Universal Phenomena in
Analytic Combinatorics

PR Abstract

From Data to Universal Laws

Our modern world relies on complex discrete structures, from vast computer networks to phylogenetic trees in biology. In many applications, we need to understand their typical behavior as they grow larger. However, computer simulations are often insufficient because the number of possible configurations in these problems quickly exceeds the number of atoms in the observable universe.

Scientific Abstract

Objective

The UNPAC project aims to uncover and utilize universal, unconventional asymptotic phenomena in large discrete structures found in combinatorics, computer science, and biology. While standard structures often follow Gaussian limit laws, we focus on the "unconventional": stretched exponentials \( a^{n^s} \), special functions (e.g., the Airy function \( \text{Ai}(z) \)), and universal phase transitions. Identifying these phenomena allows for an in-depth description of a structure's large-scale behavior, even in cases where exact enumeration is computationally intractable.

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Research Questions

We aim to characterize families of recurrences and generating functions that exhibit specific asymptotic behaviors. Our key objectives include:

• Developing ground-breaking methods for asymptotic enumeration involving stretched exponentials and special functions.
• Demonstrating the natural emergence of Airy-type limit laws in combinatorial models like Young tableaux and Directed Acyclic Graphs (DAGs).
• Enriching fundamental classes of lattice paths and DAGs to better fit applications in phylogenetics and theoretical physics.

Methodology: Analytic Combinatorics

Our approach leverages the powerful toolkit of analytic combinatorics, bridging three main pillars:

• Complex Analysis: Singularity analysis, saddle point methods, and multidimensional contour integration.
• Probability Theory: Limit laws, method of moments, and local limit theorems.
• Combinatorics: Symbolic methods, bijection construction, and multivariate generating functions.

Originality

While univariate recurrences are well-understood, the general bivariate problem remains largely open. The originality of UNPAC lies in its universal viewpoint: rather than treating strange phenomena as isolated anomalies, we are developing a unified theory. This includes proving the universality of the Airy distribution in map enumeration and introducing "Young tableaux with walls" to tackle unsolved problems in high-energy physics.

Open Positions

We are currently seeking talented researchers to join the UNPAC project at TU Wien starting in September 2026 (later start dates are also possible).

Publications