https://media.medfarm.uu.se/play/kanal/920svCopyright (C) 2026 Uppsala universitet, MedfarmDoITDumitriu and Edelman (2002) provided tridiagonal random models to compute eigenvalue distribution for Gaussian and Laguerre ensembles. By generalizing this method, Kozhan (2017) was able to find tridiagonal models corresponding to rank one additive non-hermitian perturbations of Gaussian and Wishart ensembles and computed the eigenvalue distribution for the mentioned random matrices. I will survey the above results and discuss the recent results (joint work with Kozhan (2022)) on the eigenvalue distribution of rank one Hermitian and non-Hermitian perturbations of Chiral random matrices. If time permits, I will also report new results on rank one multiplicative non-Hermitian perturbations of the ensembles mentioned above.https://media.medfarm.uu.se/play/video/15034Mon, 23 May 2022 19:10:43 +0000First I will focus on the main ideas in "Experiments with the Site Frequency Spectrum" that appeared in: The Inaugural Issue in Algebraic Biology, Bulletin of Mathematical Biology, 73.4, 829-872 (2011), https://doi.org/10.1007/s11538-010-9605-5 (jointly with eight other co-authors) with the following Abstract: "Evaluating the likelihood function of parameters in highly-structured population genetic models from extant deoxyribonucleic acid (DNA) sequences is computationally prohibitive. In such cases, one may approximately infer the parameters from summary statistics of the data such as the site-frequency-spectrum (SFS) or its linear combinations. Such methods are known as approximate likelihood or Bayesian computations. Using a controlled lumped Markov chain and computational commutative algebraic methods, we compute the exact likelihood of the SFS and many classical linear combinations of it at a non-recombining locus that is neutrally evolving under the infinitely-many-sites mutation model. Using a partially ordered graph of coalescent experiments around the SFS, we provide a decision-theoretic framework for approximate sufficiency. We also extend a family of classical hypothesis tests of standard neutrality at a non-recombining locus based on the SFS to a more powerful version that conditions on the topological information provided by the SFS." Then, I will present some generalisations of these ideas (jointly with Tanja Stadler and Amandine Veber) in "Finding the best resolution for the Kingman–Tajima coalescent: theory and applications", Journal of Mathematical Biology, 70, 1207–1247 (2015), https://doi.org/10.1007/s00285-014-0796-5 and in "Full likelihood inference from the site frequency spectrum based on the optimal tree resolution", Theoretical Population Biology, 124, 1–15 (2018), https://doi.org/10.1016/j.tpb.2018.07.002 that seem to be having some impact today in the field of theoretical and applied population genomics (including machine-learning and "AI" approaches in predictive tasks). In the last few minutes, I will outline some mathematical challenges in combinatorial stochastic control processes for statistical inference over the same probability space for empirical genome-wide phenomena based on work done jointly with Bhalchandra Thatte and Amandine Veber (and current work with Berzunza-Ojeda) on population pedigree processes with coalescence, recombination and breeding behaviour in "Ancestries of a Recombining Diploid Population", Journal of Mathematical Biology, 72.1, 363-408 (2016) http://dx.doi.org/10.1007/s00285-015-0886-zhttps://media.medfarm.uu.se/play/video/15027Thu, 12 May 2022 10:06:20 +0000Counting objects up to symmetry is a classical subject in combinatorics. In this talk, we take a probabilistic viewpoint and study the automorphism group of two types of random trees: Galton-Watson trees (a family of random rooted trees that include, for example, plane, binary and labelled trees) as well as Pólya trees (rooted, unordered and unlabelled trees). Specifically, we prove that, in both cases, the size of the automorphism group follows a log-normal distribution, asymptotically as the size of the tree goes to infinity. Our proofs use both probabilistic tools and methods from analytic combinatorics.https://media.medfarm.uu.se/play/video/15000Tue, 26 Apr 2022 16:31:08 +0000In analytic combinatorics, the problem of singularity analysis aims at deducing the asymptotic expansion of an array of numbers from the properties of their generating functions near its singularities. In the univariate case (where the array is a sequence), this problem is completely solved for all algebraic functions and the solution is captured by the classical transfer theorem of Flajolet and Odlyzko. In the multivariate case however, the solution to the problem remains elusive. In the past decade, Robin Pemantle and collaborators carried out a substantial amount of work which tackles the case of multivariate rational functions in the diagonal limit. In the bivariate setting, this means computing the asymptotic expansion of the coefficients $a_{m,n}$ of a rational function in the limit where $m,nto infty$ and $m/nto lambda$ for fixed $lambda >0$. In this talk, I will present a new result which treats a different class of multivariate functions in a more general limit regime. In the bivariate case, it provides the asymptotic expansion of $a_{m,n}$ when $m,nto infty$ and $m/n^thetato lambda$ for fixed $theta, lambda >0$. This result generalizes formally Flajolet and Odlyzko's transfer theorem. In particular, it applies to all algebraic functions which are in some sense emph{$Delta$-analytic}. (It can be shown that in the multivariate setting, the class of $Delta$-analytic algebraic functions and the class of rational functions are essentially disjoint.) Applications of the result include Ising-decorated triangulations and parking processes on random trees.https://media.medfarm.uu.se/play/video/14996Tue, 26 Apr 2022 16:25:00 +0000Abstract: Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation process derived from deleting the edges of a uniform random tree on n labelled vertices. In particular, they showed that, after proper rescaling, the above fragmentation process converges as n -> infty to the fragmentation process of the Brownian CRT obtained by cutting-down