Tsviqa Lakrec

Research

My work in dynamics and probability concerns random walks and effective equidistribution in homogeneous dynamics. In a second direction, I study positive geometry and algebraic combinatorics, especially tilings of the amplituhedron and related cluster structures.

• Homogeneous dynamics
• Ergodic theory
• Probability theory
• Random walks
• The amplituhedron
• Positive geometry

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Publications

1. C Even-Zohar, T Lakrec, M Parisi, M Sherman-Bennet, RJ Tessler, LK Williams. BCFW tilings and cluster adjacency for the amplituhedron Proceedings of the National Academy of Sciences 2025-03-21.

   arXivJournal

   Abstract

   In 2005, Britto, Cachazo, Feng, and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes in N = 4 super Yang-Mills theory. Arkani-Hamed and Trnka subsequently introduced the amplituhedron to give a geometric interpretation of the BCFW recurrence. Arkani-Hamed and Trnka conjectured that each way of iterating the BCFW recurrence gives a "triangulation" or "tiling" of the m=4 amplituhedron. In this article, we prove the BCFW tiling conjecture of Arkani-Hamed and Trnka. We also prove the cluster adjacency conjecture for BCFW tiles of the amplituhedron, which says that facets of tiles are cut out by collections of compatible cluster variables for the Grassmannian Gr4,n. Moreover we show that each BCFW tile is the subset of the Grassmannian where certain cluster variables have particular signs.

2. C Even-Zohar, T Lakrec, RJ Tessler. The Amplituhedron BCFW Triangulation Inventiones Mathematicae 239, 1009-1138 (2025).

   arXivJournal

   Abstract

   The amplituhedron Ank4 is a geometric object, introduced by Arkani-Hamed and Trnka (2013) in the study of scattering amplitudes in quantum field theories. They conjecture that Ank4 admits a decomposition into images of BCFW positroid cells, arising from the Britto--Cachazo--Feng--Witten recurrence (2005). We prove that this conjecture is true.

3. C Even-Zohar, T Lakrec, M Parisi, M Sherman-Bennet, RJ Tessler, LK Williams. A cluster of results on amplituhedron tiles Letters in Mathematical Physics 2024-09-11.

   arXivJournal

   Abstract

   The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in $\mathcal{N}=4$ super Yang--Mills theory. It generalizes cyclic polytopes and the positive Grassmannian and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the $m=4$ amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for $\text{ Gr}_{4,n}$. Secondly, we exhibit a tiling of the $m=4$ amplituhedron which involves a tile which does not come from the BCFW recurrence---the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for $\text{ Gr}_{4,n}$. This paper is a companion to our previous paper ``Cluster algebras and tilings for the $m=4$ amplituhedron.''

4. W He, T Lakrec, E Lindenstrauss. Equidistribution of affine random walks on some nilmanifolds Analysis at Large: Dedicated to the Life and Work of Jean Bourgain, 131-171.

   arXivJournal

   Abstract

   We study quantitative equidistribution in law of affine random walks on nilmanifolds, motivated by a result of Bourgain, Furman, Mozes and the third named author on the torus. Under certain assumptions, we show that a failure to having fast equidistribution is due to a failure on a factor nilmanifold. Combined with equidistribution results on the torus, this leads to an equidistribution statement on some nilmanifolds such as Heisenberg nilmanifolds. In an appendix we strengthen results of de Saxce and the first named author regarding random walks on the torus by eliminating an assumption on Zariski connectedness of the acting group.

5. W He, T Lakrec, E Lindenstrauss. Affine random walks on the torus International Mathematics Research Notices 2022 (11), 8003-8037.

   arXivJournal

   Abstract

   We consider random walks on the torus arising from the action of the group of affine transformations. We give a quantitative equidistribution result for this random walk under the assumption that the Zariski closure of the group generated by the linear part acts strongly irreducibly on $\mathbb{R}^d$ and is either Zariski connected or contains a proximal element. Specifically, we give quantitative estimates (depending only on the linear part of the random walk) for how fast the random walk equidistributes unless the initial point and the translation part of the affine transformations can be perturbed so that the random walk is trapped in a finite orbit of small cardinality. In particular, we prove that the random walk equidistributes in law to the Haar measure if and only if the random walk is not trapped in a finite orbit.

6. T Lakrec. Scenery reconstruction for random walk on random scenery systems Israel Journal of Mathematics, pages 1-52, 2022.

   arXivJournal

   Abstract

   Consider a simple random walk on $\mathbb Z$ with a random coloring of $\mathbb Z$. Look at the sequence of the first $N$ steps taken in the random walk, together with the colors of the visited locations. We call this the record. From the record one can deduce the coloring of of the interval in $Z$ that was visited, which is of size approximately $\sqrt{N}$. This is called scenery reconstruction. Now suppose that an adversary may change $\delta N$ entries in the record that was obtained. What can be deduced from the record about the scenery now? In this paper we show that it is likely that we can still reconstruct a large part of the scenery. More precisely, we show that for any $\theta<0.5,p>0,\epsilon>0$, there are $N_0$ and $\delta_0$ such that if $N>N_0$ and $\delta<\delta_0$ then with probability $>1-p$, the walk is such that we can reconstruct the coloring of $>N_\theta$ integers, up to a number of possible reconstructions that is less than $2^{\epsilon s}$, where $s$ is the number of integers whose color we reconstruct.

