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mega:seminaire [2026/10/06 15:20] – Raphaël BUTEZmega:seminaire [2026/10/07 19:13] (Version actuelle) – [Prochaine séance] Guillaume Dubach
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 Vendredi **9 Octobre**, à l'IHP. Vendredi **9 Octobre**, à l'IHP.
  
-     * 10h45-12h15:  mini cours de **[[https://ekvv.uni-bielefeld.de/pers_publ/publ/PersonDetail.jsp?personId=23134650&lang=EN|Gernot Akemann]]**  //Recent developments in non-Hermitian random matrix theory - the threefold way.//\\+     * 10h30-12h:  mini cours de **[[https://ekvv.uni-bielefeld.de/pers_publ/publ/PersonDetail.jsp?personId=23134650&lang=EN|Gernot Akemann]]**  //Recent developments in non-Hermitian random matrix theory - the threefold way.//\\
 Abstract: In this lecture I will point at some recent developments in  non-Hermitian random matrix theory. The symmetry classification of Bernard and Le Clair from the late 90's has been recently revisited by Kawabata et al., finding a total of 38 classes. Short after, based on numerics and heuristic arguments it was conjectured by Hamazaki et al. that in the limit of large matrices only three generic statistics exist in the bulk of the spectrum. The simplest representatives are given by complex symmetric, complex self-dual and complex Ginibre matrices, called class A, AI$^\dagger$ and AII$^\dagger$.  Abstract: In this lecture I will point at some recent developments in  non-Hermitian random matrix theory. The symmetry classification of Bernard and Le Clair from the late 90's has been recently revisited by Kawabata et al., finding a total of 38 classes. Short after, based on numerics and heuristic arguments it was conjectured by Hamazaki et al. that in the limit of large matrices only three generic statistics exist in the bulk of the spectrum. The simplest representatives are given by complex symmetric, complex self-dual and complex Ginibre matrices, called class A, AI$^\dagger$ and AII$^\dagger$. 
 Starting from class A which represents a determinantal point process, I will explain the relation between characteristic polynomials and planar orthogonal polynomials. Being unavailable for the latter two classes, different techniques apply and I will briefly summarise some very recent results based on the Kac-Rice formalism and replicas. A more probabilistic approach is currently open.  Starting from class A which represents a determinantal point process, I will explain the relation between characteristic polynomials and planar orthogonal polynomials. Being unavailable for the latter two classes, different techniques apply and I will briefly summarise some very recent results based on the Kac-Rice formalism and replicas. A more probabilistic approach is currently open. 
    
  
-    * 13h30-14h30: Séminaire de **[[|Georg Angermann]]** //Combinatorial Aspects of Derivatives of Pfaffian Determinants.// \\+    * 14h-15h: Séminaire de **[[|Georg Angermann]]** //Combinatorial Aspects of Derivatives of Pfaffian Determinants.// \\
 Abstract:The derivatives of a determinant or Pfaffian divided by a Vandermonde determinant occur in the study of characteristic polynomials for random matrix ensembles with unitary (determinantal) or orthogonal or symplectic (Pfaffian) structure. For the unitary case, the expansions in terms of determinants with derivative numbers varying according to row or column, have a relatively simple expression in terms of Kostka numbers. Abstract:The derivatives of a determinant or Pfaffian divided by a Vandermonde determinant occur in the study of characteristic polynomials for random matrix ensembles with unitary (determinantal) or orthogonal or symplectic (Pfaffian) structure. For the unitary case, the expansions in terms of determinants with derivative numbers varying according to row or column, have a relatively simple expression in terms of Kostka numbers.
 For the orthogonal and symplectic cases, the expansion in Pfaffians leads to more complex coefficients, as shown in collaborative work with Gernot Akemann, Mario Kieburg and Adrian Padellaro. Here we show the non-negativity of the coefficients used in this expansion by providing a combinatorial interpretation. We translate binomial coefficients to lattice path counts, and show non-negativity by a generalisation of the Lindstr\"om--Gessel--Viennot lemma. For the orthogonal and symplectic cases, the expansion in Pfaffians leads to more complex coefficients, as shown in collaborative work with Gernot Akemann, Mario Kieburg and Adrian Padellaro. Here we show the non-negativity of the coefficients used in this expansion by providing a combinatorial interpretation. We translate binomial coefficients to lattice path counts, and show non-negativity by a generalisation of the Lindstr\"om--Gessel--Viennot lemma.
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 This is joint work with Adrian Padellaro in preparation. This is joint work with Adrian Padellaro in preparation.
  
-    * 15h00-16h00:  Séminaire de **[[https://ekvv.uni-bielefeld.de/pers_publ/publ/PersonDetail.jsp?personId=148940020&lang=EN|Noah Aygün]]** //Products of characteristic polynomials in complex symmetric, self-dual, and Ginibre elliptic ensembles.// \\+    * 15h30-16h30:  Séminaire de **[[https://ekvv.uni-bielefeld.de/pers_publ/publ/PersonDetail.jsp?personId=148940020&lang=EN|Noah Aygün]]** //Products of characteristic polynomials in complex symmetric, self-dual, and Ginibre elliptic ensembles.// \\
 Abstract: It has been conjectured that generic spectral statistic in the bulk and at the edge in all 38 symmetry classes of non-Hermitian random matrix theory follows a threefold way. The simplest symmetry classes that display these limiting universality classes are Gaussian complex symmetric, complex self-dual and complex Ginibre random matrices. In the Cartan classification these are labelled as class A, AI$^\dagger$ and AII$^\dagger$, respectively.  Abstract: It has been conjectured that generic spectral statistic in the bulk and at the edge in all 38 symmetry classes of non-Hermitian random matrix theory follows a threefold way. The simplest symmetry classes that display these limiting universality classes are Gaussian complex symmetric, complex self-dual and complex Ginibre random matrices. In the Cartan classification these are labelled as class A, AI$^\dagger$ and AII$^\dagger$, respectively. 
 Despite recent progress it remains difficult to obtain general multi-point density correlation functions in the latter two classes. Here, we will consider expectation values of products of pairs of complex conjugated characteristic polynomials as observables in these three ensembles, including their elliptic deformation.  Explicit results will be presented for finite matrix size $N$ and in the large-$N$ limit.  Despite recent progress it remains difficult to obtain general multi-point density correlation functions in the latter two classes. Here, we will consider expectation values of products of pairs of complex conjugated characteristic polynomials as observables in these three ensembles, including their elliptic deformation.  Explicit results will be presented for finite matrix size $N$ and in the large-$N$ limit. 
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  • Dernière modification : 2026/10/07 19:13
  • de Guillaume Dubach