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Guy Blachar

Publications and source records attributed to Guy Blachar.

14 recordsLinked to original sources
arXiv2026

Edge-regular graphs with non-negative curvature have polynomial growth

A long-standing conjecture in the emerging discrete Bakry-\'Emery theory asserts that bounded-degree graphs satisfying $\mathrm{CD}(0,\infty)$ have polynomial growth. In the present paper, we prove this conjecture for all edge-regular graphs, and even obtain a volume doubling estimate with a constant that depends only on the degree. This is made possible thanks to the discovery of a surprising self-improvement phenomenon, which seems of independent interest: any edge-regular graph satisfying $\mathrm{CD}(\kappa,\infty)$ for some $\kappa\in\mathbb R$ must in fact satisfy $\mathrm{CD}(\kappa,n)$ for some explicit, universal and optimal dimension parameter $n$.

math.CO↗
arXiv2026

Small entropy doubling for random walks and polynomial growth

Gromov's theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. A key ingredient in its proof is the small doubling property. In this work, we study entropy analogues of this property for random walks on groups. We show that if a finitely supported symmetric random walk $R_n$ satisfies \[ \mathrm{H}(R_{2n}) \le \mathrm{H}(R_n) + \log K \] at some sufficiently large scale $n$, then the underlying group is virtually nilpotent, with bounds depending on $K$ and $\mu_{\min}$. Our approach adapts Tao's entropy Balog--Szemer\'edi--Gowers argument to unimodular locally compact groups, combined with structural results on approximate groups. As applications, we obtain entropy-based criteria for polynomial growth. We also deduce an entropy gap phenomenon: if $G$ is not virtually nilpotent, then the entropy of random walks on $G$ grows faster than a universal superlogarithmic function.

math.GR↗
arXiv2025

Phase transition for recurrence of stationary random walks on lamplighter groups

We introduce and study a class of random walks on lamplighter groups $H\wr G$, where $H$ is a nontrivial finitely generated group and $G$ is an infinite finitely generated group, called \textbf{stationary random walks}. At each step, the walk switches the lamp at its current position, moves in the base group with a drift towards the identity, and switches the lamp again at the new position. We show that when $G$ is virtually-$\mathbb{Z}$ and $H$ is finite, these walks exhibit a phase transition between recurrence and transience, while when~$G$ is not virtually-$\mathbb{Z}$ or $H$ is infinite, they are always transient. In the case $G=\mathbb{Z}$, we determine the exact critical parameter and provide a quantitative description of this phase transition.

math.PR↗
arXiv2025

The speed of random walks on semigroups

We construct, for each real number $0\leq \alpha \leq 1$, a random walk on a finitely generated semigroup whose speed exponent is $\alpha$. We further show that the speed function of a random walk on a finitely generated semigroup can be arbitrarily slow, yet tending to infinity. These phenomena demonstrate a sharp contrast from the group-theoretic setting. On the other hand, we show that the distance of a random walk on a finitely generated semigroup from its starting position is infinitely often larger than a non-constant universal lower bound, excluding a certain degenerate case.

math.GR↗
arXiv2025

Mixability of finite groups

Say that a finite group $G$ is mixable if a product of random elements, each chosen independently from two options, can distribute uniformly on $G$. We present conditions and obstructions to mixability. We show that $2$-groups, the symmetric groups, the simple alternating groups, several matrix and sporadic simple groups, and most finite Coxeter groups, are mixable. We also provide bounds on the mixing length of such groups.

math.GR↗
arXiv2024

Profinite rigidity of lamplighter groups

We show that the lamplighter groups $(\mathbb{Z}/p\mathbb{Z})^n\wr\mathbb{Z}$, where $p$ is prime and $n\ge 1$ is a positive integer, are profinitely rigid.

math.GR↗
arXiv2024

Rank-stability of polynomial equations

Extending the thoroughly studied theory of group stability, we study Ulam stability type problems for associative and Lie algebras; namely, we investigate obstacles to rank-approximation of almost solutions by exact solutions for systems of polynomial equations. This leads to a rich theory of stable associative and Lie algebras, with connections to linear soficity, amenability, growth, and group stability. We develop rank-stability and instability tests, examine the effect of algebraic constructions on rank-stability, and prove that while finite-dimensional associative algebras are rank-stable, `most' finite-dimensional Lie algebras are not.

math.RA↗
arXiv2023

Semiassociative algebras over a field

An associative central simple algebra is a form of matrices, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrices, which we are led to define and investigate. Semiassociative algebras modulo skew matrices compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.

math.RA↗
arXiv2023

Probabilistic Laws on Infinite Groups

We study the probability that certain laws are satisfied on infinite groups, focusing on elements sampled by random walks. For several group laws, including the metabelian one, we construct examples of infinite groups for which the law holds with high probability, but the group does not satisfy the law virtually. On the other hand, we show that if an infinite group satisfies the law $x^2=1$ with positive probability, then it is virtually abelian.

math.GR↗
arXiv2022

A Law of Iterated Logarithm on Lamplighter Diagonal Products

We prove a Law of Iterated Logarithm for random walks on a family of diagonal products constructed by Brieussel and Zheng (2021). This provides a wide variety of new examples of Law of Iterated Logarithm behaviours for random walks on groups. In particular, it follows that for any $\frac{1}{2}\leq β\leq 1$ there is a group $G$ and random walk $W_n$ on $G$ with $\mathbb{E}|W_n|\simeq n^β$ such that $$0<\limsup \frac{|W_n|}{n^β(\log\log n)^{1-β}}<\infty$$ and $$0<\liminf \frac{|W_n|(\log\log n)^{1-β}}{n^β}<\infty.$$

math.PR↗
arXiv2017

ELT Linear Algebra

Exploded layered tropical (ELT) algebra is an extension of tropical algebra with a structure of layers. These layers allow us to use classical algebraic results in order to easily prove analogous tropical results. Specifically we study the connection between the ELT determinant and linear dependency, and use a generalized version of Kapranov Theorem proved in [7] (called the Fundamental Theorem). In this paper we prove that an ELT matrix is singular if and only if its rows are linearly dependent and that the row rank and submatrix rank of an ELT matrix are equal. We also define an ELT rank for a tropical matrix, and prove that it is equal to its Kapranov rank. In addition, we formalize the concept of ELT inner products, and prove ELT versions of some known theorems such as Cauchy-Schwarz inequality.

math.RA↗
arXiv2017

Modules and Lie Semialgebras over Semirings with a Negation Map

In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. We define a notion of lifting a module with a negation map, similarly to the tropicalization process, and use it to prove several theorems about semirings with a negation map which possess a lift. In the context of Lie semialgebras over semirings with a negation map, we first give basic definitions, and provide parallel constructions to the classical Lie algebras. We prove an ELT version of Cartan's criterion for semisimplicity, and provide a counterexample for the naive version of the PBW Theorem.

math.RA↗
arXiv2017

ELT Linear Algebra II

This paper is a continuation of [arXiv:1603.02204]. Exploded layered tropical (ELT) algebra is an extension of tropical algebra with a structure of layers. These layers allow us to use classical algebraic results in order to easily prove analogous tropical results. Specifically we prove and use an ELT version of the transfer principal presented in [2]. In this paper we use the transfer principle to prove an ELT version of Cayley-Hamilton Theorem, and study the multiplicity of the ELT determinant, ELT adjoint matrices and quasi-invertible matrices. We also define a new notion of trace -- the essential trace -- and study its properties.

math.RA↗