Marcus Michelen

activity

20182026

most citedCentral limit theorems and the geometry of polynomials

16 citations · 27 across the 24 of their papers we have counts for

collaborators

32 papers

math.CO2026

The largest $K_{r}$ -free set of vertices in a random graph

Tom Bohman, Marcus Michelen, Dhruv Mubayi

For $r \geq 2$ and a graph $G$ , let $α_{r} (G)$ be the maximum number of vertices in a $K_{r}$ -free subgraph of $G$ . We investigate the value $α_{r} (G)$ when $G$ is the random graph $G…

math.CO2026

Comparability of random permutations in the strong Bruhat order

Nicholas Christo, Marcus Michelen

The (strong) Bruhat order for permutations provides a partial ordering defined as follows: two permutations are comparable if one can be obtained from the other by a sequence of ad…

math.CO2026

The random stable roommates problem typically has no solution

Byron Chin, Marcus Michelen

Assume that $n = 2 k$ potential roommates each have an ordered preference of the $n - 1$ others. A stable matching is a perfect matching of the $n$ roommates in which no two unmatched…

math.PR2025

A simple proof of local universality for roots of Kac polynomials

Marcus Michelen, Oren Yakir

Let $f_{n}$ be a random polynomial of degree $n$ with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of $f_{n}$ cluster uniformly around the…

math.CO2025

A polynomial improvement for the odd cycle-complete Ramsey numbers

Marcelo Campos, Matthew Jenssen, Marcus Michelen +2

We give a polynomial improvement to the cycle-complete Ramsey numbers \[ r(C_{\ell},K_k) \geq k^{1+1/(\ell- 2) + \varepsilon_{\ell} + o(1)}, \] for all fixed odd $ℓ > 7$ with $k…

math.PR2025

Convergent points for random power series on the unit circle

Marcus Michelen, Mehtaab Sawhney

Consider a random power series of the form $P (z) = \sum_{n \geq 1} ε_{n} a_{n} z^{n}$ where $a_{n} \in C$ are deterministic and $ε_{n}$ are chosen independently…

math.PR2025

Law of large numbers for the discriminant of random polynomials

Marcus Michelen, Oren Yakir

Let $f_{n}$ be a random polynomial of degree $n$ , whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let $Δ (f_{n})$ denote…

math.CO2025

A new lower bound for the Ramsey numbers $R (3, k)$

Marcelo Campos, Matthew Jenssen, Marcus Michelen +1

We prove a new lower bound for the off-diagonal Ramsey numbers, \[ R(3,k) \geq \bigg( \frac{1}{3}+ o(1) \bigg) \frac{k^2}{\log k }\, , \] thereby narrowing the gap between the uppe…

math.PR2025

Limit law for root separation in random polynomials

Marcus Michelen, Oren Yakir

Let $f_{n}$ be a random polynomial of degree $n \geq 2$ whose coefficients are independent and identically distributed random variables. We study the separation distances between roots…

math.PR2025

On random locally flat-foldable origami

Thomas C. Hull, Marcus Michelen, Corrine Yap

We develop a theory of random flat-foldable origami. Given a crease pattern, we consider a uniformly random assignment of mountain and valley creases, conditioned on the assignment…

math.CO2024

Realizability of hypergraphs and high-dimensional contingency tables with random degrees and marginals

Nicholas Christo, Marcus Michelen

A result of Deza, Levin, Meesum, and Onn shows that the problem of deciding if a given sequence is the degree sequence of a 3-uniform hypergraph is NP complete. We tackle this prob…

quant-ph2024

Entangled states are typically incomparable

Vishesh Jain, Matthew Kwan, Marcus Michelen

Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state $∣ ψ ⟩$ . Using local quantum operations on their respective subsystems, and unlimited clas…

math.MG2023★ 4 cited

A new lower bound for sphere packing

Marcelo Campos, Matthew Jenssen, Marcus Michelen +1

We show there exists a packing of identical spheres in $R^{d}$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d \to \infty$ . This improves upon previous…

math.PR2023

On random polynomials with an intermediate number of real roots

Marcus Michelen, Sean O'Rourke

For each $α \in (0, 1)$ , we construct a bounded monotone deterministic sequence $(c_{k})_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z)…

