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Gideon Amir

Publications and source records attributed to Gideon Amir.

At least 19 recordsLinked to original sources
arXiv2026

Noise Sensitivity Governed by Continuous-Time Random Walks on the Symmetric Group

We study the noise sensitivity of Boolean functions on the symmetric group, where noise is induced by running a Markov chain on the symmetric group $S_n$, focusing in particular on the case where the underlying chain is an interchange process on the complete graph $K_n$, the $d$-dimensional discrete torus or the star graph. We prove comparison results between these noise sources. We also show that the indicator of long cycles is noise-sensitive under the interchange process on each of the aforementioned graphs. In addition, we study the noise sensitivity of several fundamental functions such as the parity function and analogues of the dictator function. Furthermore, using the fact that the interchange process on the complete graph is the continuous-time random walk generated by all transpositions, we prove that noise sensitivity remains unchanged when the noise source is switched from the continuous-time random walk generated by all transpositions to that generated by all $s$-cycles ($s$ is even and $2<s\ll n$).

math.PR↗
arXiv2025

Voter Model stability with respect to conservative noises

The notions of noise sensitivity and stability were recently extended for the voter model. In this model, the vertices of a graph have opinions that are updated by uniformly selecting edges. We further extend stability results to different classes of perturbations. We consider two different types of noise: in the first one, an exclusion process is performed on the edge selections, while in the second, independent Brownian motions are applied to such a sequence. In both cases, we prove stability of the consensus opinion provided the noise is run for a short amount of time, depending on the underlying graph structure. This is done by analyzing the expected size of the pivotal set, whose definition differs from the usual one in order to reflect the change associated with these noises.

math.PR↗
arXiv2025

Convergence rate of $\ell^p$-energy minimization on graphs: sharp polynomial bounds and a phase transition at $p=3$

We consider the following dynamics on a connected graph $(V,E)$ with $n$ vertices. Given $p>1$ and an initial opinion profile $f_0:V \to [0,1]$, at each integer step $t \ge 1$ a uniformly random vertex $v=v_t$ is selected, and the opinion there is updated to the value $f_{t}(v)$ that minimizes the sum $\sum_{w \sim v} |f_t(v)-f_{t-1}(w)|^p$ over neighbours $w$ of $v$. The case $p=2$ yields linear averaging dynamics, but for all $p \ne 2$ the dynamics are nonlinear. In the limiting case $p=\infty$ (known as Lipschitz learning), $f_t(v)$ is the average of the largest and smallest values of $f_{t-1}(w)$ among the neighbours $w$ of $v$. We show that the number of steps needed to reduce the oscillation of $f_t$ below $\epsilon$ is at most $n^{\beta_p}$ (up to logarithmic factors in $n$ and $\epsilon$), where $\beta_p:=max(\frac{2p}{p-1},3)$; we prove that the exponent $\beta_p$ is optimal. The phase transition at $p=3$ is a new phenomenon. We also derive matching upper and lower bounds for convergence time as a function of $n$ and the average degree; these are the most challenging to prove.

math.PR↗
arXiv2025

Hydrodynamics and relaxation limit for multilane exclusion process and related hyperbolic systems

We investigate the hydrodynamic behavior and local equilibrium of the multilane exclusion process, whose invariant measures were studied in our previous paper \cite{mlt1a}. The dynamics on each lane follows a hyperbolic time scaling, whereas the interlane dynamics has an arbitrary time scaling. We prove the following: (i) the hydrodynamic behavior of the global density (i.e. summed over all lanes) is governed by a scalar conservation law; (ii) the latter, as well as the limit of individual lanes, is the relaxation limit of a weakly coupled hyperbolic system of balance laws that approximates the particle system. For the hydrodynamic limit, to highlight new phenomena arising in our model, a precise computation of the flux function, with the transitions between different possible shapes (and a physical interpretation thereof), is given for the two-lane model.

math.PR↗
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

Planar reinforced $k$-out percolation

We investigate the percolation properties of a planar reinforced network model. In this model, at every time step, every vertex chooses $k \ge 1$ incident edges, whose weight is then increased by 1. The choice of this $k$-tuple occurs proportionally to the product of the corresponding edge weights raised to some power $\alpha > 0$. Our investigations are guided by the conjecture that the set of infinitely reinforced edges percolates for $k = 2$ and $\alpha \gg 1$. First, we study the case $\alpha = \infty$, where we show the percolation for $k = 2$ after adding arbitrarily sparse independent sprinkling and also allowing dual connectivities. We also derive a finite-size criterion for percolation without sprinkling. Then, we extend this finite-size criterion to the $\alpha < \infty$ case. Finally, we verify these conditions numerically.

math.PR↗
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↗
arXiv2023

Majority dynamics and the median process: connections, convergence and some new conjectures

