Profile
Profile

Prof. Dr. Patrik Ferrari

Director of the Hausdorff School

E-mail: ferrari(at)uni-bonn.de
Phone: +49 228 73 2747
Homepage: https://wt.iam.uni-bonn.de/ferrari/home/
Room: 3.047
Location: Mathematics Center
Institute: Institute for Applied Mathematics
Research Area: Research Area G
Date of birth: 16.Jul 1977

Academic Career

2001

Diploma in Physics, EPF Lausanne, Switzerland

2004

Dr. rer. nat., TU Munich

2004 - 2006

Postdoc, TU Munich

2006 - 2008

Research Position, Weierstrass Institute for Applied Analysis and Stochastics, Berlin

2008 - 2009

Akademischer Oberrat, University of Bonn

Since 2009

Professor (W2), University of Bonn

Research Profile

The KPZ universality class of stochastic growth models in 1+1 dimensions consists in models with the same physical properties of the KPZ equation. Through the Feynman-Kac representation one sees that the KPZ class includes equilibrium models as directed random polymers as well. The study of special models with a determinantal structure allowed to determine the (conjectural universal) limit processes that describes the fluctuations of interfaces for KPZ models (see e.g. [3,2,1,4]). Along special space-time lines, correlations decay much more slowly than along spatial directions [12]. This property can be used to study decoupling around shocks [13]. In the last few years the number of solvable models has been extended beyond the class with determinantal correlations, leading to a number of results in agreement with the universality conjecture. For these new models, results are so-far available for one-point distributions. This is the case for the semi-discrete directed polymer [5], from which results on the distribution function of the solution of the KPZ equations are obtained [6,5].

Nevertheless, showing universality beyond integrability is still a big challenge and results are not as strong as in random matrix theory. The integrable models are an important starting point, as they could be used for perturbation theory, involving renormalization techniques. Further, if one proves universality using probabilistic argument also for non-integrable models, the identification of the limit processes goes through the solution of the integrable models. One of the major open question (regardless of the model under consideration, i.e., even for models with determinantal structure at fixed time), is the precise description of the limiting process for the time-time correlations.

Research Projects and Activities

DFG Collaborative Research Center SFB 611 “Singular Phenomena and Scaling in Mathematical Models”
Principal Investigator of Project A12 “Universality of fluctuations in mathematical models of physics”

DFG Collaborative Research Center SFB 1060 “The Mathematics of Emergent Effects”
Principal Investigator of Project B04 “Random matrices and random surfaces”

Contribution to Research Areas

Research Area G
My research deals with stochastic models of interacting particle systems and study random variables like the integrated current, which can be seen as height functions of a growing interface with stochastic dynamics. We determined the large time limit processes in the context of the exclusion process [1,2,3] and the space-time covariance [4]. For models like directed polymers at positive temperature, the random variable to be studied is the free energy [5]. The system under consideration are often discrete, but in the large time limit the fluctuation laws becomes universal. The same laws arise in the KPZ equation, which is a singular equation. Some of the models with a tunable parameter governing the asymmetry of the dynamics, can even be scaled directly to converge to the KPZ equation [6]. Recently we studied also time-time correlations [7], random but not stationary initial conditions [8], the space-time covariance structure [9] of a 2+1 dimensional model in the anisotropic KPZ class [10], and finally showed universality for flat initial conditions [11].