math.PR2023

Almost sure behavior of the zeros of iterated derivatives of random polynomials

Marcus Michelen, Xuan-Truong Vu

Let $Z_{1}, Z_{2}, \dots$ be independent and identically distributed complex random variables with common distribution $μ$ and set $P_{n} (z) := (z - Z_{1}) \dots (z - Z_{n}) .$ Recentl…

math.CO2023

The binomial random graph is a bad inducer

Vishesh Jain, Marcus Michelen, Fan Wei

For a finite graph $F$ and a value $p \in [0, 1]$ , let $I (F, p)$ denote the largest $y$ for which there is a sequence of graphs of edge density approaching $p$ so that the induced $F…

cs.DS2023

Optimal mixing of the down-up walk on independent sets of a given size

Vishesh Jain, Marcus Michelen, Huy Tuan Pham +1

Let $G$ be a graph on $n$ vertices of maximum degree $Δ$ . We show that, for any $δ > 0$ , the down-up walk on independent sets of size $k \leq (1 - δ) α_{c} (Δ) n$ mixes in time $O_{Δ,δ}(k\…

math.PR2023

Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere

Marcus Michelen, Oren Yakir

Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of $n$ points on the sphere that nearly minimizes the logarithmic energy. As a candidate startin…

cs.DS2022

Quasipolynomial-time algorithms for Gibbs point processes

Matthew Jenssen, Marcus Michelen, Mohan Ravichandran

We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a finite-range stable potential. Thi…

math.PR2022

The least singular value of a random symmetric matrix

Marcelo Campos, Matthew Jenssen, Marcus Michelen +1

Let $A$ be a $n \times n$ symmetric matrix with $(A_{i, j})_{i \leq j}$ , independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}(σ…

math.PR2021

Sums of random polynomials with differing degrees

Isabelle Kraus, Marcus Michelen, Sean O'Rourke

Let $μ$ and $ν$ be probability measures in the complex plane, and let $p$ and $q$ be independent random polynomials of degree $n$ , whose roots are chosen independently from $μ$ and…

eess.SP2021

Direct design of biquad filter cascades with deep learning by sampling random polynomials

Joseph T. Colonel, Christian J. Steinmetz, Marcus Michelen +1

Designing infinite impulse response filters to match an arbitrary magnitude response requires specialized techniques. Methods like modified Yule-Walker are relatively efficient, bu…

math.PR2021★ 1 cited

The singularity probability of a random symmetric matrix is exponentially small

Marcelo Campos, Matthew Jenssen, Marcus Michelen +1

Let $A$ be drawn uniformly at random from the set of all $n \times n$ symmetric matrices with entries in ${- 1, 1}$ . We show that \[ \mathbb{P}( \det(A) = 0 ) \leq e^{-cn},\] where…

math.PR2021

Anti-concentration of random variables from zero-free regions

Marcus Michelen, Julian Sahasrabudhe

This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let $X$ be a random variab…

The largest Kr​-free set of vertices in a random graph

Comparability of random permutations in the strong Bruhat order

The random stable roommates problem typically has no solution

A simple proof of local universality for roots of Kac polynomials

A polynomial improvement for the odd cycle-complete Ramsey numbers

Convergent points for random power series on the unit circle

Law of large numbers for the discriminant of random polynomials

A new lower bound for the Ramsey numbers R(3,k)

Limit law for root separation in random polynomials

On random locally flat-foldable origami

Realizability of hypergraphs and high-dimensional contingency tables with random degrees and marginals

Entangled states are typically incomparable

A new lower bound for sphere packing

On random polynomials with an intermediate number of real roots

Almost sure behavior of the zeros of iterated derivatives of random polynomials

The binomial random graph is a bad inducer

Optimal mixing of the down-up walk on independent sets of a given size

Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere

Quasipolynomial-time algorithms for Gibbs point processes

The least singular value of a random symmetric matrix

Sums of random polynomials with differing degrees

Direct design of biquad filter cascades with deep learning by sampling random polynomials

The singularity probability of a random symmetric matrix is exponentially small

Anti-concentration of random variables from zero-free regions

The largest $K_{r}$ -free set of vertices in a random graph

A new lower bound for the Ramsey numbers $R (3, k)$