arXiv subjects

Nina Gantert

Publications and source records attributed to Nina Gantert.

At least 19 recordsLinked to original sources

arXiv2026

When does propagation of chaos in the critical Curie-Weiss model stop?

We study increasing propagation of chaos for the critical Curie- Weiss model (i.e. the mean-field Ising model at critical inverse temperature 1, with no external field). We give a simple way to see that for windows of size k(N) of smaller order than sqrt{N} we still have propagation of chaos (reproving earlier results, see e.g.[1]), while for k(N) of order sqrt{N} the propagation of chaos breaks down. The law of a single spin converges to π, the Bernoulli law with parameter 1/2. If k(N) = alpha sqrt{N}, we give an explicit formula for the limiting distance in total variation of the law of the first k(N) spins with respect to the k-fold product of pi, as a function of alpha. For even larger window sizes, the distribution of the spins has, in the thermodynamical limit, maximal distance to the k-fold product of pi. One of the ingredients of the proof is a result about the unimodality/non- unimodality of the law of the number of positive spins among the first k = k(N) spins, which may be of independent interest.

math.PR↗

arXiv2026

Biased branching random walks on Bienaymé--Galton--Watson trees

We study $λ$-biased branching random walks on Bienaymé--Galton--Watson trees in discrete time. We consider the maximal displacement at time $n$, $\max_{\vert u \vert =n} \vert X(u) \vert$, and show that it almost surely grows at a deterministic, linear speed. We characterize this speed with the help of the large deviation rate function of the $λ$-biased random walk of a single particle. A similar result is given for the minimal displacement at time $n$, $\min_{\vert u \vert =n} \vert X(u) \vert$.

math.PR↗

arXiv2026

Large deviations for sums of multivariate stretched-exponential random variables: the few-big-jumps principle

Large deviations for sums of i.i.d.\ random variables with stretched-exponential tails (also called Weibull or semi-exponential tails) have been well understood since the 60's, going back to Nagaev's seminal work. Many extensions in the $1$-dimensional setting have been developed since then, showing that such deviations are typically governed by a single big jump. In higher dimensions, a corresponding theory has remained largely undeveloped. This work provides such a multivariate extension and establishes large deviation results for sums of i.i.d.\ random vectors in $\mathbb{R}^k$ under fairly general assumptions. Roughly speaking, for some $α\in(0,1)$, the log-probability of one random vector divided by $x$ exceeding a threshold $t$ in all components behaves asymptotically, for large $x$, as $x^α$ times a negative infimum of a function $\mathcal{J}$. We prove large deviation results for sums of i.i.d.\ copies, where the rate function is given by a minimization of at most $k$ summands of $\mathcal{J}$. This establishes a few-big-jumps principle that generalizes the classical $1$-dimensional phenomenon: the deviation is typically realized by \emph{at most} $k$ independent vectors. The results are applied to absolute powers of multivariate Gaussian vectors as well as to various other examples. They also allow us to study random projections of high-dimensional $\ell_p^N$-balls, revealing interesting insights about the appearance of light- and heavy-tailed distributions in high-dimensional geometry.

math.PR↗

arXiv2025

The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$

We consider discrete-time branching random walks with a radially symmetric distribution. Independently of each other individuals generate offspring whose relative locations are given by a copy of a radially symmetric point process $\mathcal{L}$. The number of particles at time $t$ form a supercritical Galton-Watson process. We investigate the maximal distance to the origin of such branching random walks. Conditioned on survival, we show that, under some assumptions on $\mathcal{L}$, it grows in the same way as for branching Brownian motion or a broad class of one-dimensional branching random walks: the first term is linear in time and the second logarithmic. The constants in front of these terms are explicit and depend only on the mean measure of $\mathcal{L}$ and dimension. Our main tool in the proof is a ballot theorem with moving barrier which may be of independent interest.

