The Intertwining Property for Laguerre Processes with a Fixed Parameter

Abstract We investigate the intertwining of Laguerre processes of parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> in different dimensions. We introduce a Feller kernel that depends on $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and intertwines the $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -Laguerre process in $$N+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> dimensions and that in N dimensions. When $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> is a non-negative integer, the new kernel is interpreted in terms of the conditional distribution of the squared singular values: if the singular values of a unitarily invariant random matrix of order $$(N+\alpha +1) \times (N+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are fixed, then the those of its $$(N+\alpha ) \times N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> truncation matrix are given by the new kernel.