Search ResultsShowing 1-16 of 16
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arXiv:2310.06745 (Published 2023-10-10)
Parisi formula for balanced Potts spin glass
Comments: 61 pagesThe Potts spin glass is a generalization of the Sherrington--Kirkpatrick (SK) model that allows for spins to take more than two values. Based on a novel synchronization mechanism, Panchenko (2018) showed that the limiting free energy is given by a Parisi-type variational formula. The functional order parameter in this formula is a probability measure on a monotone path in the space of positive-semidefinite matrices. By comparison, the order parameter for the SK model is much simpler: a probability measure on the unit interval. Nevertheless, a longstanding prediction by Elderfield and Sherrington (1983) is that the order parameter for the Potts spin glass can be reduced to that of the SK model. We prove this prediction for the balanced Potts spin glass, where the model is constrained so that the fraction of spins taking each value is asymptotically the same. It is generally believed that the limiting free energy of the balanced model is the same as that of the unconstrained model, in which case our results reduce the functional order parameter of Panchenko's variational formula to probability measures on the unit interval. The intuitive reason -- for both this belief and the Elderfield--Sherrington prediction -- is that no spin value is a priori preferred over another, and the order parameter should reflect this inherent symmetry. This paper rigorously demonstrates how symmetry, when combined with synchronization, acts as the desired reduction mechanism. Our proof requires that we introduce a generalized Potts spin glass model with mixed higher-order interactions, which is interesting it its own right. We prove that the Parisi formula for this model is differentiable with respect to inverse temperatures. This is a key ingredient for guaranteeing the Ghirlanda--Guerra identities without perturbation, which then allow us to exploit symmetry and synchronization simultaneously.
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arXiv:2309.04454 (Published 2023-09-08)
An upper bound on geodesic length in 2D critical first-passage percolation
Comments: 62 pages, 14 figuresWe consider i.i.d. first-passage percolation (FPP) on the two-dimensional square lattice, in the critical case where edge-weights take the value zero with probability $1/2$. Critical FPP is unique in that the Euclidean lengths of geodesics are superlinear, rather than linear, in the distance between their endpoints. This fact was speculated by Kesten in 1986 but not confirmed until 2019 by Damron and Tang, who showed a lower bound on geodesic length that is polynomial with degree strictly greater than $1$. In this paper we establish the first non-trivial upper bound. Namely, we prove that for a large class of critical edge-weight distributions, the shortest geodesic from the origin to a box of radius $R$ uses at most $R^{2+\epsilon}\pi_3(R)$ edges with high probability, for any $\epsilon > 0$. Here $\pi_3(R)$ is the polychromatic 3-arm probability from classical Bernoulli percolation; upon inserting its conjectural asymptotic, our bound converts to $R^{4/3 + \epsilon}$. In any case, it is known that $\pi_3(R) \lesssim R^{-\delta}$ for some $\delta > 0$, and so our bound gives an exponent strictly less than $2$. In the special case of Bernoulli($1/2$) edge-weights, we replace the additional factor of $R^\epsilon$ with a constant and give an expectation bound.
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arXiv:2307.10531 (Published 2023-07-20)
Intertwining the Busemann process of the directed polymer model
Comments: 79 pagesWe study the Busemann process of the planar directed polymer model with i.i.d. weights on the vertices of the planar square lattice, both the general case and the solvable inverse-gamma case. We demonstrate that the Busemann process intertwines with an evolution obeying a version of the geometric Robinson-Schensted-Knuth correspondence. In the inverse-gamma case this relationship enables an explicit description of the distribution of the Busemann process: the Busemann function on a nearest-neighbor edge has independent increments in the direction variable, and its distribution comes from an inhomogeneous planar Poisson process. Various corollaries follow, including that each nearest-neighbor Busemann function has the same countably infinite dense set of discontinuities in the direction variable. This contrasts with the known zero-temperature last-passage percolation cases, where the analogous sets are nowhere dense but have a dense union. The distribution of the asymptotic competition interface direction of the inverse-gamma polymer is discrete and supported on the Busemann discontinuities. Further implications follow for the eternal solutions and the failure of the one force-one solution principle for the discrete stochastic heat equation solved by the polymer partition function.
