arXiv subjects

Pax Kivimae

Publications and source records attributed to Pax Kivimae.

8 recordsLinked to original sources

arXiv2026

Wandering Exponents and the Free Energy of the High-Dimensional Elastic Polymer

We study the behavior of the elastic polymer, a model of a directed polymer in a continuous Gaussian random environment that is independent in time and correlated in space, as the dimension of the environment is taken to infinity. We give an explicit asymptotic formula for the free energy, which is given in terms of the distribution of the inner product of two sampled configurations, which we also obtain an implicit formula for. From this, we provide an explicit characterization of both the low- and high-temperature phases of this model in terms of the spatial correlation function of the environment. We find asymptotics for the wandering exponent when the spatial correlation function is either an exponential or a power-law decay. Our results show that when the correlations are either suitably weak or short ranged, the model is asymptotically diffusive. On the other hand, for suitably strong long ranged correlations, the model is asymptotically superdiffusive. Moreover, we show that this transition coincides exactly with another transition where the model goes from being one-step replica symmetry breaking to full-step replica symmetry breaking. This rigorously confirms many of the findings of Mezard and Parisi [53] in the physics literature.

math.PR↗

arXiv2024

The Larkin Mass and Replica Symmetry Breaking in the Elastic Manifold

This is the second of a series of three papers about the Elastic Manifold model. This classical model proposes a rich picture due to the competition between the inherent disorder and the smoothing effect of elasticity. In this paper, we analyze our variational formula for the free energy obtained in our first companion paper [16]. We show that this variational formula may be simplified to one which is solved by a unique saddle point. We show that this saddle point may be solved for in terms of the corresponding critical point equation. Moreover, its terms may be interpreted in terms of natural statistics of the model: namely the overlap distribution and effective radius of the model at a given site. Using this characterization, obtain a complete characterization of the replica symmetry breaking phase. From this we are able to confirm a number of physical predictions about this boundary, namely those involving the Larkin mass [6, 53, 54], an important critical mass for the system. The zero-temperature Larkin mass has recently been shown to be the topological trivialization threshold, following work of Fyodorov and Le Doussal [37, 38], made rigorous by the first author, Bourgade and McKenna [12, 13].

math.PR↗

arXiv2024

The Free Energy of the Elastic Manifold

This is the first of a series of three papers about the Elastic Manifold model. This classical model proposes a rich picture due to the competition between the inherent disorder and the smoothing effect of elasticity. In this paper, we prove a Parisi formula, i.e. we compute the asymptotic quenched free energy and show it is given by the solution to a certain variational problem. This work comes after a long and distinguished line of work in the Physics literature, going back to the 1980's (including the foundational work by Daniel Fisher [29], Marc Mezard and Giorgio Parisi [50, 51], and more recently by Yan Fyodorov and Pierre Le Doussal [34, 35]. Even though the mathematical study of Spin Glasses has seen deep progress in the recent years, after the celebrated work by Michel Talagrand [67, 68], the Elastic Manifold model has been studied from a mathematical perspective, only recently and at zero temperature. The annealed topological complexity has been computed, by the first author with Paul Bourgade and Benjamin McKenna [15, 16]. Here we begin the study of this model at positive temperature by computing the quenched free energy. We obtain our Parisi formula by first applying Laplace's method to reduce the question to a related new family of spherical Spin Glass models with an elastic interaction. The upper bound is then obtained through an interpolation argument initially developed by Francisco Guerra [42] for the study of Spin Glasses. The lower bound follows by adapting the cavity method along the lines explored by Wei-Kuo Chen [23] and the multi-species synchronization method of Dmitry Panchenko [55]. In our next papers [19, 20] we will analyze the consequences of this Parisi formula.

math.PR↗

arXiv2024

Moments of Characteristic Polynomials of Non-Symmetric Random Matrices

We study the moments of the absolute characteristic polynomial of the real elliptic ensemble, including the case of the real Ginibre ensemble. We obtain asymptotics for all integral moments inside the real bulk to order 1 + o(1). In particular, for the real Ginibre ensemble, this extends known computations for even moments, and confirms a recent conjecture of Serebryakov and Simm [48] in the integral case. For the elliptic case, this generalizes computations of first two moments by Fyodorov [25] and Fyodorov and Tarnowski [31]. We additionally find uniform asymptotics for the multi-point correlations of the absolute characteristic polynomial. Our proof relies on a relation between expectations for the absolute characteristic polynomial and the real correlation functions, as well as an algebraic method of obtaining asymptotics for the behavior of these correlation functions near the diagonal.

math.PR↗

arXiv2022

Concentration of Equilibria and Relative Instability in Disordered Non-Relaxational Dynamics

We consider a system of random autonomous ODEs introduced by Cugliandolo et al. [22], which serves as a non-relaxational analog of the gradient flow for the spherical p-spin model. The asymptotics for the expected number of equilibria in this model was recently computed by Fyodorov [32] in the high-dimensional limit, followed a similar computation for the expected number of stable equilibria by Garcia [38]. We show that for $p > 9$ the number of equilibria, as well as the number of stable equilibria, concentrate around their respective averages, generalizing recent results of Subag and Zeitouni [61, 64] in the relaxational case. In particular, we confirm that this model undergoes a transition from relative to absolute instability, in the sense of Ben Arous, Fyodorov, and Khoruzhenko [11].

math.PR↗

arXiv2022

Gaussian Multiplicative Chaos for Gaussian Orthogonal and Symplectic Ensembles

We study the characteristic polynomials of both the Gaussian Orthogonal and Symplectic Ensembles. We show that for both ensembles, powers of the absolute value of the characteristic polynomials converge in law to Gaussian multiplicative chaos measures after normalization for sufficiently small real powers. The main tool is a new asymptotic relation between the fractional moments of the absolute characteristic polynomials of Gaussian Orthogonal, Unitary, and Symplectic Ensembles.

math.PR↗

arXiv2022

The Ground State Energy and Concentration of Complexity in Spherical Bipartite Models

We establish an asymptotic formula for the ground-state energy of the spherical pure $(p,q)$-spin glass model for $p,q\ge 97$. We achieve this through understanding the concentration of the complexity of critical points with values within a region of the ground state energy. More specifically, we show that the second moment of this count coincides with the square of the first moment up to a sub-exponential factor.

math.PR↗

arXiv2020

Critical Fluctuations for the Spherical Sherrington-Kirkpatrick Model in an External Field

We prove the existence of a critical regime for the fluctuations of the ground-state energy of the spherical Sherrington-Kirkpatrick model in an external field, confirming predictions given in [3,12]. We also establish a critical regime for the fluctuations in a model with a critical Ferromagnetic interaction term, producing a three-parameter family of distributions generalizing the two-parameter family given in [4]. These results are both established in the generality of a $β$-ensemble analogue of the spherical Sherrington-Kirkpatrick model, which subsumes the complex and quarternionic generalizations.

math.PR↗