Random Structures

Counting independent sets in percolated graphs via the Ising model Given a graph $G$, we form a random subgraph $G_p$ by including each edge of $G$ independently with probability $p$. We provide an asymptotic expansion of the expected number of independent sets in random subgraphs of regular bipartite graphs satisfying certain vertex-isoperimetric properties, extending the work of Kronenberg and Spinka on the percolated hypercube. Combining graph containers with the cluster expansion from statistical physics, we give an expansion of the partition function of the Ising model in certain range of the parameters. Among other applications, we obtain results for even tori of growing side-length. As a tool, we prove a refined container lemma for the Ising model, which mildly improves recent bounds of Jenssen, Malekshahian, and Park.

Read the full paper here: arxiv.org/abs/2504.08715

22.05.2025 14:39 — 👍 0    🔁 0    💬 0    📌 0

A dark blue background with title of the Preprint Counting Independent sets in percolated graphs via the Ising model and an illustration of the Ising model figure from the research

Our paper on percolated graphs is a compelling example of how combining ideas from combinatorics, probability, and statistical physics can lead to new insights.



Our approach is based on the initial insight that, in expectation, the problem behaves like the Ising Model.

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22.05.2025 14:39 — 👍 1    🔁 0    💬 1    📌 0

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Excited to announce Nathanael has been named a Fellow of the IMS for his contributions to probability theory and mathematical physics.

“I couldn't be prouder to have been named a Fellow. The IMS publishes the best journals in the field, making sure these are priced affordably and inclusively.”

06.05.2025 11:44 — 👍 1    🔁 0    💬 0    📌 0

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View from Facculty of Mathematics of Uni Vienna

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We hosted Andres Contreras Hip from Chicago for a 2-week research visit in Vienna! He joined us for maths collaboration, talks, and even visited the Budapest-Vienna Seminar.
Andres concluded: “Great math, food & music in a superb city.”
Swipe to see Vienna & Budapest through his eyes!

24.04.2025 06:43 — 👍 0    🔁 0    💬 0    📌 0

Nathanael and Prof Sun on campus at PKU

PKU campus in spring

Nathanael at the Institute of Mathematics in Beijing

A german quote "We must know, we will know" above the door to a lecture hall at PKU

Nathanael just returned from a research trip to China, where he met Prof Sun at PKU.

He said:
“I was hugely impressed by the scientific level, which is on a strikingly exponential upward trajectory. China is building a world-class scientific environment, and probability community in particular."

17.04.2025 10:59 — 👍 0    🔁 0    💬 0    📌 0

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We're a research cluster uniting experts from @univie.ac.at , @tuwien.at & @tugraz.bsky.social to explore the intersection of #probability and #combinatorics & tackle some of maths' most fascinating problems!

Generously funded by @fwf-at.bsky.social

#RandomStructures #Mathematics #MathResearch

17.04.2025 08:38 — 👍 2    🔁 0    💬 0    📌 0