Where appropriate, doing and showing simulations of the models appearing in the articles is encouraged.

Goal of the oral presentations (25 minutes+questions)

The aim is to present the model in an understandable way, explain the question(s) of interest, state the main result(s) and give some important ideas.

Grading criteria:
- respect of the duration
- clarity of the presentation
- synthesis capacity
- subject expertise
- (optional) personal initiative (additional examples, simulations, etc.)

The language of presentation is English.

Some advice:
- rehearse your presentation
- look at the audience
- read Talks are not the same as papers (Terence Tao's blog)

Grading criteria

The oral presentations will take place on Monday, December 7th and Tuesday, December 15th from 13:30 to 17:00

Topics for presentations

• Topic 1 : Wigner's semi-circular law.
  • Document : Lecture notes on random matrix theory (C. Bordenave). The goal is to present "Lecture 1", which establishes a limit theorem for eigenvalues of certain random matrices.
  • Notions involved: caracterisation of measures by their moments, convergence of random measures, combinatorics (Catalan numbers).
  • Estimated effort: ★★
• Topic 2 : Local convergence of large random trees
  • Document : Local limits of conditioned Galton-Watson trees: the infinite spine case. (R. Abraham and J.-F. Delmas Electron. J. Probab. 19 (2014)). The goal is to study the article up to section 4.3 (includes), which establishes a limit theorem for large random trees, seen from their root.
  • Notions involved: definition of local convergence for graphs, monotone class theorem, polish metric space (complete having a countable dense set), random walks.
  • Estimated effort : ★★★
• Topic 3 : Phase transition in percolation.
  • Document : A new proof of the sharpness of the phase transition for Bernoulli percolation on Z^d (H. Duminil-Copin, V. Tassion, L’Enseignement Mathématique (2) 62 (2016), 199–206). The goal is to establish a phase transition for a percolation model.
  • Notions involved: percolation, independence, differential inequality
  • Estimated effort: ★
• Topic 4 : Directed polymers in random environment.
  • Document: A collection of stochastic models (sections 19.1 et 19.2) (D. Chafai, F. Malrieu). The goal is to present Theorem 19.1, which establishes a phase transition for a model of random walks in random environment.
  • Notions involved: Martingales, conditional expectation
  • Estimated effort: ★★
• Topic 5 : Unimodular graphs.
  • Document : Stationary random graphs (Chapitre 2 et section 1.2) (lecture notes by N. Curien). This topic is slightly more "abstract" : the goal is to define the concept of random (infinite) graph with a distinguished vertex, where the distinguished vertex is "chosen uniformly at random", and to give examples and applications.
  • Notions involved: random graphs, notion of local convergence, monotone class theorem, polish metric spaces (complete having a countable dense sequence)
  • Estimated effort: ★★★
• Topic 6 : Exchangeable partitions.
  • Document : Exchangeable coalescents (Sections 1.1 et 1.2) (lecture notes by J. Bertoin). The goal is to present Theorem 1.1, which gives a characterisation of random partitions of the nonnegative integers whose law is invariant under every finite permutation.
  • Notions involved: i.i.d. random variables, random measures
  • Estimated effort: ★★
• Topic 7 : Zeros of Brownian motion.
  • Document : Fractals in Probability and Analysis (section 6.10 et sections 1.1,1.2) (C.J. Bishop et Y. Peres, Cambridge University Press). The goal is to study the structure of the set of zeros of Brownian motion, and to show in particular that its fractal dimension is almost surely equal to 1/2.
  • Notions involved: Brownian motion (a certain familiarity with Brownian motion is recommended, since we will study it in the EA only in November), fractal dimension, strong Markov property.
  • Estimated effort: ★★★
• Topic 8 : Random fractals.
  • Document : Fractal geometry: mathematical foundations and applications (chapter 15, chapter 2, also some other parts) (K. Falconer, Wiley). The goal is to compute the fractal Hausdorff dimension of random Cantor type sets (Theorem 15.1) using martingale techniques.
  • Notions involved: fractal dimension, martingales, conditional expectation.
  • Estimated effort: ★★★
• Topic 9 : Points of increase of Brownian motion.
  • Document : Fractals in Probability and Analysis (section 6.11 et 6.12) (C.J. Bishop et Y. Peres, Cambridge University Press). The goal is to prove Corollary 6.12.2, which says that almost surely, Brownian motion does not have any point of local increase.
  • Notions involved: random walks, independence (most of the contents uses only random walks and does not require Brownian motion).
  • Estimated effort: ★★
• Topic 10 : Minimum of a branching random walk.
