arXiv subjects

Lorenzo Taggi

Publications and source records attributed to Lorenzo Taggi.

At least 19 recordsLinked to original sources

arXiv2026

Dimer model and random lattice permutations with general weights

The dimer model and random lattice permutations are two fundamental objects at the interface of probability, combinatorics, and mathematical physics. We study these models on finite periodic boxes in $\mathbb{Z}^d$ within a common framework. For the dimer model, edges connecting arbitrary vertices carry a weight which depends on their relative displacement, and dimer configurations are weighted through their occupied edges. Superimposing two independent perfect matchings gives the double-dimer model, whose configurations are collections of disjoint loops. Permutations, instead, are weighted through the spatial displacement of their jumps. For broad classes of weights of finite or infinite range we prove long-range order and the occurrence of macroscopic loops. This extends nearest-neighbour results to arbitrary-range edges and jumps. In particular, long-range weights yield long-range order and macroscopic loops already in dimensions $d=1,2$. In dimension two, this behaviour is qualitatively different from that of the nearest-neighbour model. We complement these results with sharp absence criteria.

math.PR↗

arXiv2026

The number of particles in activated random walk on the complete graph

We consider an elementary model for self-organised criticality, the activated random walk on the complete graph. We introduce a discrete time Markov chain as follows. At each time step, we add an active particle at a random vertex and let the system stabilise following the activated random walk dynamics, obtaining a particle configuration with all sleeping particles. Particles visiting a boundary vertex are removed from the system. We characterise the support of the stationary distribution of this Markov chain, showing that, with high probability, the number of particles concentrates around the value $ρ_c N + a \sqrt{N \log N}$ with fluctuations of order at most $o( \sqrt{N \log N })$, where $N$ is the number of vertices. Due to the mean-field nature of the model, we are able to determine precisely the critical density $ρ_c= \fracλ{1+λ}$, where $λ$ is the sleeping rate, as well as the constant $a = \sqrtλ / (1 + λ) $ characterising the lower order shift. Our approach utilises results about super-martingales associated with activated random walk that are of independent interest.

math.PR↗

arXiv2025

Mermin-Wagner theorem for dimers, monomer double-dimers, and spatial random permutations

We study a generalisation of the double-dimer model that encompasses several models of interest, including the monomer double-dimer model, spatial random permutations, the dimer model, and the spin $O(N)$ model, and which is also related to the loop $O(N)$ model. We show that on two-dimensional-like graphs (such as slabs), both the correlation function and the probability that a loop visits two vertices decay to zero as the distance between the vertices diverges. Our approach is based on the introduction of a new complex spin representation for all models in this class, together with a new proof of the Mermin-Wagner theorem that does not require positivity of the Gibbs measure. Even for the well-studied dimer and double-dimer models our results are new: since they do not rely on exact solvability or Kasteleyn's theorem, they apply beyond the planar-graph setting.

math.PR↗

arXiv2025

Internal Diffusion Limited Aggregation with Critical Branching Random Walks

Internal Diffusion Limited Aggregation is an interacting particle system that describes the growth of a random cluster governed by the boundary harmonic measure seen from an internal point. Our paper studies IDLA in $\mathbb{Z}^d$ driven by critical branching random walks. We prove that, unlike classical IDLA, this process exhibits a phase transition in the dimension. More precisely, we establish the existence of a spherical shape theorem in dimension $d\geq 3$ and the absence of a spherical shape theorem for $d \leq 2$. Our bounds on the inner and outer worst deviations are of polynomial nature, which we expect to be a feature of this model.

math.PR↗

arXiv2025

The critical density of the Stochastic Sandpile Model

We study the stochastic sandpile model on $\mathbb{Z}^d$ and demonstrate that the critical density is strictly less than one in all dimensions. This generalizes a previous result by Hoffman, Hu, Richey, and Rizzolo (2022), which was limited to the one-dimensional case. In addition, we show that the critical density is strictly positive on any vertex-transitive graph, extending the earlier result of Sidoravicius and Teixeira (2018) and providing a simpler proof.

