arXiv subjects

Tuan-Minh Nguyen

Publications and source records attributed to Tuan-Minh Nguyen.

12 recordsLinked to original sources

arXiv2026

Recurrence and transience of random walks with drift $ρx^α/t^β$

Menshikov and Volkov [Electron. J. Probab. 13 (2008)] studied recurrence and transience of a class of Markovian random walks on $\mathbb R_+$ whose conditional drift depends on both time and position and is of order $ρx^αt^{-β}$ with $ρ>0$. The case on the critical line $2β-α=1$, with $α\in(-1,1)\setminus\{0\}$, remained open. We prove recurrence in this remaining case. Furthermore, we establish recurrence and transience criteria that complete the classification for $-1<α<1$ and $β\ge 0$, without assuming the Markov property and under weaker assumptions on the increments than those imposed by Menshikov and Volkov.

math.PR↗

arXiv2026

Superdiffusivity of random walks on the three-dimensional randomly oriented Manhattan lattice

We study the superdiffusive behavior of random walks on the randomly oriented Manhattan lattice, i.e., the $d$-dimensional integer lattice $\mathbb{Z}^d$ where each axis-aligned line is independently assigned a random direction (forward or backward) with equal probability. The walker takes nearest-neighbor steps, choosing an axis randomly and moving along the assigned direction of that axis's line, with equal probabilities for each axis. We show that, in the critical dimension $d=3$, the diffusion coefficient of the random walk diverges in the Tauberian sense as $\sqrt{\log t}$ with a multiplicative correction $(\log\log t)^{\pm(2+\varepsilon)}$ as time $t\to\infty$. This gives an answer to a conjecture by Ledger, Tóth and Valkó (2018).

math.PR↗

arXiv2026

"True" self-avoiding walks on general trees

We study the asymptotic behavior of ``true" self-avoiding random walks on general infinite locally finite trees. In this model, the walk starts at the root and, at each step, from its current vertex chooses a neighboring edge to traverse with probability proportional to the current weight of that edge, where the weight of each edge after being traversed $n$ times is given by $w(n)=\exp(-βn)$. We show that the process exhibits a sharp phase transition between recurrence and transience. The critical value is determined by the branching-ruin number of the tree, which coincides with the Hausdorff dimension of the boundary of the tree under a suitable metric. We prove that the walk is almost surely transient when the branching-ruin number is greater than $1/2$, and recurrent when it is less than $1/2$. This resolves an open question posed by Kosygina.

math.PR↗

arXiv2026

Limit theorems for random walks with spatio-temporal drift

We study a class of discrete-time random walks in $\mathbb{R}^d$ whose conditional drift decays polynomially in time and grows polynomially with the distance from the origin to the current position. This class is related to several models of self-interacting random processes. We determine the asymptotic behavior of the walk under the assumption that its increments have moments of order $p$ for some $p>2$. In the linear case, where the drift depends linearly on the current position, we establish a phase transition in the convergence in distribution of the normalized process to Gaussian limits. In the nonlinear case, we identify three distinct regimes separated by a critical line and show that the normalized process exhibits qualitatively different behaviors in each regime, including convergence in distribution to a Gaussian law, convergence to a non-Gaussian limit given by the stationary distribution of a stochastic differential equation, and almost sure localization on a hypersphere.

math.PR↗

arXiv2026

Once-excited random walks on general trees

We study once-excited random walks on general trees, modeled by placing a single "cookie" at each vertex. Each cookie acts as a metaphorical reward that is consumed upon the first visit to the vertex where the cookie is placed. On that initial visit, the walk is in an excited state and behaves like a biased random walk. Once the cookie is consumed, the process reverts to a symmetric random walk on all subsequent visits. We consider a random environment in which the bias parameters are independent random variables. We prove that the process exhibits a sharp phase transition between transience and recurrence on general trees with polynomial growth, where the critical threshold is determined by the branching-ruin number of the tree.

math.PR↗

arXiv2024

Strongly vertex-reinforced jump process on graphs with bounded degree

We study asymptotic behaviours of a non-linear vertex-reinforced jump process defined on an arbitrary infinite graph with bounded degree. We prove that if the reinforcement function $w$ is reciprocally integrable and non-decreasing, then the process visits only a finite number of vertices. In the case where $w$ is approximately equal to a super-linear polynomial, we show that the process eventually gets stuck on a star-shaped subgraph and there is exactly one vertex with unbounded local time.

