@article{bates_morrison_rogers_serafini_sood_2025, title={A new combinatorial interpretation of partial sums of $m$-step Fibonacci numbers}, url={https://arxiv.org/abs/2503.11055}, DOI={10.48550/ARXIV.2503.11055}, publisher={arXiv}, author={Bates, Erik and Morrison, Blan and Rogers, Mason and Serafini, Arianna and Sood, Anav}, year={2025}, month={Mar} }
 @article{bates_2024, title={Empirical Measures, Geodesic Lengths, and a Variational Formula in First-Passage Percolation}, url={http://dx.doi.org/10.1090/memo/1460}, DOI={10.1090/memo/1460}, abstractNote={This monograph resolves—in a dense class of cases—several open problems concerning geodesics in i.i.d. first-passage percolation on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our primary interest is in the empirical measures of edge-weights observed along geodesics from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n bold-italic xi"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold-italic">ξ</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">n{\boldsymbol \xi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic xi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold-italic">ξ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\boldsymbol \xi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these empirical measures converge weakly to a deterministic limit as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, answering a question of Hoffman. These families include arbitrarily small <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-perturbations of any given distribution, almost every finitely supported distribution, uncountable collections of continuous distributions, and certain discrete distributions whose atoms can 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and distributions given by a density function that is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-times differentiable. All results also hold for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic xi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold-italic">ξ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\boldsymbol \xi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-directed infinite geodesics. In comparison, we show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is replaced by the infinite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ary tree, then any distribution for the weights 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 regime since the seminal 1965 manuscript of Hammersley and Welsh.}, journal={Memoirs of the American Mathematical Society}, author={Bates, Erik}, year={2024}, month={Jan} }
 @book{bates_2024, title={Empirical measures, geodesic lengths, and a variational formula in first-passage percolation}, ISBN={9781470467913}, publisher={American Mathematical Society}, author={Bates, E.}, year={2024} }
 @article{bates_sohn_2024, title={Parisi Formula for Balanced Potts Spin Glass}, volume={405}, ISSN={["1432-0916"]}, url={https://doi.org/10.1007/s00220-024-05100-9}, DOI={10.1007/s00220-024-05100-9}, number={10}, journal={COMMUNICATIONS IN MATHEMATICAL PHYSICS}, author={Bates, Erik and Sohn, Youngtak}, year={2024}, month={Oct} }
 @article{bates_harper_shen_sorensen_2023, title={An upper bound on geodesic length in 2D critical first-passage percolation}, url={https://arxiv.org/abs/2309.04454}, DOI={10.48550/ARXIV.2309.04454}, abstractNote={We 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.}, publisher={arXiv}, author={Bates, Erik and Harper, David and Shen, Xiao and Sorensen, Evan}, year={2023} }
 @article{bates_wai-tong_fan_seppäläinen_2023, title={Intertwining the Busemann process of the directed polymer model}, url={https://arxiv.org/abs/2307.10531}, DOI={10.48550/ARXIV.2307.10531}, abstractNote={We 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.}, publisher={arXiv}, author={Bates, Erik and Wai-Tong and Fan and Seppäläinen, Timo}, year={2023} }
 @article{bates_sohn_2022, title={Crisanti–Sommers Formula and Simultaneous Symmetry Breaking in Multi-species Spherical Spin Glasses}, url={https://doi.org/10.1007/s00220-022-04421-x}, DOI={10.1007/s00220-022-04421-x}, journal={Communications in Mathematical Physics}, author={Bates, Erik and Sohn, Youngtak}, year={2022}, month={Sep} }
 @article{bates_sohn_2022, title={Free energy in multi-species mixed p-spin spherical models}, volume={27}, url={http://dx.doi.org/10.1214/22-ejp780}, DOI={10.1214/22-ejp780}, abstractNote={We 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.}, number={none}, journal={Electronic Journal of Probability}, publisher={Institute of Mathematical Statistics}, author={Bates, Erik and Sohn, Youngtak}, year={2022}, month={Jan} }
 @article{bates_ganguly_hammond_2022, title={Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape}, volume={27}, url={http://dx.doi.org/10.1214/21-ejp706}, DOI={10.1214/21-ejp706}, abstractNote={Within 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 [27] of Dauvergne, Ortmann, and Virág, this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, 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 x1 and x2, and consider geodesics traveling (x1,0)→(y,1) and (x2,0)→(y,1). We prove that the set of y∈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)∈R2 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 [10]; 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 [40].}, number={none}, journal={Electronic Journal of Probability}, publisher={Institute of Mathematical Statistics}, author={Bates, Erik and Ganguly, Shirshendu and Hammond, Alan}, year={2022}, month={Jan} }
 @article{bates_podder_2021, title={Avoidance couplings on non‐complete graphs}, volume={59}, url={http://dx.doi.org/10.1002/rsa.20999}, DOI={10.1002/rsa.20999}, abstractNote={Abstract A 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 noncolliding 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.}, number={1}, journal={Random Structures & Algorithms}, publisher={Wiley}, author={Bates, Erik and Podder, Moumanti}, year={2021}, month={Aug}, pages={25–52} }