the Brownian CRT along its skeleton in a Poisson manner. In this talk, we will discuss the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree t_n conditioned on having n vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index alpha in (1,2]. The main result establishes that, after rescaling, the fragmentation process of t_n converges, as n -> infty, to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an alpha-stable Lévy tree. We will also explain how one can construct the latter by considering the partitions of the unit interval induced by the normalized alpha-stable Lévy excursion with a deterministic drift. In particular, the above extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT. The approach uses the Prim's algorithm (or Prim-Jarník algorithm) to define a consistent exploration process that encodes the fragmentation process of t_n. We will discuss the key ideas of the proof. Joint work with Cecilia Holmgren (Uppsala University)https://media.medfarm.uu.se/play/video/14658Thu, 28 Oct 2021 13:24:28 +0000Abstract: How much does the chromatic number of the random graph G(n, 1/2) vary? A classic result of Shamir and Spencer shows that it is contained in some sequence of intervals of length about n^(1/2). Until recently, however, no non-trivial lower bounds on the fluctuations of the chromatic number of a random graph were known, even though the question was raised by Bollobás many years ago. I will talk about the main ideas needed to prove that, at least for infinitely many n, the chromatic number of G(n, 1/2) is not concentrated on fewer than n^(1/2-o(1)) consecutive values. I will also discuss the Zigzag Conjecture, made recently by Bollobás, Heckel, Morris, Panagiotou, Riordan and Smith: this proposes that the correct concentration interval length 'zigzags' between n^(1/4+o(1)) and n^(1/2+o(1)), depending on n. Joint work with Oliver Riordan.https://media.medfarm.uu.se/play/video/14666Thu, 04 Nov 2021 13:27:42 +0000A random recursive tree is a rooted tree constructed by successively choosing a vertex uniformly at random and adding a child to the selected vertex. A random preferential attachment tree is grown in a similar manner, but the vertex selection is made proportional to a linear function of the number of children of a vertex. Preferential attachment trees are the tree version of the Barabasi-Albert preferential attachment model. We consider a red-blue colouring of the vertices of preferential attachment trees, which we call a broadcasting induced colouring: the root is either red or blue with equal probability, while for a fixed value p between 0 and 1, every other vertex is assigned the same colour as its parent with probability p and the other colour otherwise. In this talk I will discuss properties of preferential attachment trees with broadcasting induced colourings, including limit laws for the number of vertices, clusters (maximal monochromatic subtrees) and leaves of each colour. The main focus of the talk will be on the size of the root cluster, that is, the maximal monochromatic subtree containing the root.https://media.medfarm.uu.se/play/video/14732Thu, 25 Nov 2021 11:33:16 +0000Many well-known combinatorial sequences satisfy some sort of recurrence relations. In this talk, we discuss a special class of such sequences, so-called q-recursive sequences. For an integer q at least 2, a q-recursive sequence is defined by recurrence relations on subsequences of indices modulo some fixed power of q. Precise asymptotic results for these sequences are obtained via a detour to q-regular sequences in the sense of Allouche and Shallit. It turns out that many combinatorial sequences are in fact q-recursive. We conclude the talk by studying some specific q-recursive sequences in detail. This is joint work with Clemens Heuberger and Daniel Krenn.https://media.medfarm.uu.se/play/video/14791Tue, 14 Dec 2021 15:25:02 +0000Multitudinous probabilistic and combinatorial objects are associated with generating functions satisfying a composition scheme F(z) = G(H(z)). The analysis becomes challenging when this scheme is critical (i.e., G and H are simultaneously singular). Motivated by many examples (random mappings, planar maps, directed lattice paths), we consider a natural extension of this scheme, namely F(z, u) = G(u H(z))M(z). We also consider a variant of this scheme, which allows us to analyse the number of H-components of a given size in F. We prove that these two models lead to a rich world of limit laws, where we identify the key rôle played by a new universal three-parameter law: the beta-Mittag-Leffler distribution, which is essentially the product of a beta and a Mittag-Leffler distribution. We also prove (double) phase transitions, additionally involving Boltzmann and mixed Poisson distributions. In all cases we obtain moment convergence and local limit theorems. We present several applications of our results for, e.g., random walks, trees, Pólya urns, and the Chinese restaurant process. Among other things, this allows us to answer a problem of Janson on the limit distribution of triangular Pólya urns.https://media.medfarm.uu.se/play/video/14885Thu, 03 Feb 2022 12:20:40 +0000https://media.medfarm.uu.se/play/video/14641Wed, 20 Oct 2021 23:42:17 +0000https://media.medfarm.uu.se/play/video/14642Thu, 21 Oct 2021 00:44:50 +0000Abstract: We consider simply generated random plane trees with n vertices and k_n leaves, sampled from a sequence of weights. Motivated by questions on random planar maps, we will focus on the asymptotic behaviour of the largest degree. Precisely we will give conditions on both the number of leaves and the weight sequence that ensure the convergence in distribution of the associated Lukasiewicz path (or depth-first walk) to the Brownian excursion. This should also provide a first step towards the convergence of their height of contour function. The proof scheme is to reduce step by step to simpler and simpler objects and we will discuss excursion and bridge paths, non decreasing paths conditioned by their tip, and finally estimates of the form of the local limit theorem which may be of independent interest.https://media.medfarm.uu.se/play/video/14640Wed, 20 Oct 2021 17:59:36 +0000