We consider the median dynamics process in general graphs. In this model, each vertex has an independent initial opinion uniformly distributed in the interval [0,1] and, with rate one, updates its opinion to coincide with the median of its neighbors. This process provides a continuous analog of binary majority dynamics. We deduce properties of median dynamics through this connection and raise new conjectures regarding the behavior of majority dynamics on general graphs. We also prove these conjectures on some graphs where majority dynamics has a simple description.

math.PR↗
arXiv2022

Fire retainment on Cayley graphs

We study the fire-retaining problem on groups, a quasi-isometry invariant introduced by Martínez-Pedroza and Prytula [8], related to the firefighter problem. We prove that any Cayley graph with degree-$d$ polynomial growth does not satisfy $\{f(n)\}$-retainment, for any $f(n) = o(n^{d-2})$, matching the upper bound given for the firefighter problem for these graphs. In the exponential growth regime we prove general lower bounds for direct products and wreath products. These bounds are tight, and show that for exponential-growth groups a wide variety of behaviors is possible. In particular, we construct, for any $d\geq 1$, groups that satisfy $\{n^{d}\}$-retainment but not $o(n^d)$-retainment, as well as groups that do not satisfy sub-exponential retainment.

math.GR↗
arXiv2022

Dynamical noise sensitivity for the voter model

We study noise sensitivity of the consensus opinion of the voter model on finite graphs, with respect to noise affecting the initial opinions and noise affecting the dynamics. We prove that the final opinion is stable with respect to small perturbations of the initial configuration, and is sensitive to perturbations of the dynamics governing the evolution of the process. Our proofs rely on the duality relationship between the voter model and coalescing random walks, and on a precise description of this evolution when we have coupled dynamics.

math.PR↗
arXiv2022

The branching number of intermediate growth trees

We introduce an "intermediate branching number"(IBN) which captures the branching of intermediate growth trees, similar in spirit to the well-studied branching number of exponential growth trees. We show that the IBN is the critical threshold for several random processes on trees, and analyze the IBN on some examples of interest. Our main result is an algorithm to find spherically symmetric trees with large IBN inside some permutation wreath products. We demonstrate the usefulness of these trees to the study of intermediate growth groups by using them to get the first tight bounds for the firefighter problem on some inetrmediate growth groups.

math.PR↗
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↗
arXiv2022

Changeover phenomenon in randomly colored Potts models

A hybrid Potts model where a random concentration $p$ of the spins assume $q_0$ states and a random concentration $1-p$ of the spins assume $q>q_0$ states is introduced. It is known that when the system is homogeneous, with an integer spin number $q_0$ or $q$, it undergoes a second or a first order transition, respectively. It is argued that there is a concentration $p^\ast$ such that the transition nature of the model is changed at $p^\ast$. This idea is demonstrated analytically and by simulations for two different types of interaction: the usual square lattice nearest neighboring and mean field all-to-all. Exact expressions for the second order critical line in concentration-temperature parameter space of the mean field model together with some other related critical properties, are derived.

cond-mat.stat-mech↗
arXiv2021

Amenability of quadratic automaton groups

We give lower bounds for the electrical resistance between vertices in the Schreier graphs of the action of the linear (degree 1) and quadratic (degree 2) mother groups on the orbit of the zero ray. These bounds, combined with results of \cite{JNS} show that every quadratic activity automaton group is amenable. The resistance bounds use an apparently new "weighted" version of the Nash-Williams criterion which may be of independent interest.

math.GR↗
arXiv2021

Invariant measures for multilane exclusion process

We consider the simple exclusion process on Z x {0, 1}, that is, an ''horizontal ladder'' composed of 2 lanes, depending on 6 parameters. Particles can jump according to a lane-dependent translation-invariant nearest neighbour jump kernel, i.e. ''horizontally'' along each lane, and ''vertically'' along the scales of the ladder. We prove that generically, the set of extremal invariant measures consists of (i) translation-invariant product Bernoulli measures; and, modulo translations along Z: (ii) at most two shock measures (i.e. asymptotic to Bernoulli measures at $\pm$$\infty$) with asymptotic densities 0 and 2; (iii) at most one (outside degenerate cases) shock measure with a density jump of magnitude 1. We fully determine this set for a range of parameter values. In fact, outside degenerate cases, there is at most one shock measure of type (iii). Our results can be generalized in several directions using the same approach and answer certain open questions formulated in \cite{ligd} as a step towards the process on $\mathbb{Z}^2$.

math.PR↗
arXiv2021

Percolation phase transition on planar spin systems

In this article we study the continuity and sharpness of the phase transition for percolation models defined on top of planar spin systems. The two examples that we treat in detail concern the Glauber dynamics for the Ising model and a Dynamic Bootstrap process. For both of these models we prove that their phase transition is continuous and sharp, providing also quantitative estimates on the two point connectivity. The techniques that we develop in this work can be applied to a variety of different percolation models based on spin-flip dynamics. We also discuss some of the problems that can be tackled in a similar fashion.

math.PR↗