 @article{bates_2021, title={Full-path localization of directed polymers}, volume={26}, url={http://dx.doi.org/10.1214/21-ejp641}, DOI={10.1214/21-ejp641}, abstractNote={Certain 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 εN, where ε>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.}, number={none}, journal={Electronic Journal of Probability}, publisher={Institute of Mathematical Statistics}, author={Bates, Erik}, year={2021}, month={Jan} }
 @article{bates_chatterjee_2020, title={Fluctuation lower bounds in planar random growth models}, volume={56}, url={http://dx.doi.org/10.1214/19-aihp1043}, DOI={10.1214/19-aihp1043}, abstractNote={Nous montrons des bornes inférieures de $\sqrt{\log n}$ pour l’ordre des fluctuations de trois modèles planaires de croissance (percolation de premier passage, percolation de dernier passage et polymères dirigés) sans autre hypothèse sur la loi des poids des sommets ou des arêtes que les conditions minimales permettant d’éviter les cas pathologiques. De telles bornes étaient connues auparavant seulement pour certaines classes restreintes de lois. De surcroît, nous montrons que l’exposant des fluctuations autour de la forme limite pour la percolation de premier passage est au moins $1/8$, ce qui étend des résultats précédents à des lois plus générales.}, number={4}, journal={Annales de l'Institut Henri Poincaré, Probabilités et Statistiques}, publisher={Institute of Mathematical Statistics}, author={Bates, Erik and Chatterjee, Sourav}, year={2020}, month={Nov} }
 @article{bates_chatterjee_2020, title={Localization in Gaussian disordered systems at low temperature}, volume={48}, url={http://dx.doi.org/10.1214/20-aop1436}, DOI={10.1214/20-aop1436}, abstractNote={For 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.}, number={6}, journal={The Annals of Probability}, publisher={Institute of Mathematical Statistics}, author={Bates, Erik and Chatterjee, Sourav}, year={2020}, month={Nov} }
 @article{bates_chatterjee_2020, title={The endpoint distribution of directed polymers}, volume={48}, url={http://dx.doi.org/10.1214/19-aop1376}, DOI={10.1214/19-aop1376}, abstractNote={Probabilistic 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.}, number={2}, journal={The Annals of Probability}, publisher={Institute of Mathematical Statistics}, author={Bates, Erik and Chatterjee, Sourav}, year={2020}, month={Mar} }
 @article{bates_sauermann_2019, title={An Upper Bound on the Size of Avoidance Couplings}, volume={28}, url={http://dx.doi.org/10.1017/s0963548318000500}, DOI={10.1017/s0963548318000500}, abstractNote={We 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 ({\it Electron.~Commun.~Probab.~18}, 2013). The proof considers couplings of i.i.d.~sequences of Bernoulli random variables satisfying a similar avoidance property, for which there is separate interest. Our bound in this setting should be closer to optimal.}, number={3}, journal={Combinatorics, Probability and Computing}, publisher={Cambridge University Press (CUP)}, author={BATES, ERIK and SAUERMANN, LISA}, year={2019}, month={May}, pages={325–334} }
 @phdthesis{bates_2019, title={Localization and free energy asymptotics in disordered statistical mechanics and random growth models}, url={https://arxiv.org/abs/1906.07780}, DOI={10.48550/ARXIV.1906.07780}, abstractNote={This 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.}, school={arXiv}, author={Bates, Erik}, year={2019} }
 @article{bates_sloman_sohn_2019, title={Replica Symmetry Breaking in Multi-species Sherrington–Kirkpatrick Model}, volume={174}, url={https://doi.org/10.1007/s10955-018-2197-4}, DOI={10.1007/s10955-018-2197-4}, number={2}, journal={Journal of Statistical Physics}, publisher={Springer Science and Business Media LLC}, author={Bates, Erik and Sloman, Leila and Sohn, Youngtak}, year={2019}, month={Jan}, pages={333–350} }
 @article{bates_2018, title={Localization of directed polymers with general reference walk}, volume={23}, url={http://dx.doi.org/10.1214/18-ejp158}, DOI={10.1214/18-ejp158}, abstractNote={Directed 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.}, number={none}, journal={Electronic Journal of Probability}, publisher={Institute of Mathematical Statistics}, author={Bates, Erik}, year={2018}, month={Jan} }
 @article{malaia_bates_seitzman_coppess_2016, title={Altered brain network dynamics in youths with autism spectrum disorder}, volume={234}, url={http://dx.doi.org/10.1007/s00221-016-4737-y}, DOI={10.1007/s00221-016-4737-y}, abstractNote={The heterogeneity of behavioral manifestation of autism spectrum disorders (ASDs) requires a model which incorporates understanding of dynamic differences in neural processing between ASD and typically developing (TD) populations. We use network approach to characterization of spatiotemporal dynamics of EEG data in TD and ASD youths. EEG recorded during both wakeful rest (resting state) and a social-visual task was analyzed using cross-correlation analysis of the 32-channel time series to produce weighted, undirected graphs corresponding to functional brain networks. The stability of these networks was assessed by novel use of the L1-norm for matrix entries (edit distance). There were a significantly larger number of stable networks observed in the resting condition compared to the task condition in both populations. In resting state, stable networks persisted for a significantly longer time in children with ASD than in TD children; networks in ASD children also had larger diameter, indicative of long-range connectivity. The resulting analysis combines key features of microstate and network analyses of EEG.}, number={12}, journal={Experimental Brain Research}, publisher={Springer Science and Business Media LLC}, author={Malaia, Evie A. and Bates, Erik and Seitzman, Benjamin and Coppess, Katherine}, year={2016}, month={Dec}, pages={3425–3431} }