In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius

We consider first-passage percolation with i.i.d. non-negative weights coming from some continuous distribution under a moment condition. We review recent results in the study of geodesics in first-passage percolation and study their implications for the multi-type Richardson model. In two dimensions this establishes a dual relation between the existence of infinite geodesics and coexistence among competing types. The argument amounts to making precise the heuristic that infinite geodesics can be thought of as ‘highways to infinity’. We explain the limitations of the current techniques by presenting a partial result in dimensions d > 2.

An important but little-studied property of spin glasses is the stability of their ground states to changes in one or a finite number of couplings. It was shown in earlier work that, if multiple ground states are assumed to exist, then fluctuations in their energy differences—and therefore the possibility of multiple ground states—are closely related to the stability of their ground states. Here we examine the stability of ground states in two models, one of which is presumed to have a ground state structure that is qualitatively similar to other realistic short-range spin glasses in finite dimensions.

For different reversible Markov kernels on finite state spaces, we look for families of probability measures for which the time evolution almost remains in their convex hull. Motivated by signal processing problems and metastability studies we are interested in the case when the size of such families is smaller than the size of the state space, and we want such distributions to be with “small overlap” among them. To this aim we introduce a squeezing function to measure the common overlap of such families, and we use random forests to build random approximate solutions of the associated intertwining equations for which we can bound from above the expected values of both squeezing and total variation errors. We also explain how to modify some of these approximate solutions into exact solutions by using those eigenvalues of the associated Laplacian with the largest size.

We study a dependent site percolation model on the n-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.

For directed last passage percolation on ℤ 2 $$\mathbb {Z}^2$$ with exponential passage times on the vertices, let T n denote the last passage time from (0, 0) to (n, n). We consider asymptotic two point correlation functions of the sequence T n. In particular we consider Corr(T n, T r) for r ≤ n where r, n →∞ with r ≪ n or n − r ≪ n. Establishing a conjecture from Ferrari and Spohn (SIGMA 12:074, 2016), we show that in the former case Corr ( T n , T r ) = Θ ( ( r n ) 1 ∕ 3 ) $$\mathrm {Corr}(T_{n}, T_{r})=\varTheta ((\frac {r}{n})^{1/3})$$ whereas in the latter case 1 − Corr ( T n , T r ) = Θ ( ( n − r n ) 2 ∕ 3 ) $$1-{\mathrm {Corr}}(T_{n}, T_{r})=\varTheta ((\frac {n-r}{n})^{2/3})$$ . The argument revolves around finer understanding of polymer geometry and is expected to go through for a larger class of integrable models of last passage percolation. As a by-product of the proof, we also get quantitative estimates for locally Brownian nature of pre-limits of Airy2 process coming from exponential LPP, a result of independent interest.

For two-dimensional percolation at criticality, we discuss the inequality α 4 > 1 for the polychromatic four-arm exponent (and stronger versions, the strongest so far being α 4 ≥ 1 + α 2 2 $$\alpha _4 \geq 1 + \frac {\alpha _2}{2}$$ , where α 2 denotes the two-arm exponent). We first briefly discuss five proofs (some of them implicit and not self-contained) from the literature. Then we observe that, by combining two of them, one gets a completely self-contained (and yet quite short) proof.

A noise reinforced Brownian motion is a centered Gaussian process B ̂ = ( B ̂ ( t ) ) t ≥ 0 $$\hat B=(\hat B(t))_{t\geq 0}$$ with covariance ?? ( B ̂ ( t ) B ̂ ( s ) ) = ( 1 − 2 p ) − 1 t p s 1 − p for 0 ≤ s ≤ t , $$\displaystyle \mathbb {E}(\hat B(t)\hat B(s))=(1-2p)^{-1}t^ps^{1-p} \quad \text{for} \quad 0\leq s \leq t, $$ where p ∈ (0, 1∕2) is a reinforcement parameter. Our main purpose is to establish a version of Donsker’s invariance principle for a large family of step-reinforced random walks in the diffusive regime, and more specifically, to show that B ̂ $$\hat B$$ arises as the universal scaling limit of the former. This extends known results on the asymptotic behavior of the so-called elephant random walk.

