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Gil Alon

Publications and source records attributed to Gil Alon.

17 recordsLinked to original sources
arXiv2026

Aldous-type Spectral Gaps in Unitary Groups

Aldous' spectral gap conjecture, proven by Caputo, Liggett and Richthammer, states the following: for any set of transpositions in the symmetric group $\mathrm{Sym}(n)$, the spectral gap of the corresponding random walk on the group -- an $n!$-state process -- coincides with that of the corresponding random walk of a single element -- an $n$-state process. This paper presents an analog of this conjecture in the unitary group $\mathrm{U}(n)$, and proves it in several non-trivial cases. The phenomenon we discover is that for some natural families of probability distributions on $\mathrm{U}(n)$, the spectral gap of the corresponding random walk, which has a continuous state space, is identical to that of a discrete KMP process (also known as the uniform reshuffling process) with two indistinguishable particles on a hypergraph on $n$ vertices -- a discrete Markov chain with $\binom{n+1}{2}$ states.

math.PR↗
arXiv2025

Spectral gap for the signed interchange process with arbitrary sets

In 2020, F. Cesi introduced a random walk on the hyperoctahedral group $B_n$ and analysed its spectral gap when the allowed generators are transpositions and diagonal elements corresponding to singletons. In this paper we extend the allowed generators to transpositions and any diagonal elements, and characterise completely the set of representations from which the spectral gap arises. This settles a conjecture posed in Cesi's paper.

math.PR↗
arXiv2025

The Combinatorial Nullstellensatz, Chevalley-Warning Theorem and weak Finitesatz in skew polynomial rings

We study zeros of polynomials in the multivariate skew polynomial ring $D[x_1,\ldots,x_n; \sigma]$, where $\sigma$ is an automorphism of a division ring $D$. We prove a generalization of Noga Alon's celebrated Combinatorial Nullstellensatz for such polynomials. In the case where $D$ is a finite field, we prove skew analogues of the Chevalley--Warning theorem, Ax's Lemma, and the weak case of Terjanian's Finitesatz.

math.AC↗
arXiv2024

On the geometry of zero sets of central quaternionic polynomials II

Following the work of the first and last authors [2], we further analyze the structure of a zero set of a left ideal in the ring of central polynomials over the quaternion algebra H. We describe the "algebraic hull" of a point in H^n and prove it is a product of spheres. Using this description we give a new proof to a conjecture of Gori, Sarfatti and Vlacci. We also show that the main result of [2] does not extend to general division algebras.

math.RA↗
arXiv2024

On the geometry of zero sets of central quaternionic polynomials

Let R be the ring of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if a polynomial p in R vanishes at all the common zeros of I in H^n with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in H^n. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.

math.RA↗
arXiv2024

On the stopping time of the Collatz map in $\mathbb{F}_2[x]$

We study the stopping time of the Collatz map for a polynomial $f \in \mathbb{F}_2[x]$, and bound it by $O({\rm deg} (f)^{1.5})$, improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.

math.CO↗
arXiv2023

On the Aldous-Caputo Spectral Gap Conjecture for Hypergraphs

In their celebrated paper (arXiv:0906.1238), Caputo, Liggett and Richthammer proved Aldous' conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying random walk. A crucial ingredient in the proof was the Octopus Inequality - a certain inequality of operators in the group ring $\mathbb{R}[\mathrm{Sym}_n]$ of the symmetric group. Here we generalize the Octopus Inequality and apply it to generalize the Caputo-Liggett-Richthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.

math.GR↗
arXiv2020

Diagnosis Prevalence vs. Efficacy in Machine-learning Based Diagnostic Decision Support

Many recent studies use machine learning to predict a small number of ICD-9-CM codes. In practice, on the other hand, physicians have to consider a broader range of diagnoses. This study aims to put these previously incongruent evaluation settings on a more equal footing by predicting ICD-9-CM codes based on electronic health record properties and demonstrating the relationship between diagnosis prevalence and system performance. We extracted patient features from the MIMIC-III dataset for each admission. We trained and evaluated 43 different machine learning classifiers. Among this pool, the most successful classifier was a Multi-Layer Perceptron. In accordance with general machine learning expectation, we observed all classifiers' F1 scores to drop as disease prevalence decreased. Scores fell from 0.28 for the 50 most prevalent ICD-9-CM codes to 0.03 for the 1000 most prevalent ICD-9-CM codes. Statistical analyses showed a moderate positive correlation between disease prevalence and efficacy (0.5866).

stat.AP↗
arXiv2019

A Quaternionic Nullstellensatz

We prove a Nullstellensatz for the ring of polynomial functions in n non-commuting variables over Hamilton's ring of real quaternions. We also characterize the generalized polynomial identities in n variables which hold over the quaternions, and more generally, over any division algebra.

math.RA↗
arXiv2018

The mean-field quantum Heisenberg ferromagnet via representation theory

We use representation theory to write a formula for the magnetisation of the quantum Heisenberg ferromagnet. The core new result is a spectral decomposition of the function $\alpha_k 2^{\alpha_1+\dotsb+\alpha_n}$ where $\alpha_k$ is the number of cycles of length k of a permutation. In the mean-field case, we simplify the formula further, arriving at a closed-form expression for the magnetisation, which allows to analyse the phase transition.

math.PR↗
arXiv2018

Comparing with octopi

Operator inequalities with a geometric flavour have been successful in studying mixing of random walks and quantum mechanics. We suggest a new way to extract such inequalities using the octopus inequality of Caputo, Liggett and Richthammer.

math.PR↗
arXiv2017

Automorphisms of Products of Drinfeld Half Planes

The Drinfeld upper half-planes play the role of symmetric spaces in the $p$-adic analytic world. We find the automorphism group of a product of such spaces, where each may be defined over a different field. We deduce a rigidity theorem for quotients of such products by discrete and torsion free groups.

math.NT↗
arXiv2014

Eigenvalues of the Adin-Roichman Matrices

We find the spectrum of the Walsh-Hadamard type matrices defined by R.Adin and Y.Roichman in their recent work on character formulas and descent sets for the symmetric group.

math.CO↗
arXiv2013

Semicharacters of Groups

We define the notion of a semicharacter of a group G : A function from the group to C*, whose restriction to any abelian subgroup is a homomorphism. We conjecture that for any finite group, the order of the group of semicharacters is divisible by the order of the group. We prove that the conjecture holds for some important families of groups, including the Symmetric groups and the groups GL(2,q).

math.GR↗
arXiv2011

Ordering the representations of S_n using the interchange process

Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett and Richthammer, we define an associated order on the irreducible representations of S_n. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the "Aldous order" completely is a generalized question. We show a few additional entries in this order.

math.PR↗
arXiv2010

The probability of long cycles in interchange processes

We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time t, and estimates for the cases of shorter cycles.

math.PR↗