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Crisanti-Sommers formula and simultaneous symmetry breaking in multi-species spherical spin glasses
Comments: 49 pages, 1 figure; corrected (1.8) & (2.3) to allow for first-order term in covariance function; to appear in Comm. Math. PhysThere is a rich history of expressing the limiting free energy of mean-field spin glasses as a variational formula over probability measures on $[0,1]$, where the measure represents the similarity (or "overlap") of two independently sampled spin configurations. At high temperatures, the formula's minimum is achieved at a measure which is a point mass, meaning sample configurations are asymptotically orthogonal up to a magnetic field correction. At low temperatures, though, a very different behavior emerges known as replica symmetry breaking (RSB). The deep wells in the energy landscape create more rigid structure, and the optimal overlap measure is no longer a point mass. The exact size of its support remains in many cases an open problem. Here we consider these themes for multi-species spherical spin glasses. Following a companion work in which we establish the Parisi variational formula, here we present this formula's Crisanti-Sommers representation. In the process, we gain new access to a problem unique to the multi-species setting. Namely, if RSB occurs for one species, does it necessarily occur for other species as well? We provide sufficient conditions for the answer to be yes. For instance, we show that if two species share any quadratic interaction, then RSB for one implies RSB for the other. Moreover, the level of symmetry breaking must be identical, even in cases of full RSB. In the presence of an external field, any type of interaction suffices.
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Free energy in multi-species mixed $p$-spin spherical models
Comments: 75 pages; corrected (1.14) to allow for first-order term in covariance functionJournal: Electron. J. Probab. 27 (2022), paper no. 52DOI: 10.1214/22-EJP780Keywords: spin spherical models, free energy, parisi formula, convexity assumption, multi-species spherical spin glassesTags: journal articleWe prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed $p$-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space. This space is not related to the model's spherical structure and can be carried over to other multi-species settings.
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arXiv:2006.12580 (Published 2020-06-22)
Empirical distributions, geodesic lengths, and a variational formula in first-passage percolation
Comments: 93 pages, 5 figuresCategories: math.PRThis article resolves, in a dense set of cases, several open problems concerning geodesics in i.i.d. first-passage percolation on $\mathbb{Z}^d$. Our primary interest is in the empirical measures of edge-weights observed along geodesics from $0$ to $n\xi$, where $\xi$ is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as $n \to \infty$, answering a question of Hoffman. These families include arbitrarily small $L^\infty$-perturbations of any given distribution, almost every finitely supported distribution, uncountable collections of continuous distributions, and certain discrete distributions whose atoms have any prescribed sequence of probabilities. Moreover, the constructions are explicit enough to guarantee examples possessing certain features, for instance: both continuous and discrete distributions whose support is all of $[0,\infty)$, and distributions given by a density function that is $k$-times differentiable. All results also hold for $\xi$-directed infinite geodesics. In comparison, we show that if $\mathbb{Z}^d$ is replaced by the infinite $d$-ary tree, then any weight distribution admits a unique limiting empirical measure along geodesics. In both the lattice and tree cases, our methodology is driven by a new variational formula for the time constant, which requires no assumptions on the edge-weight distribution. Incidentally, this variational approach also allows us to obtain new convergence results for geodesic lengths, which have been unimproved in the subcritical case since the seminal 1965 manuscript of Hammersley and Welsh.
