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Universality
- M. Hairer
Universality M. Hairer University of Warwick New fellows seminar - - PowerPoint PPT Presentation
Universality M. Hairer University of Warwick New fellows seminar - - PowerPoint PPT Presentation
Universality M. Hairer University of Warwick New fellows seminar Universality Experimental / numerical fact: large-scale behaviour of systems displaying random behaviour self-similar and independent of microscopic description. Universality
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Universality
Experimental / numerical fact: large-scale behaviour of systems displaying random behaviour self-similar and independent of microscopic description. Example: Random walks / central limit theorem
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Universality
Experimental / numerical fact: large-scale behaviour of systems displaying random behaviour self-similar and independent of microscopic description. Example: Surface growth (KPZ)
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Universality
Experimental / numerical fact: large-scale behaviour of systems displaying random behaviour self-similar and independent of microscopic description. Example: Surface growth (KPZ)
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Heuristic explanation
Schematic evolution in “space of models” under rescaling: Want to understand these “fixed points” and this picture! Fixed points are universal scale-invariant models. Mathematically tractable when fixed point is Gaussian, very hard
- therwise. (Conformal invariance helps a lot in 2D.)
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Heuristic explanation
Schematic evolution in “space of models” under rescaling: Want to understand these “fixed points” and this picture! Fixed points are universal scale-invariant models. Mathematically tractable when fixed point is Gaussian, very hard
- therwise. (Conformal invariance helps a lot in 2D.)
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Heuristic explanation
Schematic evolution in “space of models” under rescaling: Want to understand these “fixed points” and this picture! Fixed points are universal scale-invariant models. Mathematically tractable when fixed point is Gaussian, very hard
- therwise. (Conformal invariance helps a lot in 2D.)
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Intermediate situation
Sometimes Gaussian and non-Gaussian fixed points coexist. Schematic evolution under rescaling: G N Can we understand red line and blue region? Weaker form of universality.
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Intermediate situation
Sometimes Gaussian and non-Gaussian fixed points coexist. Schematic evolution under rescaling: G N Can we understand red line and blue region? Weaker form of universality.
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Interface fluctuation models
Gaussian fixed point: Edwards-Wilkinson universality class. Exponents 1/2 in space and 1/4 in time. Gaussian fluctuations with explicit description. Nonlinear fixed point: KPZ universality class. Exponents 2/3 in space and 1/3 in time. Fluctuations described by random matrix (Tracy-Widom) distributions. No full description yet. Crossover regime: (red line) KPZ equation: ∂th = ∂2
xh + (∂xh)2 + ξ .
Behaves like EW at small scales and KPZ at large scales. Recent contribution: build robust solution theory allowing to show universality results.
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Interface fluctuation models
Gaussian fixed point: Edwards-Wilkinson universality class. Exponents 1/2 in space and 1/4 in time. Gaussian fluctuations with explicit description. Nonlinear fixed point: KPZ universality class. Exponents 2/3 in space and 1/3 in time. Fluctuations described by random matrix (Tracy-Widom) distributions. No full description yet. Crossover regime: (red line) KPZ equation: ∂th = ∂2
xh + (∂xh)2 + ξ .
Behaves like EW at small scales and KPZ at large scales. Recent contribution: build robust solution theory allowing to show universality results.
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Interface fluctuation models
Gaussian fixed point: Edwards-Wilkinson universality class. Exponents 1/2 in space and 1/4 in time. Gaussian fluctuations with explicit description. Nonlinear fixed point: KPZ universality class. Exponents 2/3 in space and 1/3 in time. Fluctuations described by random matrix (Tracy-Widom) distributions. No full description yet. Crossover regime: (red line) KPZ equation: ∂th = ∂2
xh + (∂xh)2 + ξ .
Behaves like EW at small scales and KPZ at large scales. Recent contribution: build robust solution theory allowing to show universality results.
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Main problem
Problem: Functions are not smooth. Usual definition of smoothness: Obviously bad idea!
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Main problem
Problem: Functions are not smooth. Usual definition of smoothness: Obviously bad idea!
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Main problem
Problem: Functions are not smooth. Usual definition of smoothness: Obviously bad idea!
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Main problem
Problem: Functions are not smooth. Usual definition of smoothness: Obviously bad idea!
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A possible solution
Main idea: construct purpose-built objects and describe solution in terms of these: Problem of ill-posed operations boils down to constructing them at the level of these objects.
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