Preprints

1. C Even-Zohar, T Lakrec, M Parisi, M Sherman-Bennet, RJ Tessler, LK Williams. Cluster algebras and tilings for the m=4 amplituhedron arXiv preprint arXiv:2310.17727.

   arXiv

   Abstract

   The amplituhedron $\mathcal{A}_{n,k,m}(Z)$ is the image of the positive Grassmannian $\mathrm{Gr}^{\ge0}_{k,n}$ under the amplituhedron map $\mathrm{Gr}^{\ge0}_{k,n}\rightarrow\mathrm{Gr}^{\ge0}_{k,k+m}$ induced by a positive linear map $Z:\mathbb{R}^n\rightarrow\mathbb{R}^{k+m}$. It was originally introduced in physics in order to give a geometric interpretation of scattering amplitudes. More specifically, one can compute scattering amplitudes in $\mathcal{N}=4$ SYM by decomposing the amplituhedron into 'tiles' (closures of images of $4k$-dimensional cells of $\mathrm{Gr}^{\ge0}_{k,n}$ on which the amplituhedron map is injective) and summing up the 'volumes' of the tiles. Such a decomposition into tiles is called a tiling. In this article we deepen our understanding of tiles and tilings of the $m=4$ amplituhedron. We prove the cluster adjacency conjecture for BCFW tiles of $\mathcal{A}_{n,k,4}(Z)$, which says that facets of BCFW tiles are cut out by collections of compatible cluster variables for Gr4,n. We also give an explicit description of each BCFW tile as the subset of $\mathrm{Gr}^{\ge0}_{k,k+4}$ where certain cluster variables have particular signs. And we prove the BCFW tiling conjecture, which says that any way of iterating the BCFW recurrence gives rise to a tiling of the amplituhedron $\mathcal{A}_{n,k,4}(Z)$. Along the way we construct many explicit seeds for the $\mathrm{Gr}^{\ge0}_{n,4}$ comprised of high-degree cluster variables, which may be of independent interest in the study of cluster algebras.

2. C Even-Zohar, T Lakrec, RJ Tessler. Spectral Analysis of Word Statistics arXiv preprint arXiv:2012.00742.

   arXiv

   Abstract

   Given a random text over a finite alphabet, we study the frequencies at which fixed-length words occur as subsequences. As the data size grows, the joint distribution of word counts exhibits a rich asymptotic structure. We investigate all linear combinations of subword statistics, and fully characterize their different orders of magnitude using diverse algebraic tools. Moreover, we establish the spectral decomposition of the space of word statistics of each order. We provide explicit formulas for the eigenvectors and eigenvalues of the covariance matrix of the multivariate distribution of these statistics. Our techniques include and elaborate on a set of algebraic word operators, recently studied and employed by Dieker and Saliola (Adv Math, 2018). Subword counts find applications in Combinatorics, Statistics, and Computer Science. We revisit special cases from the combinatorial literature, such as intransitive dice, random core partitions, and questions on random walk. Our structural approach describes in a unified framework several classical statistical tests. We propose further potential applications to data analysis and machine learning.

Conference proceedings

1. C Even-Zohar, T Lakrec, M Parisi, M Sherman-Bennet, RJ Tessler, LK Williams. Cluster algebras and tilings for the m=4 amplituhedron FPSAC 2024, talk.

2. C Even-Zohar, T Lakrec, RJ Tessler. The Amplituhedron BCFW Triangulation FPSAC 2023, talk.

3. C Even-Zohar, T Lakrec, RJ Tessler. Spectral Analysis of Word Statistics FPSAC 2021, poster.

Students

• Mikael Jonas Konrad, M.Sc. 2022-2023, joint supervision with Alex Gorodnik.

Teaching

• Seminaire Groupes, Geometrie, combinatoire (Student seminar, University of Geneva, Spring 2026)
• Introduction to Dynamical Systems (Course, University of Geneva, Spring 2025)
• Algebraic Combinatorics (Course, University of Zurich, Spring 2024)
• Topics in Dynamics and Expansion (Student seminar, University of Zurich, Spring 2023)
• Spectral graph theory (Student seminar, University of Zurich, Fall 2022)
• Additive Combinatorics (Student seminar, University of Zurich, Spring 2022)
• Dynamical Systems and Ergodic Theory (TA, University of Zurich, Fall 2021)

Contact

E-mail: tsviqa@gmail.com
Office: 3-17
Office hours: by appointment