Stability and chaos in dynamical last passage percolation

Abstract

Many complex statistical mechanical models have intricate energy landscapes. The ground state, or lowest energy state, lies at the base of the deepest valley. In examples such as spin glasses and Gaussian polymers, there are many valleys; the abundance of near-ground states (at the base of valleys) indicates the phenomenon of chaos, under which the ground state alters profoundly when the model's disorder is slightly perturbed. In this article, we compute the critical exponent that governs the onset of chaos in a dynamic manifestation of a canonical model in the Kardar-Parisi-Zhang [KPZ] universality class, Brownian last passage percolation [LPP]. In this model in its static form, semi-discrete polymers advance through Brownian noise, their energy given by the integral of the white noise encountered along their journey. A ground state is a geodesic, of extremal energy given its endpoints. We perturb Brownian LPP by evolving the disorder under an Ornstein-Uhlenbeck flow. We prove that, for polymers of length $n$, a sharp phase transition marking the onset of chaos is witnessed at the critical time $n^{-1/3}$. Indeed, the overlap between the geodesics at times zero and $t > 0$ that travel a given distance of order $n$ will be shown to be of order $n$ when $t\ll n^{-1/3}$; and to be of smaller order when $t\gg n^{-1/3}$. We expect this exponent to be shared among many interface models. The present work thus sheds light on the dynamical aspect of the KPZ class; it builds on several recent advances. These include Chatterjee's harmonic analytic theory [Cha14] of equivalence of superconcentration and chaos in Gaussian spaces; a refined understanding of the static landscape geometry of Brownian LPP developed in the companion paper [GH20]; and, underlying the latter, strong comparison estimates of the geodesic energy profile to Brownian motion in [CHH19].

Figures (10)

• Figure 1: Last passage percolation with uniform $U[0,1]$ weights is dynamically updated according to independent unit Poisson processes at each vertex. Depicted are snapshots at times $0.07$ and $0.3$ of a given dynamical simulation. The geodesic, blue at time zero, evolves to its present red state in each case. Since $1000^{-1/3} = 0.1$, the left sketch depicts a subcritical scenario and the right sketch a supercritical one, with the transition from high to low overlap evident in the images.
• Figure 2: In the left sketch, the lattice $\mathbb{Z}^2$ has been rotated counterclockwise by forty-five degrees, and contracted by a factor of $2^{1/2}$. The geodesic thus passes from $(0,0)$ to $(0,n)$. The formerly anti-diagonal midlife line $y = n/2$ witnesses the geodesic's passage at location $z_{{\rm max}}$. In the middle sketch, the inverted profile $-Z^\swarrow_{n,n}$ has been translated vertically so as to touch, but not to cross, the profile $Z^\nearrow_{0,0}$. Horizontal coordinates of contact between the two graphs are locations of passage for geodesics through the midlife line $y = n/2$. The two profiles make jumps valued in $\{-1,0,1\}$ and resemble random walk, in a similiar fashion to Bernoulli-$p$ measure being invariant for the totally asymmetric simple exclusion process. The routed weight profile $Z$ is depicted in the right sketch: its maximizers are the same locations of passage.
• Figure 3: Let $(n,s_1,s_2)$ be a compatible triple and let $x, y \in \mathbb{R}$. The endpoints of the geodesic in the left sketch are such that, when the scaling map $R_n$ is applied to produce the right sketch, the result is the $n$-polymer $\rho_n [ (x,s_1) \to (y,s_2) ]$ from $(x,s_1)$ to $(y,s_2)$.
• Figure 4: In the left sketch, there is one long excursion between the red polymer at time zero and the blue polymer at time $t$, so that the brown proxy $\rho_n^{t \to 0}$ (which merges at both ends with the red curve) is composed of two time-zero polymers that abut an element (marked with a small square) of $\rho_n^t$ at height one-half. The middle sketch illustrates behaviour for certain weight profiles that is typically consistent with the left sketch: above, near touch, namely a close encounter between the indicated transformations of narrow weight profiles routed at $(0,0)$ or $(0,1)$; below, and equivalently, twin peaks, namely a value at $\rho_n^t(1/2)$ for the time-zero routed weight profile $x \to Z^0_n(x,1/2)$ rivalling the peak at $\rho_n^0(1/2)$ to a height of the order $\tau^{1/2}$ seen in (\ref{['e.mimic']}). The right sketch indicates the polymers at time zero and time $t$ in the case of several excursions of roughly equal duration that is the subject of Subsection \ref{['s.severalexcursions']}. The three vertical double-arrowed intervals illustrate excursions of duration of order $2^{-\ell}$. The proxy $\rho_n^{t \to 0}$ is not drawn, but it interpolates by means of time-zero polymers the various small crosses. Let $h$ denote the $y$-coordinate of one of the displayed dotted horizontal intervals $I$; and let $x$ denote the horizontal coordinate of the cross that lies on $I$. Associated to $I$ is a shortfall in weight of the proxy relative to the time-zero polymer $\rho_n^0$, since the value $Z^0_n(x,h)$ lies below the supremum of the routed weight profile $Z^0_n(\cdot,h)$.
• Figure 5: In this instance, as typically, the polymer $\rho_n [ (x,h_1 ) \to (y,h_2) ]$ visits the bold horizontal intervals $J \times \{ h_1^+[\eta] \}$ and $K \times \{ h_2^-[\eta]\}$ at the start and end of its prime.

Theorems & Definitions (61)

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