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arXiv:2001.11524 (Published 2020-01-30)
Avoidance couplings on non-complete graphs
Comments: 27 pages, 5 figuresA coupling of random walkers on the same finite graph, who take turns sequentially, is said to be an avoidance coupling if the walkers never collide. Previous studies of these processes have focused almost exclusively on complete graphs, in particular how many walkers an avoidance coupling can include. For other graphs, apart from special cases, it has been unsettled whether even two non-colliding simple random walkers can be coupled. In this article, we construct such a coupling on (i) any $d$-regular graph avoiding a fixed subgraph depending on $d$; and (ii) any square-free graph with minimum degree at least three. A corollary of the first result is that a uniformly random regular graph on $n$ vertices admits an avoidance coupling with high probability.
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arXiv:1912.04164 (Published 2019-12-09)
Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape
Comments: 39 pages, 9 figuresWithin the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work arXiv:1812.00309 of Dauvergne, Ortmann, and Vir\'ag, this object was constructed and shown to be the limit after parabolic correction of one such model: Brownian last passage percolation. This limit object, called the directed landscape, admits geodesic paths between any two space-time points $(x,s)$ and $(y,t)$ with $s<t$. In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations $x_1$ and $x_2$, and consider geodesics traveling $(x_1,0)\to (y,1)$ and $(x_2,0)\to (y,1)$. We prove that the set of $y\in\mathbb{R}$ for which these geodesics coalesce only at time $1$ has Hausdorff dimension one-half. Second, we consider endpoints $(x,0)$ and $(y,1)$ between which there exist two geodesics intersecting only at times $0$ and $1$. We prove that the set of such $(x,y)\in\mathbb{R}^2$ also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in arXiv:1904.01717; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in arXiv:1709.04110.
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arXiv:1910.12012 (Published 2019-10-26)
Full-path localization of directed polymers
Comments: 26 pages, 2 figuresCertain polymer models are known to exhibit path localization in the sense that at low temperatures, the average fractional overlap of two independent samples from the Gibbs measure is bounded away from $0$. Nevertheless, the question of where along the path this overlap takes place has remained unaddressed. In this article, we prove that on linear scales, overlap occurs along the entire length of the polymer. Namely, we consider time intervals of length $\varepsilon N$, where $\varepsilon>0$ is fixed but arbitrarily small. We then identify a constant number of distinguished trajectories such that the Gibbs measure is concentrated on paths having, with one of these distinguished paths, a fixed positive overlap simultaneously in every such interval. This result is obtained in all dimensions for a Gaussian random environment by using a recent non-local result as a key input.
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arXiv:1906.07780 (Published 2019-06-18)
Localization and free energy asymptotics in disordered statistical mechanics and random growth models
Comments: Ph.D. thesis (Stanford University, 2019), xii+294 pages; Chapter 2 gives a unified presentation of arXiv:1612.03443 and arXiv:1708.03713, Chapter 3 includes content of arXiv:1906.05502, Chapter 4 includes content of arXiv:1810.03215, Chapter 5 includes content of arXiv:1810.03656, MATLAB code in appendixKeywords: random growth models, free energy asymptotics, disordered statistical mechanics, localization, popular zero-temperature polymer modelsTags: dissertationThis dissertation develops, for several families of statistical mechanical and random growth models, techniques for analyzing infinite-volume asymptotics. In the statistical mechanical setting, we focus on the low-temperature phases of spin glasses and directed polymers, wherein the ensembles exhibit localization which is physically phenomenological. We quantify this behavior in several ways and establish connections to properties of the limiting free energy. We also consider two popular zero-temperature polymer models, namely first- and last-passage percolation. For these random growth models, we investigate the order of fluctuations in their growth rates, which are analogous to free energy.
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arXiv:1906.05502 (Published 2019-06-13)
Localization in Gaussian disordered systems at low temperature
Comments: 64 pagesFor a broad class of Gaussian disordered systems at low temperature, we show that the Gibbs measure is asymptotically localized in small neighborhoods of a small number of states. From a single argument, we obtain (i) a version of "complete" path localization for directed polymers that is not available even for exactly solvable models; and (ii) a result about the exhaustiveness of Gibbs states in spin glasses not requiring the Ghirlanda-Guerra identities.