We study first passage percolation on the plane for a family of invariant, ergodic measures on ℤ 2 $$\mathbb {Z}^2$$ . We prove that for all of these models the asymptotic shape is the ℓ 1 ball and that there are exactly four infinite geodesics starting at the origin a.s. In addition we determine the exponents for the variance and wandering of finite geodesics. We show that the variance and wandering exponents do not satisfy the relationship of χ = 2ξ − 1 which is expected for independent first passage percolation.

We consider Activated Random Walks on ℤ $$\mathbb {Z}$$ with totally asymmetric jumps and critical particle density, with different time scales for the progressive release of particles and the dissipation dynamics. We show that the cumulative flow of particles through the origin rescales to a pure-jump self-similar process which we describe explicitly.

The balanced excited random walk, introduced by Benjamini, Kozma and Schapira in 2011, is defined as a discrete time stochastic process in ℤ d $$\mathbb Z^d$$ , depending on two integer parameters 1 ≤ d 1, d 2 ≤ d, which whenever it is at a site x ∈ ℤ d $$x\in \mathbb Z^d$$ at time n, it jumps to x ± e i with uniform probability, where e 1, …, e d are the canonical vectors, for 1 ≤ i ≤ d 1, if the site x was visited for the first time at time n, while it jumps to x ± e i with uniform probability, for 1 + d − d 2 ≤ i ≤ d, if the site x was already visited before time n. Here we give an overview of this model when d 1 + d 2 = d and introduce and study the cases when d 1 + d 2 > d. In particular, we prove that for all the cases d ≥ 5 and most cases d = 4, the balanced excited random walk is transient.

This article deals with limit theorems for certain loop variables for loop soups whose intensity approaches infinity. We first consider random walk loop soups on finite graphs and obtain a central limit theorem when the loop variable is the sum over all loops of the integral of each loop against a given one-form on the graph. An extension of this result to the noncommutative case of loop holonomies is also discussed. As an application of the first result, we derive a central limit theorem for windings of loops around the faces of a planar graph. More precisely, we show that the winding field generated by a random walk loop soup, when appropriately normalized, has a Gaussian limit as the loop soup intensity tends to ∞, and we give an explicit formula for the covariance kernel of the limiting field. We also derive a Spitzer-type law for windings of the Brownian loop soup, i.e., we show that the total winding around a point of all loops of diameter larger than δ, when multiplied by 1 ∕ log δ $$1/\log \delta $$ , converges in distribution to a Cauchy random variable as δ → 0. The random variables analyzed in this work have various interpretations, which we highlight throughout the paper.

The Derrida–Retaux recursive system was investigated by Derrida and Retaux (J Stat Phys 156:268–290, 2014) as a hierarchical renormalization model in statistical physics. A prediction of Derrida and Retaux (J Stat Phys 156:268–290, 2014) on the free energy has recently been rigorously proved (Chen et al., The Derrida–Retaux conjecture on recursive models. https://arxiv.org/abs/1907.01601 ), confirming the Berezinskii–Kosterlitz–Thouless-type phase transition in the system. Interestingly, it has been established in the paper by Chen et al. that the prediction is valid only under a certain integrability assumption on the initial distribution, and a new type of universality result has been shown when this integrability assumption is not satisfied. We present a unified approach for systems satisfying a certain domination condition, and give an upper bound for derivatives of all orders of the moment generating function. When the integrability assumption is not satisfied, our result allows to identify the large-time order of magnitude of the product of the moment generating functions at criticality, confirming and completing a previous result in Collet et al. (Commun Math Phys 94:353–370, 1984).

We introduce a general class of time inhomogeneous random walks on the N-hypercube. These random walks are reversible with an N-product Bernoulli stationary distribution and have a property of local change of coordinates in a transition. Several types of representations for the transition probabilities are found. The paper studies cut-off for the mixing time. We observe that for a sub-class of these processes with long range (i.e. non-local) there exists a critical value of the range that allows an almost-perfect mixing in at most two steps. That is, the total variation distance between the two steps transition and stationary distributions decreases to zero as the dimension of the hypercube N increases. Notice that a well-known result (Theorem 1 in [6]) shows that there does not exist a random walk on Abelian groups (such as the hypercube) which mixes perfectly in exactly two steps.