math.PR↗

arXiv2024

Coexistence, enhancements and short loops in random walk loop soups

We consider a general random walk loop soup which includes, or is related to, several models of interest, such as the Spin O(N) model, the double dimer model and the Bose gas. The analysis of this model is challenging because of the presence of spatial interactions between the loops. For this model it is known from Quitmann and Taggi (2022) that macroscopic loops occur in dimension three and higher when the inverse temperature is large enough. Our first result is that, on the $d$ dimensional lattice, the presence of repulsive interactions is responsible for a shift of the critical inverse temperature, which is strictly greater than $\frac{1}{2d}$, the critical value in the non interacting case. Our second result is that a positive density of microscopic loops exists for all values of the inverse temperature. This implies that, in the regime in which macroscopic loops are present, microscopic and macroscopic loops coexist. Moreover, we show that, even though the increase of the inverse temperature leads to an increase of the total loop length, the density of microscopic loops is uniformly bounded from above in the inverse temperature. Our last result is confined to the special case in which the random walk loop soup is the one associated to the Spin O(N) model with arbitrary integer values of $N \geq 2$ and states that, on $\mathbb{Z}^2$, the probability that two vertices are connected by a loop decays at least polynomially fast with their distance.

math.PR↗

arXiv2023

Essential enhancements in Abelian networks: continuity and uniform strict monotonicity

We prove that in wide generality the critical curve of the activated random walk model is a continuous function of the deactivation rate, and we provide a bound on its slope which is uniform with respect to the choice of the graph. Moreover, we derive strict monotonicity properties for the probability of a wide class of `increasing' events,extending previous results of Rolla and Sidoravicius (2012). Our proof method is of independent interest and can be viewed as a reformulation of the `essential enhancements' technique -- which was introduced for percolation -- in the framework of Abelian networks.

math.PR↗

arXiv2023

Macroscopic loops in the $3d$ double-dimer model

The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of self-avoiding loops. Our first result is that in $\mathbb{Z}^d$, $d>2$, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from Betz and Taggi (2019) and Taggi (2021) that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is macroscopic. Our paper shows that, when $d >2$, such a critical threshold is strictly positive. In other words, the self-avoiding walk is macroscopic even in the presence of a positive density of monomers.

math-ph↗

arXiv2023

Prudent walk in dimension six and higher

We study the high-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. We prove that the prudent self-avoiding walk converges to Brownian motion under diffusive scaling if the dimension is large enough. The same result is true for weakly prudent walk in dimension d>5. A challenging property of the high-dimensional prudent walk is the presence of an infinite-range self-avoidance constraint. Interestingly, as a consequence of such a strong self-avoidance constraint, the upper critical dimension of the prudent walk is five, and thus greater than for the classical self-avoiding walk.

math.PR↗

arXiv2023

Macroscopic loops in the Bose gas, Spin O(N) and related models

We consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model, random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in $\mathbb{Z}^d$, $d \geq 3$, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate $\mathbb{Z}^d$ by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hard-core constraints.

math.PR↗

arXiv2021

Active phase for activated random walks on $\mathbb{Z}^d$, $ d \geq 3$, with density less than one and arbitrary sleeping rate

It has been conjectured that the critical density of the Activated Random Walk model is strictly less than one for any value of the sleeping rate. We prove this conjecture on $\mathbb{Z}^d$ when $d \geq 3$ and, more generally, on graphs where the random walk is transient. Moreover, we establish the occurrence of a phase transition on non-amenable graphs, extending previous results which require that the graph is amenable or a regular tree.

math.PR↗

arXiv2021

Exponential decay of transverse correlations for O(N) spin systems and related models

We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N > 1. Our result is new when N > 3, in which case no Lee-Yang theorem is available, it is an alternative to Lee-Yang when N = 2, 3, and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a `colour-switch' lemma, and a sampling procedure which allows us to bound from above the `typical' length of the open paths.

math.PR↗

arXiv2020

Site-monotonicity properties for reflection positive measures with applications to quantum spin systems

