rate was unaffected in the same interval across all centers. Conclusions: Our study confirmed a decrease in admission rate of stroke patients in a large area of northern Italy during the lockdown period, especially during the first dramatic phase. Overall, there was no decrease in thrombolysis rate, confirming an effect of emergency care system for stroke patients. Instead, the significant decrease in thrombectomy rate during lockdown addresses some considerations of local and regional stroke networks during COVID-19 pandemic evolution