# Loren Coquille's webpage

I moved to the Hausdorff Center für Mathematik at the University of Bonn, Germany.
See http://wt.iam.uni-bonn.de/coquille/home

Until August 2013 I was PhD student at the Mathematics department of the University of Geneva.
My main interests are somewhere in the intersection between physical mathematics and mathematical physics...

## Research and publications

### Two notes on the discrete Gaussian Free Field with disordered pinning on $\mathbb Z^d$, $d\geq 2$.

Joint work with P. Miłoś.
We study the discrete massless Gaussian Free Field on $\mathbb Z^d$, $d\geq2$, in the presence of a disordered square-well potential supported on a finite strip around zero. The disorder is introduced by reward/penalty interaction coefficients, which are given by i.i.d. random variables.
In the first note, we prove under minimal assumptions on the law of the environment, that the quenched free energy associated to this model exists in $\mathbb R^+$, is deterministic, and strictly smaller than the annealed free energy whenever the latter is strictly positive.
In the second note, we consider Bernoulli reward/penalty coefficients $b\cdot e_x+h$ with $e_x\sim \text{Bernoulli}_{1/2}(-1,+1)$ for all $x\in\mathbb Z^d$, and $b>0$, $h\in\mathbb R$. We prove that in the plane $(b,h)$, the quenched critical line (separating the phases of positive and zero free energy) lies strictly below the line $h=0$, showing in particular that there exists a non trivial region where the field is localized though repulsed on average by the environment.

A note on the discrete Gaussian Free Field with disordered pinning on $\mathbb Z^d$, $d\geq 2$. L. Coquille, P. Miłoś, March 2013.
Accepted for publication in Stoch. Proc. Appl.
DOI 10.1016/j.spa.2013.04.022
[Preprint]

A second note on the discrete Gaussian Free Field with disordered pinning on $\mathbb Z^d$, $d\geq 2$. L. Coquille, P. Miłoś, March 2013.
[Preprint]

### On the Gibbs states of the non-critical Potts model on $\mathbb Z^2$.

Joint work with H. Duminil-Copin, D. Ioffe, and Y. Velenik.
We prove that all Gibbs states of the supercritical $q$-state nearest neighbour Potts model on $\mathbb Z^2$ are convex combinations of the $q$ pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature $\beta>\beta_c(q)=\log(1+\sqrt{q})$.

On the Gibbs states of the non-critical Potts model on $\mathbb Z^2$.
L. Coquille, H. Duminil-Copin, D. Ioffe, Y. Velenik. May 2012.
Accepted for publication in Probab. Theory Relat. Fields.
DOI 10.1007/s00440-013-0486-z
[Preprint]

### A finite-volume version of Aizenman-Higuchi theorem for the $2d$ Ising model.

Joint work with Y. Velenik.
In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model at inverse temperature $\beta \geq 0$ are of the form $\alpha \mu_\beta^+ +(1-\alpha)\mu_\beta^-$, where $\mu_\beta^+$ and $\mu_\beta^-$ are the two pure phases and $0 \leq \alpha \leq 1$. We present here a new approach to this result, with a number of advantages: (i) We obtain a finite-volume, quantitative analogue (implying the classical claim); (ii) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (iii) this new approach seems substantially more robust.

A finite-volume version of Aizenman-Higuchi theorem for the $2d$ Ising model.
L. Coquille, Y. Velenik. April 2010.
Probab. Theory Relat. Fields 153 25-44 (2012).
DOI: 10.1007/s00440-011-0339-6
[Preprint] [Poster] [Slides]

## Formation

• September 2013 - today : Postdoc at the Hausdorff Center für Mathematik at the University of Bonn, Germany.
• August 2009 - August 2013 : PhD in Mathematics under the supervision of Y. Velenik, in University of Geneva.
Title : "Flowers, Forests and Fields in Physics".
[PDF] (in english), [Slides] (in french).
• June 2009 : Master's degree in Theoretical Physics under the supervision of Y. Velenik.
Dissertation topic : "The random cluster model, Ornstein-Zernike asymptotics of the 2-point function above the critical temperature".
[PDF] (in french), [Slides] (in french).
• June 2008 : Bachelor's degree in Mathematics, University of Geneva.
• June 2007 : Bachelor's degree in Physics, University of Geneva.

[PDF] De la non-localité ou le problème EPR2. Theoretical tutorial (in french), under the supervision of C. Branciard, in N. Gisin's group, February 2009.
Preprint (Branciard, Gisin, Scarani).

[PDF] Etude de l’équivalence et de la synthèse entre mécanique matricielle de Heisenberg et mécanique ondulatoire de Schrödinger. Historical work (in french), written for the lecture of History and Philosophy of Quantum Mechanics, given by J. Lacki, October 2006.

## Teaching experience

2012-2013
• Teaching assistant for the course "Models on lattices and phase transitions" by Yvan Velenik, MSc mathematics.
• Member of the popularization team of the math department.

2011-2012
• Co-organizer of the Probability and Statistics seminar for students "Markov chains and fields - theory and applications".

2010-2011
• Teaching assistant for the course "Probability and Statistics" by Yvan Velenik, BSc mathematics, 2nd year.
• Teaching assistant for the course "Analysis III (Measure theory)" by Sasha Friedli, BSc mathematics, 3rd year.
• Teaching assistant for the course "Selected topics of Probability Theory B" by Yvan Velenik, MSc mathematics.