Selected Publications

[1] Alexei Borodin, Patrik L. Ferrari, Michael Prähofer, Tomohiro Sasamoto
Fluctuation properties of the TASEP with periodic initial configuration
J. Stat. Phys. , 129: (5-6): 1055--1080
2007
[2] Alexei Borodin, Patrik L. Ferrari, Tomohiro Sasamoto
Transition between {Airy_1} and {Airy_2} processes and TASEP fluctuations
Comm. Pure Appl. Math. , 61: (11): 1603--1629
2008
[3] Jinho Baik, Patrik L. Ferrari, Sandrine Péché
Limit process of stationary TASEP near the characteristic line
Comm. Pure Appl. Math. , 63: (8): 1017--1070
2010
[4] Patrik L. Ferrari, Herbert Spohn
Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process
Comm. Math. Phys. , 265: (1): 1--44
2006
[5] Alexei Borodin, Ivan Corwin, Patrik Ferrari
Free energy fluctuations for directed polymers in random media in 1+1 dimension
Comm. Pure Appl. Math. , 67: (7): 1129--1214
2014
[6] Alexei Borodin, Ivan Corwin, Patrik Ferrari, Bálint Vetö
Height fluctuations for the stationary KPZ equation
Math. Phys. Anal. Geom. , 18: (1): Art. 20, 95
2015
[7] Patrik L. Ferrari, Herbert Spohn
On time correlations for KPZ growth in one dimension
SIGMA Symmetry Integrability Geom. Methods Appl. , 12: : Paper No. 074, 23
2016
[8] S. Chhita, P.L. Ferrari, H. Spohn
Limit distributions for KPZ growth models with spatially homogeneous random initial conditions
preprint: arXiv:1611.06690; To appear in Ann. Appl. Probab.
0
[9] A. Borodin, I. Corwin, P.L. Ferrari
Anisotropic (2+1)d growth and Gaussian limits of q-Whittaker processes
preprint, arXiv:1612.00321; To appear in Probab. Theory Relat. Fields
0
[10] Alexei Borodin, Patrik L. Ferrari
Anisotropic growth of random surfaces in 2+1 dimensions
Comm. Math. Phys.
,
325: (2): 603--684
2014
[11] P.L. Ferrari, A. Occelli
Universality of the GOE Tracy-Widom distribution for TASEP with arbitrary particle density
preprint, arXiv:1704.01291
0
[12] Ivan Corwin, Patrik L. Ferrari, Sandrine Péché
Universality of slow decorrelation in KPZ growth
Ann. Inst. Henri Poincaré Probab. Stat.
, 48: (1): 134--150
2012
[13] Patrik L. Ferrari, Peter Nejjar
Anomalous shock fluctuations in TASEP and last passage percolation models
Probab. Theory Related Fields , 161: (1-2): 61--109
2015
[14] Patrik L. Ferrari, Herbert Spohn
Step fluctuations for a faceted crystal
J. Statist. Phys. , 113: (1-2): 1--46
2003

Publication List

MathSciNet Publication List (external link)

Editorships

• Annals of Applied Probability (since 2013)
• Mathematical Physics, Analysis and Geometry (since 2013)

Awards

2001

Award for the second best general exams average of the complete academic program at EPFL (over all departments)

2004

Distinction “Summa Cum Laude” for the PhD thesis

2009

Heinz Maier-Leibnitz Prize 2009 of the German Research Foundation (DFG)

Selected Invited Lectures

2008

Lecture of 11.5h on Random Matrices and Related Problems at the Beg Rohu Summer School in Bretagne, France

2009

Minicourse of 4h on Dimers and orthogonal polynomials: connections with random matrices at the Workshop Dimer models and random tilings, Institut Henri Poincaré, Paris, France

2011

Short lecture of 6h on Random Matrices and Interacting Particle Systems at the Finnish Center of Excellence in Analysis and Dynamics Research, Helsinki, Finland

2013

Minicourse at the School/Workshop “Random Matrices and Growth Models”, ICTP, Trieste, Italy

2013

Advanced course at the Alea in Europe School, Marseille, France

Offers

2008

Professor (W2), University of Bochum

2008

Professor (W2), University of Bonn

2011

Professor (W3), University of Leipzig

Selected PhD students

René Frings (2014): “Interlacing Patterns in Exclusion Processes and Random Matrices”

Peter Nejjar (2015): “Shock Fluctuations in KPZ Growth Models”,
now Postdoc, Institute of Science and Technology, Austria

Supervised Theses

  • Master theses: 15, currently 1
  • Diplom theses: 8
  • PhD theses: 2, currently 1
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