math.PR↗

arXiv2025

Poisson Representable Processes

Motivated by Alain-Sol Sznitman's interlacement process, we consider the set of $\{0,1\}$-valued processes which can be constructed in an analogous way, namely as a union of sets coming from a Poisson process on a collection of sets. Our main focus is to determine which processes are representable in this way. Some of our results are as follows. (1) All positively associated Markov chains and a large class of renewal processes are so representable. (2) Whether an average of two product measures, with close densities, on $n$ variables, is representable is related to the zeroes of the polylogarithm functions. (3) Using (2), we show that a number of tree-indexed Markov chains as well as the Ising model on $ \mathbb{Z}^d ,$ $d\geq 2,$ for certain parameters are not so representable. (4) The collection of permutation invariant processes that are representable corresponds exactly to the set of infinitely divisible random variables on $[0,\infty]$ via a certain transformation. (5) The supercritical (low temperature) Curie-Weiss model is not representable for large~$n$.

math.PR↗

arXiv2024

The TASEP on Galton-Watson trees

We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are generated at the root. Particles can only jump away from the root, and they jump from $x$ to $y$ at rate $r_{x,y}$ provided $y$ is empty. Starting from the all empty initial condition, we show that the distribution of the configuration at time $t$ converges to an equilibrium. We study the current and give conditions on the transition rates such that the current is of linear order or such that there is zero current, i.e. the particles block each other. A key step, which is of independent interest, is to bound the first generation at which the particle trajectories of the first $n$ particles decouple.

math.PR↗

arXiv2024

Biased random walk on dynamical percolation

We study biased random walks on dynamical percolation on $\mathbb{Z}^d$. We establish a law of large numbers and an invariance principle for the random walk using regeneration times. Moreover, we verify that the Einstein relation holds, and we investigate the speed of the walk as a function of the bias. While for $d=1$ the speed is increasing, we show that in general this fails in dimension $d \geq 2$. As our main result, we establish two regimes of parameters, separated by an explicit critical curve, such that the speed is either eventually strictly increasing or eventually strictly decreasing. This is in sharp contrast to the biased random walk on a static supercritical percolation cluster, where the speed is known to be eventually zero.

math.PR↗

arXiv2024

The averaging process on infinite graphs

We consider the averaging process on an infinite connected graph with bounded degree and independent, identically distributed starting values or initial opinions. Assuming that the law of the initial opinion of a vertex has a finite second moment, we show that the opinions of all vertices converge in $L^2$ to the first moment of the law of the initial opinions. A key tool in the proof is the Sharing a drink procedure introduced by Olle Häggström.

math.PR↗

arXiv2024

Branching random walk and log-slowly varying tails

We study a branching random walk with independent and identically distributed, heavy tailed displacements. The offspring law is supercritical and satisfies the Kesten-Stigum condition. We treat the case when the law of the displacements does not lie in the max-domain of attraction of an extreme value distribution. Hence, the classical extreme value theory, which is often deployed in this kind of models, breaks down. We show that if the tails of the displacements are such that the absolute value of the logarithm of the tail is a slowly varying function, one can still effectively analyse the extremes of the process. More precisely, after a non-linear transformation the extremes of the branching random walk process converge to a cluster Cox process.

math.PR↗

arXiv2024

Survival and complete convergence for a branching annihilating random walk

We study a discrete-time branching annihilating random walk (BARW) on the $d$-dimensional lattice. Each particle produces a Poissonian number of offspring with mean $μ$ which independently move to a uniformly chosen site within a fixed distance $R$ from their parent's position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. We prove that for any $μ>1$ the process survives when $R$ is sufficiently large. For fixed $R$ we show that the process dies out if $μ$ is too small or too large. Furthermore, we exhibit an interval of $μ$-values for which the process survives and possesses a unique non-trivial ergodic equilibrium for $R$ sufficiently large. We also prove complete convergence for that case.

math.PR↗

arXiv2024

Random planar trees and the Jacobian conjecture

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem involving shuffling subtrees of $d$-Catalan trees, i.e. planar $d$-ary trees. We also show that, if one can construct a certain Markov chain on large $d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we use the local limit theory of large random trees to show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high degree terms.

math.CO↗

arXiv2024

The extremal point process for branching random walk with stretched exponential displacements

We investigate a branching random walk where the displacements are independent from the branching mechanism and have a stretched exponential distribution. We describe the positions of the particles in the vicinity of the rightmost particle in terms of point process convergence. As a consequence we give a~new limit theorem for the position of the rightmost particle. Our methods rely on providing precise large deviations for sums of i.i.d. random variables with stretched exponential distributions outside the so-called one big jump regime.

