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Singularity confinement for a class of m-th order difference equations of combinatorics
Mark Adler1, Pierre van Moerbeke, Pol Vanhaecke
1Department of Mathematrics, Brandeis University, Waltham, MA 02454, USA
Summary
This study reveals that integrals over the unitary group U(n) follow difference equations with a discrete Painlevé property. This property ensures solutions remain finite, even after developing temporary poles.
Area of Science:
- Mathematical Physics
- Representation Theory
- Combinatorics
Background:
- Integrals over the unitary group U(n) are crucial in various mathematical fields.
- Generating functions for problems like longest increasing subsequences are linked to these integrals.
- Nonlinear, non-autonomous difference equations model complex systems.
Purpose of the Study:
- To analyze the behavior of difference equations satisfied by integrals over the unitary group U(n).
- To investigate the presence and implications of the discrete Painlevé property in these equations.
- To establish a connection between discrete and continuous time systems regarding singularity confinement.
Main Methods:
- Derivation of nonlinear, non-autonomous difference equations for integrals over U(n).
- Application of the discrete Painlevé analysis to these difference equations.
- Exploitation of the relationship between discrete difference equations and the continuous-time Toeplitz lattice.
Main Results:
- A large class of integrals over U(n) satisfy specific difference equations.
- These difference equations are proven to possess the discrete Painlevé property.
- The property ensures that solutions, after potentially developing poles, return to a finite state ('singularity confinement').
Conclusions:
- The discrete Painlevé property of the difference equations is inherited from the continuous-time Toeplitz lattice.
- This finding provides a deeper understanding of the integrability and behavior of these mathematical systems.
- The results have implications for random matrix theory and combinatorics, particularly concerning longest increasing subsequences.
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