We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application of such a general theorem, we derive site-monotonicity properties for the spin-spin correlation of the quantum Heisenberg antiferromagnet and $XY$ model, we prove that such spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates -- improving previous positivity results which hold for the Cesàro sum -- and we derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model, lattice permutations, thus extending the previous results of \textit{Lees and Taggi (2019)}.

math.PR↗

arXiv2019

Uniformly positive correlations in the dimer model and phase transition in lattice permutations on $\mathbb{Z}^d$, $d > 2$, via reflection positivity

Our first main result is that correlations between monomers in the dimer model in $\mathbb{Z}^d$ do not decay to zero when $d > 2$. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully-packed loops, such a walk is `long' and the distance between its end-points grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some `virtual' vertices.

math.PR↗

arXiv2019

Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N

We provide a uniformly-positive point-wise lower bound for the two-point function of the classical spin $O(N)$ model on the torus of $\mathbb{Z}^d$, $d \geq 3$, when $N \in \mathbb{N}_{>0}$ and the inverse temperature $β$ is large enough. This is a new result when $N>2$ and extends the classical result of Fröhlich, Simon and Spencer (1976). Our bound follows from a new site-monotonicity property of the two-point function which is of independent interest and holds not only for the spin $O(N)$ model with arbitrary $N \in \mathbb{N}_{>0}$, but for a wide class of systems of interacting random walks and loops, including the loop $O(N)$ model, random lattice permutations, the dimer model, the double dimer model, and the loop representation of the classical spin $O(N)$ model.

math.PR↗

arXiv2019

Interacting self-avoiding polygons

We consider a system of self-avoiding polygons interacting through a potential that penalizes or rewards the number of mutual touchings and we provide an exact computation of the critical curve separating a regime of long polygons from a regime of localized polygons. Moreover, we prove the existence of a sub-region of the phase diagram where the self-avoiding polygons are space filling and we provide a non-trivial characterization of the regime where the polygon length admits uniformly bounded exponential moments.

math.PR↗

arXiv2019

Abelian oil and water dynamics does not have an absorbing-state phase transition

The oil and water model is an interacting particle system with two types of particles and a dynamics that conserves the number of particles, which belongs to the so-called class of Abelian networks. Widely studied processes in this class are sandpiles models and activated random walks, which are known (at least for some choice of the underlying graph) to undergo an absorbing-state phase transition. This phase transition characterizes the existence of two regimes, depending on the particle density: a regime of fixation at low densities, where the dynamics converges towards an absorbing state and each particle jumps only finitely many times, and a regime of activity at large densities, where particles jump infinitely often and activity is sustained indefinitely. In this work we show that the oil and water model is substantially different than sandpiles models and activated random walks, in the sense that it does not undergo an absorbing-state phase transition and is in the regime of fixation at all densities. Our result works in great generality: for any graph that is vertex transitive and for a large class of initial configurations.

math.PR↗

arXiv2018

Shifted critical threshold in the loop $O(n)$ model at arbitrary small $n$

In the loop $O(n)$ model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional to $${λ^{\# \mbox{edges}} n^{\# \mbox{loops}},}$$ where $λ, n \in [0, \infty)$. Let $μ$ be the connective constant of the lattice and, for any $n \in [0, \infty)$, let $λ_c(n)$ be the largest value of $λ$ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that $λ_c(n) =1/μ$ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and that for any $n \geq 0$, $λ_c(n) \geq 1/μ$. In this note we prove that, \begin{align*} λ_c(n) & > 1/μ\, \, \, \, \, \, \, \, \, \, \, \mbox{whenever $n >0$}, \\ λ_c(n) & \geq 1/μ\, + \, c_0 \, n \, + \, O(n^2), \end{align*} on $\mathbb{Z}^d$, with $d \geq 2$, and on the hexagonal lattice, where $c_0>0$. This means that, when $n$ is positive (even arbitrarily small), as a consequence of the mutual repulsion between the loops, a phase transition can only occur at a strictly larger critical threshold than in the self-avoiding walk.

math.PR↗