math.PR↗

arXiv2023

Interacting Edge-Reinforced Random Walks

We consider the edge-reinforced random walk with multiple (but finitely many) walkers which influence the edge weights together. The walker which moves at a given time step is chosen uniformly at random, or according to a fixed order. First, we consider 2 walkers with linear reinforcement on a line graph comprising three nodes. We show that the edge weights evolve similarly to the setting with a single walker which corresponds to a Pólya urn. In particular, the left edge weight proportion is a martingale at certain stopping times, showing that a (random) limiting proportion exists. We then look at an arbitrary number of walkers on Z with very general reinforcement. We show that in this case, the behaviour is also the same as for a single walker: either all walkers are recurrent or all walkers have finite range. In the particular case of reinforcements of "sequence type", we give a criterion for recurrence.

math.PR↗

arXiv2023

Biased Random Walk on Spanning Trees of the Ladder Graph

We consider a specific random graph which serves as a disordered medium for a particle performing biased random walk. Take a two-sided infinite horizontal ladder and pick a random spanning tree with a certain edge weight $c$ for the (vertical) rungs. Now take a random walk on that spanning tree with a bias $β>1$ to the right. In contrast to other random graphs considered in the literature (random percolation clusters, Galton-Watson trees) this one allows for an explicit analysis based on a decomposition of the graph into independent pieces. We give an explicit formula for the speed of the biased random walk as a function of both the bias $β$ and the edge weight $c$. We conclude that the speed is a continuous, unimodal function of $β$ that is positive if and only if $β< β_c^{(1)}$ for an explicit critical value $β_c^{(1)}$ depending on $c$. In particular, the phase transition at $β_c^{(1)}$ is of second order. We show that another second order phase transition takes place at another critical value $β_c^{(2)}<β_c^{(1)}$ that is also explicitly known: For $β<β_c^{(2)}$ the times the walker spends in traps have second moments and (after subtracting the linear speed) the position fulfills a central limit theorem. We see that $β_c^{(2)}$ is smaller than the value of $β$ which achieves the maximal value of the speed. Finally, concerning linear response, we confirm the Einstein relation for the unbiased model ($β=1$) by proving a central limit theorem and computing the variance.

math.PR↗

arXiv2022

The tail of the length of an excursion in a trap of random size

Consider a random walk with a drift to the right on $\{0,\ldots,k\}$ where $k$ is random and geometrically distributed. We show that the tail $P[T>t]$ of the length $T$ of an excursion from $0$ decreases up to constants like $t^{-\varrho}$ for some $\varrho>0$ but is not regularly varying. We compute the oscillations of $t^\varrho\,P[T>t]$ as $t\to\infty$ explicitly.

math.PR↗

arXiv2022

Mixing times for the simple exclusion process with open boundaries

We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the process exhibits pre-cutoff and in some cases cutoff. Our main contribution is to study mixing times for the asymmetric simple exclusion process with open boundaries. We show that the order of the mixing time can be linear or exponential in the size of the segment depending on the choice of the boundary parameters, proving a strikingly different (and richer) behavior for the simple exclusion process with open boundaries than for the process on the closed segment. Our arguments combine coupling, second class particle and censoring techniques with current estimates. A novel idea is the use of multi-species particle arguments, where the particles only obey a partial ordering.

math.PR↗

arXiv2022

The maximum of a branching random walk with stretched exponential tails

We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $\mathbb{P}[X \geq t] \sim a \exp(-λt^r)$ for some constants $a, λ> 0$ where $r \in (0,1)$. We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost-sure limit theorems, convergence in law and some integral tests. The limit theorems reveal interesting differences betweens the two regimes $ r \in (0, 2/3)$ and $ r \in (2/3, 1)$, with yet different limits in the boundary case $r = 2/3$.

math.PR↗

arXiv2021

Cutoff on trees is rare

We study the simple random walk on trees and give estimates on the mixing and relaxation time. Relying on a recent characterization by Basu, Hermon and Peres, we give geometric criteria, which are easy to verify and allow to determine whether the cutoff phenomenon occurs. We thoroughly discuss families of trees with cutoff, and show how our criteria can be used to prove the absence of cutoff for several classes of trees, including spherically symmetric trees, Galton-Watson trees of a fixed height, and sequences of random trees converging to the Brownian CRT.

math.PR↗