We study the critical case of first-passage percolation in two dimensions. Letting (t e) be i.i.d. nonnegative weights assigned to the edges of ℤ 2 $$\mathbb {Z}^2$$ with ℙ ( t e = 0 ) = 1 ∕ 2 $$\mathbb {P}(t_e=0)=1/2$$ , consider the induced pseudometric (passage time) T(x, y) for vertices x, y. It was shown in [4] that the growth of the sequence ?? T ( 0 , ∂ B ( n ) ) $$\mathbb {E}T(0,\partial B(n))$$ (where B(n) = [−n, n]2) has the same order (up to a constant factor) as the sequence ?? T inv ( 0 , ∂ B ( n ) ) $$\mathbb {E}T^{\text{inv}}(0,\partial B(n))$$ . This second passage time is the minimal total weight of any path from 0 to ∂B(n) that resides in a certain embedded invasion percolation cluster. In this paper, we show that this constant factor cannot be taken to be 1. That is, there exists c > 0 such that for all n, ?? T inv ( 0 , ∂ B ( n ) ) ≥ ( 1 + c ) ?? T ( 0 , ∂ B ( n ) ) . $$\displaystyle \mathbb {E}T^{\text{inv}}(0,\partial B(n)) \geq (1+c) \mathbb {E}T(0,\partial B(n)). $$ This result implies that the time constant for the model is different than that for the related invasion model, and that geodesics in the two models have different structure.

We study the spectrum of a random multigraph with a degree sequence D n = ( D i ) i = 1 n $${\mathbf {D}}_n=(D_i)_{i=1}^n$$ and average degree 1 ≪ ω n ≪ n, generated by the configuration model, and also the spectrum of the analogous random simple graph. We show that, when the empirical spectral distribution (ESD) of ω n − 1 D n $$\omega _n^{-1} {\mathbf {D}}_n $$ converges weakly to a limit ν, under mild moment assumptions (e.g., D i∕ω n are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to ν ⊠ σ SC $$\nu \boxtimes \sigma _{{\text{SC}}}$$ , the free multiplicative convolution of ν with the semicircle law. Relating this limit with a variant of the Marchenko–Pastur law yields the continuity of its density (away from zero), and an effective procedure for determining its support. Our proof of convergence is based on a coupling between the random simple graph and multigraph with the same degrees, which might be of independent interest. We further construct and rely on a coupling of the multigraph to an inhomogeneous Erdős-Rényi graph with the target ESD, using three intermediate random graphs, with a negligible fraction of edges modified in each step.

We study the size of the near-critical window for Bernoulli percolation on ℤ d $${\mathbb {Z}}^d$$ . More precisely, we use a quantitative Grimmett–Marstrand theorem to prove that the correlation length, both below and above criticality, is bounded from above by exp ( C ∕ | p − p c | 2 ) $$\exp (C/|p-p_c|{ }^2)$$ . Improving on this bound would be a further step towards the conjecture that there is no infinite cluster at criticality on ℤ d $${\mathbb {Z}}^d$$ for every d ≥ 2.

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these results.

We establish central limit theorems for the position and velocity of the charged particle in the mechanical particle model introduced by Fontes, Jordão Neves and Sidoravicius (2000).

We consider a random walk on a multidimensional integer lattice with random bounds on local times, conditioned on the event that it hits a high level before its death. We introduce an auxiliary “core” process that has a regenerative structure and plays a key role in our analysis. We obtain a number of representations for the distribution of the random walk in terms of the similar distribution of the “core” process. Based on that, we prove a number of limiting results by letting the high level to tend to infinity. In particular, we generalise results for a simple symmetric one-dimensional random walk obtained earlier in the paper by Benjamini and Berestycki (J Eur Math Soc 12(4):819–854, 2010).

We present a simple model of a random walk with partial memory, which we call the random memory walk. We introduce this model motivated by the belief that it mimics the behavior of the once-reinforced random walk in high dimensions and with small reinforcement. We establish the transience of the random memory walk in dimensions three and higher, and show that its scaling limit is a Brownian motion.