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arXiv:1810.03656 (Published 2018-10-08)
Fluctuation lower bounds in planar random growth models
Comments: 28 pages, 1 figureCategories: math.PRWe prove $\sqrt{\log n}$ lower bounds on the order of growth fluctuations in three planar growth models (first-passage percolation, last-passage percolation, and directed polymers) under no assumptions on distribution of vertex or edge weights other than the minimum conditions required for avoiding pathologies. Such bounds were previously known only for certain restrictive classes of distributions. In addition, the first-passage shape fluctuation exponent is shown to be at least $1/8$, extending previous results to more general distributions.
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arXiv:1810.03215 (Published 2018-10-07)
Replica symmetry breaking in multi-species Sherrington--Kirkpatrick model
Comments: 18 pagesIn the Sherrington--Kirkpatrick (SK) and related mixed $p$-spin models, there is interest in understanding replica symmetry breaking at low temperatures. For this reason, the so-called AT line proposed by de Almeida and Thouless as a sufficient (and conjecturally necessary) condition for symmetry breaking, has been a frequent object of study in spin glass theory. In this paper, we consider the analogous condition for the multi-species SK model, which concerns the eigenvectors of a Hessian matrix. The analysis is tractable in the two-species case with positive definite variance structure, for which we derive an explicit AT temperature threshold. To our knowledge, this is the first non-asymptotic symmetry breaking condition produced for a multi-species spin glass. As possible evidence that the condition is sharp, we draw further parallel with the classical SK model and show coincidence with a separate temperature inequality guaranteeing uniqueness of the replica symmetric critical point.
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arXiv:1712.00210 (Published 2017-12-01)
An upper bound on the size of avoidance couplings on $K_n$
Comments: 7 pagesCategories: math.PRWe show that a coupling of non-colliding simple random walkers on the complete graph on $n$ vertices can include at most $n - \log n$ walkers. This improves the only previously known upper bound of $n-2$ due to Angel, Holroyd, Martin, Wilson, and Winkler (Electron. Commun. Probab. 18, 2013).
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arXiv:1708.03713 (Published 2017-08-11)
Localization of directed polymers with general reference walk
Comments: 41 pagesDirected polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference walk. Some finer results are known for the so-called long-range directed polymer in which the reference walk lies in the domain of attraction of an $\alpha$-stable process. In this note, low-temperature localization properties recently proved for the classical case are shown to be true with any reference walk. First, it is proved that the polymer's endpoint distribution is asymptotically purely atomic, thus strengthening the best known result for long-range directed polymers. A second result proving geometric localization along a positive density subsequence is new to the general case. The proofs use a generalization of the approach introduced by the author with S. Chatterjee in a recent manuscript on the quenched endpoint distribution; this generalization allows one to weaken assumptions on the both the walk and the environment. The methods of this paper also give rise to a variational formula for free energy which is analogous to the one obtained in the simple random walk case.
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arXiv:1612.03443 (Published 2016-12-11)
The endpoint distribution of directed polymers
Comments: 84 pages. Comments are welcomeProbabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called "partitioned subprobability measure", to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on large deviations of the occupation measure of Brownian motion, we define a suitable topology on the space of partitioned subprobability measures and prove that this topology is compact. Then using a variant of the cavity method from the theory of spin glasses, we show that any limit law of a sequence of endpoint distributions must satisfy a fixed point equation on this abstract space, and that the limiting free energy of the model can be expressed as the solution of a variational problem over the set of fixed points. As a first application of the theory, we prove that in an environment with finite exponential moment, the endpoint distribution is asymptotically purely atomic if and only if the system is in the low temperature phase. The analogous result for a heavy-tailed environment was proved by Vargas in 2007. As a second application, we prove a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. All our results hold in arbitrary dimensions, and make no use of integrability.