2009-2010
• Teaching assistant for the course "Probability and Statistics" by Yvan Velenik, BSc mathematics, 2nd year.
• Teaching assistant for the course "Selected topics of Probability Theory A" by Yvan Velenik, MSc mathematics.

2006-2007
• Monitor for the course "Algebra I", BSc mathematics, 1st year.
• Monitor for the course "Analysis I", BSc mathematics, 1st year.

## Talks

• 21 June 2013 : Defense of my PhD. [Slides]
• 14 June 2013 : Invited speaker at the conference "Branching, diffusions and Gaussian free fields", Marseille, France.
• 23 January 2013 : Talk for the "UBC Probability Seminar", University of British Columbia, Vancouver, Canada.
• 22 October 2012 : Talk for the "Journée de Probabilités Lyon-Genève", ICJ Université Lyon 1, Lyon, France.
• 14 September 2012 : Talk for the "Séminaire informel de Probabilités", ENS, Paris, France.
• 11 June 2012 : Talk for the "Horowitz Seminar on Probability, Ergodic Theory and Dynamical Systems", Tel Aviv, Israel.
• 28 May 2012 : Invited speaker at "The 2012 Meeting of the Israeli Mathematical Union", Bar Ilan University, Ramat Gan, Israel.
• 26 April 2012 : Talk for the "Groupe de travail Modélisation", LPMA - Chevaleret, Paris 7, France.
• 27 February 2012 : Invited speaker at the "YEP IX" conference, Eurandom, Eindhoven, Holland.
• 5 October 2011 : Talk for the Probability seminar, EPF-Lausanne, Switzerland.
• 8 September 2011 : Talk for the "8th Swiss Graduate Colloquium", Bâle, Switzerland.
• 20 May 2011 : Presentation of a poster at "Les Houches'summer school", Les Houches, France. [Poster]
• 22 February 2011 : Talk for the mathematical physics seminar, Physics department, University of Geneva, Switzerland.
• 16 December 2010 : Presentation of a poster for the "ESF conference", Eurandom, Eindhoven, Holland.
• 2 July 2010 : Talk for the Probability seminar, LATP-CMI Marseille, France.
• 25 May 2010 : Talk for the Probability seminar, Technion Institute of Technology, Haifa, Israel.
• 25 June 2009 : Presentation of my Master thesis. [Slides]

## Long duration stays

• March-July 2012 : 5 month stay at the Technion Institute of Technology, Haifa, Israel. Collaboration with D. Ioffe and N. Crawford.
• May 2010 : One month stay at the Technion Institute of Technology, Haifa, Israel. Collaboration with D. Ioffe and N. Crawford.

## Attendance at conferences

• 10-14 June 2013 : Conference "Branching, diffusions and Gaussian free fields", Marseille, France.
• 11-15 February 2013 : "Ecole d'hiver de Probabilités - Systèmes dynamiques et désordonnés", CIRM, Luminy, France.
• 4-8 February 2013 : "Ecole d'hiver de Probabilités - Mécanique statistique de l'équilibre", CIRM, Luminy, France.
I was teaching assistant for the mini-course "Introduction to statistical mechanics" given by Yvan Velenik. Here are the exercise sheets.
• 18-25 June 2012 : "St-Petersburg School in Probability and Statistical Physics", St-Petersburg, Russia.
• 3-10 May 2012 : "ISF Workshop on Interactions Between Asymptotic Geometric Analysis and Mathematical Physics", Eilat and Haifa, Israel.
• 27 February - 2 March 2012 : Young European Probabilists ("YEP IX") conference about "Two dimensional statistical mechanics", Eindhoven, Holland.
• 2-15 January 2012 : PASI conference about "Topics in percolative and disordered systems", Santiago, Chile and Buenos Aires, Argentina.
• 8-9 September 2011 : "8th Swiss Graduate Colloquium", Bâle, Switzerland.
• 13-17 June 2011 : La Pietra Summer school, Firenze, Italy.
• 17-27 May 2011 : Summer school "Vicious walkers and random matrices", Les Houches, France.
• 12-17 December 2010 : ESF Mathematics conference on "Combinatorics and Analysis in Spatial Probability", Eindhoven, Holland.
• 14-16 October 2010 : "Workshop on Probabilistic Techniques in Statistical Mechanics" in celebration of E. Bolthausen's 65th birthday, Berlin, Germany.
• 11 July - 7 August 2010 : Clay Summer School, Buzios, Brasil.
• 16 October 2009 : Hypathie Seminar, Marseille, France.
• 6-9 September 2009 : Borel Seminar "Statistical mechanics and combinatorics", Les Diablerets, Switzerland.

## Popularization

• I am "proof-reader" for the collection of books "Le monde est mathématique" edited by Le Monde.
• I worked with the "Musée d'Histoire des Sciences" of Geneva for the organization of an exibition about Probability.
Dates of the exibition : February 1st 2012 - April 7th 2013. See "Les jeux sont faits - hasard et probabilités".
• I am "proof-reader" for the mathematics popularization website Images des Mathématiques.
• Since January 2012, I am a member of the popularization team of the math department (University of Geneva).