We show that the loop O(n) model on the hexagonal lattice exhibits exponential decay of loop sizes whenever n > 1 and x < 1 3 + ε ( n ) $$x<\tfrac {1}{\sqrt {3}}+\varepsilon (n)$$ , for some suitable choice of ε(n) > 0.It is expected that, for n ≤ 2, the model exhibits a phase transition in terms of x, that separates regimes of polynomial and exponential decay of loop sizes. In this paradigm, our result implies that the phase transition for n ∈ (1, 2] occurs at some critical parameter x c(n) strictly greater than that x c ( 1 ) = 1 ∕ 3 $$x_c(1) = 1/\sqrt {3}$$ . The value of the latter is known since the loop O(1) model on the hexagonal lattice represents the contours of the spin-clusters of the Ising model on the triangular lattice.The proof is based on developing n as 1 + (n − 1) and exploiting the fact that, when x < 1 3 $$x<\tfrac {1}{\sqrt {3}}$$ , the Ising model exhibits exponential decay on any (possibly non simply-connected) domain. The latter follows from the positive association of the FK-Ising representation.

The method of ‘coupling from the past’ permits exact sampling from the invariant distribution of a Markov chain on a finite state space. The coupling is successful whenever the stochastic dynamics are such that there is coalescence of all trajectories. The issue of the coalescence or non-coalescence of trajectories of a finite state space Markov chain is investigated in this note. The notion of the ‘coalescence number’ k(μ) of a Markovian coupling μ is introduced, and results are presented concerning the set K(P) of coalescence numbers of couplings corresponding to a given transition matrix P.

We consider a one-dimensional system of particles, moving at constant velocities chosen independently according to a symmetric distribution on {−1, 0, +1}, and annihilating upon collision—with, in case of triple collision, a uniformly random choice of survivor among the two moving particles. When the system contains infinitely many particles, whose starting locations are given by a renewal process, a phase transition was proved to happen (see Haslegrave et al., Three-speed ballistic annihilation: phase transition and universality, 2018) as the density of static particles crosses the value 1∕4. Remarkably, this critical value, along with certain other statistics, was observed not to depend on the distribution of interdistances. In the present paper, we investigate further this universality by proving a stronger statement about a finite system of particles with fixed, but randomly shuffled, interdistances. We give two proofs, one by an induction allowing explicit computations, and one by a more direct comparison. This result entails a new nontrivial independence property that in particular gives access to the density of surviving static particles at time t in the infinite model. Finally, in the asymmetric case, further similar independence properties are proved to keep holding, including a striking property of gamma distributed interdistances that contrasts with the general behavior.

We investigate the effect of disorder on the Curie-Weiss model with Glauber dynamics. In particular, we study metastability for spin-flip dynamics on the Erdős-Rényi random graph ERn(p) with n vertices and with edge retention probability p ∈ (0, 1). Each vertex carries an Ising spin that can take the values − 1 or + 1. Single spins interact with an external magnetic field h ∈ (0, ∞), while pairs of spins at vertices connected by an edge interact with each other with ferromagnetic interaction strength 1∕n. Spins flip according to a Metropolis dynamics at inverse temperature β. The standard Curie-Weiss model corresponds to the case p = 1, because ERn(1) = K n is the complete graph on n vertices. For β > β c and h ∈ (0, pχ(βp)) the system exhibits metastable behaviour in the limit as n →∞, where β c = 1∕p is the critical inverse temperature and χ is a certain threshold function satisfying limλ→∞ χ(λ) = 1 and limλ↓1 χ(λ) = 0. We compute the average crossover time from the metastable set (with magnetization corresponding to the ‘minus-phase’) to the stable set (with magnetization corresponding to the ‘plus-phase’). We show that the average crossover time grows exponentially fast with n, with an exponent that is the same as for the Curie-Weiss model with external magnetic field h and with ferromagnetic interaction strength p∕n. We show that the correction term to the exponential asymptotics is a multiplicative error term that is at most polynomial in n. For the complete graph K n the correction term is known to be a multiplicative constant. Thus, apparently, ERn(p) is so homogeneous for large n that the effect of the fluctuations in the disorder is small, in the sense that the metastable behaviour is controlled by the average of the disorder. Our model is the first example of a metastable dynamics on a random graph where the correction term is estimated to high precision.

We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model (PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to locally tree-like finite random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model, i.e., uniform simple random graphs with a prescribed degree sequence.

We study a random walk (Markov chain) in an unbounded planar domain bounded by two curves of the form x 2 = a + x 1 β + $$x_2 = a^+ x_1^{\beta ^+}$$ and x 2 = − a − x 1 β − $$x_2 = -a^- x_1^{\beta ^-}$$ , with x 1 ≥ 0. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From the vicinity of the upper and lower sections of the boundary, the walk drifts back into the interior at a given angle α + or α − to the relevant inwards-pointing normal vector. Here we focus on the case where α + and α − are equal but opposite, which includes the case of normal reflection. For 0 ≤ β +, β − < 1, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.