math.PR↗

arXiv2024

Finding pure Nash equilibria in large random games

Best Response Dynamics (BRD) is a class of strategy updating rules to find Pure Nash Equilibria (PNE) in a game. At each step, a player is randomly picked, and the player switches to a "best response" strategy based on the strategies chosen by others, so that the new strategy profile maximises their payoff. If no such strategy exists, a different player will be chosen randomly. When no player wants to change their strategy anymore, the process reaches a PNE and will not deviate from it. On the other hand, either PNE may not exist, or BRD could be "trapped" within a subgame that has no PNE. We consider a random game with $N$ players, each with two actions available, and i.i.d. payoffs, in which the payoff distribution may have an atom, i.e. ties are allowed. We study a class of random walks in a random medium on the $N$-dimensional hypercube induced by the random game. The medium contains two types of obstacles corresponding to PNE and traps. The class of processes we analyze includes BRD, simple random walks on the hypercube, and many other nearest neighbour processes. We prove that, with high probability, these processes reach a PNE before hitting any trap.

math.PR↗

arXiv2022

Speed of excited random walks with long backward steps

We study a model of multi-excited random walk with non-nearest neighbour steps on $\mathbb Z$, in which the walk can jump from a vertex $x$ to either $x+1$ or $x-i$ with $i\in \{1,2,\dots,L\}$, $L\ge 1$. We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton-Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh [Probab. Theory Related Fields (2008), 141 (3-4)], we extend their result (w.r.t the case $L=1$) to our model. More specifically, we show that in the regime of transience to the right, the walk has positive speed if and only if the expected total drift $δ>2$. This confirms a special case of a conjecture proposed by Davis and Peterson.

math.PR↗

arXiv2021

Long range one-cookie random walk with positive speed

We study one-dimensional excited random walks with non-nearest neighbor jumps. When the process is at a vertex that has not been visited before, its next transition has a positive drift to the right, possibly with long jumps. Whenever the process visits a vertex that has already been visited in the past, its next transition is the one of a simple symmetric random walk. We give a sufficient condition for the process to have positive speed.

math.PR↗

arXiv2021

Vertex-reinforced jump process on the integers with nonlinear reinforcement

We consider a non-linear vertex-reinforced jump process (VRJP($w$)) on $\mathbb{Z}$ with an increasing measurable weight function $w:[1,\infty)\to [1,\infty)$ and initial weights equal to one. Our main goal is to study the asymptotic behaviour of VRJP($w$) depending on the integrability of the reciprocal of $w$. In particular, we prove that if $\int_1^{\infty} \frac{\text{d}u}{w(u)} =\infty$ then the process is recurrent, i.e. it visits each vertex infinitely often and all local times are unbounded. On the other hand, if $\int_1^{\infty} \frac{\text{d} u}{w(u)} <\infty$ and there exists a $ρ>0$ such that $t \mapsto w(t)^ρ\int_t^{\infty}\frac{\text{d}u}{w(u)}$ is non-increasing then the process will eventually get stuck on exactly three vertices, and there is only one vertex with unbounded local time. We also show that if the initial weights are all the same, VRJP on $\mathbb{Z}$ cannot be transient, i.e. there exists at least one vertex that is visited infinitely often. Our results extend the ones previously obtained by Davis and Volkov [Probab. Theory Relat. Fields (2002)] who showed that VRJP with linear reinforcement on $\mathbb{Z}$ is recurrent.

math.PR↗

arXiv2021

Strongly vertex-reinforced jump process on a complete graph

The aim of our work is to study vertex-reinforced jump processes with super-linear weight function $w(t) = t^α$ , for some $α>1$. On any complete graph $G = (V, E)$, we prove that there is one vertex $v \in V$ such that the total time spent at $v$ almost surely tends to infinity while the total time spent at the remaining vertices is bounded.

math.PR↗

arXiv2019

On a class of random walks in simplexes

We study the limit behaviour of a class of random walk models taking values in the $d$-dimensional unit standard simplex, $d\ge 1$, defined as follows. From an interior point $z$, the process chooses one of the $d+1$ vertices of the simplex, with probabilities depending on $z$, and then the particle randomly jumps to a new location $z'$ on the segment connecting $z$ to the chosen vertex. In some specific cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are, in fact, Dirichlet. We also consider a related history-dependent random walk model in $[0,1]$ based on an urn-type scheme. We show that this random walk converges in distribution to the arcsine law.

math.PR↗