  • Document : Branching random walks (section 1.2, 1.3, 1.4) (Z. Shi). The goal is to present Theorem 1.3, which involves the minimum of a branching random walk (which can model a population which reproduces and moves on the real line)
  • Notions involved: random walks, subadditive ergodic theorem
  • Estimated effort: ★
• Topic 11 : Voronoi percolation.
  • Document : Exponential decay of connection probabilities for subcritical Voronoi percolation in R^d (H. Duminil-Copin, A. Raoufi, V. Tassion, Probab. Theory Relat. Fields (2019) 173: 479). The goal is to establish a phase transition pour a percolation model on Voronoi cells constructed from a Poisson process.
  • Notions involved: percolation, randomized algorithms
  • Estimated effort: ★★★★
• Topic 12 : Random walks in random environment
  • Document : Lecture Notes on Random Walks in Random Environments (sections 1 et 2) (J. Peterson). The goal is to study the behavior of a random walks on the integers, which makes jumps of amplitude 1 with randomly chosen probabilities (Theorem 2.1 and if possible Theorem 2.4)
  • Notions involved: random walks, Markov chains
  • Estimated effort: ★★
• Topic 13 : Exclusion process.
  • Document : Translation Invariant Exclusion Processes (T. Seppäläinen). The goal is to construct a Markov process on Z^d (the vertices are called sites), where particles jump at random times towards non-occupied sites. The fact that there are infinitely many sites makes the construction delicate.
  • Notions involved: Markov process, Poisson process, Borel-Cantelli lemma
  • Estimated effort: ★★
• Topic 14 : Lévy-Cramer theorem.
  • Document : Three remarkable properties of the Normal distribution (E. Benhamou, B. Guez, N. Paris). The goal is to show that if the sum of two random independent real-valued random variables is Gaussian, then both are Gaussian (Theorem 2.1).
  • Notions involved: characteristic functions, complex analysis
  • Estimated effort: ★
• Topic 15 : Non-differentiability of Brownian motion.
  • Document : Brownian motion (section 1.3) (P. Mörters, Y. Peres). The goal is to show that almost surely, Brownian motion is nowhere differentiable (Theorem 1.27, Theorem 1.30).
  • Notions involved: Brownian motion, Borel-Cantelli lemma
  • Estimated effort: ★
• Topic 16 : Population genetics.
• Document A collection of stochastic models (sections 12.2 & 12.3) (D. Chafai, F. Malrieu) The goal is to present results on the Moran and Wright-Fisher model.
• Notions involved: Markov chains, martingales.
• Estimated effort: ★★
• Topic 17 : Largest roots of random polynomials.
• Document : The largest root of random Kac polynomials is heavy tailed (R. Butez). The goal is to study the largest and smallest root in modulus of random polynomials where the coefficients are i.i.d.
• Notions involved: Borel-Cantelli lemmas, complex analysis (Rouché's theorem)
• Estimated effort: ★★
• Topic 18 : Cutoff for deck shuffling
• Document [in French]: Chapitres Choisis de Théorie des Probabilités (chapitre 13) (Y. Velenik) [Automatic translation using ChatGPT-6 Astra Ultra with no human intervention]. The goal is to study the number of times needed to shuffle a deck (by taking a card at the top of the deck and inserting it in a random position) so that it is "well"-shuffled.
• Notions involved: Markov chains
• Estimated effort: ★★
• Topic 19 : The Ergodic theorem
• Document [in French] : Chapitres Choisis de Théorie des Probabilités (chapitre 15) (Y. Velenik) [Automatic translation using ChatGPT-6 Astra Ultra with no human intervention]. The goal is to present Birkhoff's ergodic theorem, which can be seen as an extension of the strong law of large numbers.
• Notions involved: limit theorems, conditional expectations
• Estimated effort: ★★
• Topic 20 : The voter model
• Document: Probability on graphs (Section 10.3) (G. Grimmett). The goal is to study if consensus is reached for the votel model in Z^d, which is a simple model for the propagation of an opinion.
• Notions involved: Markov chains, random walks
• Estimated effort: ★★
• Topic 21 : Elephant random walks
• Document: A martingale approach for the elephant random walk (B. Bercu). The goal is to study limit theorems (Theorems 3.3, 3.6, 3.7) for the elephant random walk, which is a non-Markovian random walk with a long-term memory effect, using martingale convergence theorems (taken for granted).
• Notions involved: limit theorems, martingales
• Estimated effort: ★★
• Topic 22 : The Page-Rényi process
• Document: The Page-Rényi parking process (L. Gérin). Cars of length two successively park uniformly at random on pairs of adjacent places, until only isolated places remain. The foal is to show that the proportion of the interval occupied by cars goes to 1 - e^(−2), when the length of the interval goes to infinity (Theorem 1).
• Notions involved: limit theorems, random permutations, coupling.
• Estimated effort: ★★