Benjamini et al. (Inst Hautes Études Sci Publ Math 90:5–43, 2001) showed that weighted majority functions of n independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability ??, the probability p ?? that the weighted majority changes is at most C?? 1∕4. They asked what is the best possible exponent that could replace 1∕4. We prove that the answer is 1∕2. The upper bound obtained for p ?? is within a factor of π ∕ 2 + o ( 1 ) $$\sqrt {\pi /2}+o(1)$$ from the known lower bound when ?? → 0 and n?? →∞.

We discuss anisotropic scaling limits of long-range dependent linear random fields X on ℤ 2 $$\mathbb {Z}^2$$ with arbitrary dependence axis (direction in the plane along which the moving-average coefficients decay at a smallest rate). The scaling limits V γ X $$V^X_{\gamma }$$ are random fields on ℝ + 2 $$\mathbb {R}^2_+$$ defined as the limits (in the sense of finite-dimensional distributions) of partial sums of X taken over rectangles with sides increasing along horizontal and vertical directions at rates λ and λ γ respectively as λ →∞ for arbitrary fixed γ > 0. The scaling limits generally depend on γ and constitute a one-dimensional family { V γ X , γ > 0 } $$\{V^X_{\gamma }, \gamma >0\}$$ of random fields. The scaling transition occurs at some γ 0 X > 0 $$\gamma ^X_0 >0$$ if V γ X $$V^X_\gamma $$ are different and do not depend on γ for γ > γ 0 X $$ \gamma > \gamma ^X_0 $$ and γ < γ 0 X $$\gamma < \gamma ^X_0$$ . We prove that the fact of ‘oblique’ dependence axis (or incongruous scaling) dramatically changes the scaling transition in the above model so that γ 0 X = 1 $$\gamma _0^X = 1$$ independently of other parameters, contrasting the results in Pilipauskaitė and Surgailis (2017) on the scaling transition under congruous scaling.

The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process that is conjectured to be at the core of the KPZ universality class. In this article we study two aspects the KPZ fixed point that share the same Brownian limiting behaviour: the local space regularity and the long time evolution. Most of the results that we will present here were obtained by either applying explicit formulas for the transition probabilities or applying the coupling method to discrete approximations. Instead we will use the variational description of the KPZ fixed point, allowing us the possibility of running the process starting from different initial data (basic coupling), to prove directly the aforementioned limiting behaviours.

It is proved that in non-collapse quantum mechanics the state of a closed system can always be expressed as a superposition of states all of which describe histories that conform to Born’s probability rule. This theorem allows one to see Born probabilities in non-collapse quantum mechanics as an appropriate predictive tool, implied by the theory, provided an appropriate version of the superposition principle is included in its axioms

We prove that the one dimensional Multi-Particle Diffusion Limited Aggregation model has linear growth whenever the particle density exceeds 1 answering a question of Kesten and Sidoravicius. As a corollary we prove linear growth in all dimensions d when the particle density is at least 1.

We consider random interlacements on ℤ d $${\mathbb Z}^d$$ , d ≥ 3. We show that the percolation function that to each u ≥ 0 attaches the probability that the origin does not belong to an infinite cluster of the vacant set at level u, is C 1 on an interval [ 0 , u ^ ) $$[0,\widehat {u})$$ , where u ^ $$\widehat {u}$$ is positive and plausibly coincides with the critical level u ∗ for the percolation of the vacant set. We apply this finding to a constrained minimization problem that conjecturally expresses the exponential rate of decay of the probability that a large box contains an excessive proportion ν of sites that do not belong to an infinite cluster of the vacant set. When u is smaller than u ^ $$\widehat {u}$$ , we describe a regime of “small excess” for ν where all minimizers of the constrained minimization problem remain strictly below the natural threshold value u ∗ − u $$\sqrt {u}_* - \sqrt {u}$$ for the variational problem.

We discuss random geometric structures obtained by percolation of Brownian loops, in relation to the Gaussian Free Field, and how their existence and properties depend on the dimension of the ambient space. We formulate a number of conjectures for the cases d = 3, 4, 5 and prove some results when d > 6.