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Motivated by the Lee--Yang approach to phase transitions, we study the
partition function of the Generalized Random Energy Model (GREM) at complex
inverse temperature $\beta$. We compute the limiting log-partition function and
describe the fluctuations of the partition function. For the GREM with $d$
levels, in total, there are $\frac 12 (d+1)(d+2)...
Context in source publication
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... complex plane phase diagram of the REM has been described by Derrida [15]; see also [25]. There are three phases, see Figure 2, which we will denote by (a) E k (expectation dominated phase), (b) F k (fluctuations dominated phase), (c) G k ("glassy phase" = extreme values dominated phase). Concretely, the phases are given by ...
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Citations
... In the physics literature, Derrida has initiated the study of random energy models (REM) as toy models of mean-field disordered systems and in particular those at complex temperatures: [9,10,27]. See [23,16,17,13,14] for the rigorous analysis of the REM, GREM and BBM at complex temperatures. A natural analogue in the context of log-correlated Gaussian fields is the so-called Gaussian multiplicative chaos. ...
We identify the fluctuations of the partition function of the continuous random energy model on a Galton-Watson tree in the so-called weak correlation regime. Namely, when the ``speed functions'', that describe the time-inhomogeneous variance, lie strictly below their concave hull and satisfy a certain weak regularity condition. We prove that the phase diagram coincides with the one of the random energy model. However, the fluctuations are different and depend on the slope of the covariance function at $0$ and the final time $t$.
... Remark. The appearance of the random variance in Theorem 1.4 (and in the following ones) is in sharp contrast with the REM [21] and generalized REM [22], where CLTs with deterministic variance hold for β in the strip |σ| < 1/ √ 2. ...
We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and on the boundaries I/III and II/III, we prove a central limit theorem with a random variance. In Phase I and on the boundary I/II, we prove an a.s. and $L^1$ martingale convergence. All results are shown for any given correlation between the real and imaginary parts of the random energy.
... The same method can be applied to various ensembles of random algebraic polynomials. A similar method was used in [Sta11], [LMS12] for complex zeros of random Taylor series near the circle of convergence, in [Shi12] for certain series with random coefficients, and in [KK14a,KK14b] for complex zeros of the partition function of the (Generalized) Random Energy Model. Unlike in these works, we study real zeros. ...
Consider a random trigonometric polynomial $X_n: \mathbb R \to \mathbb R$ of
the form $$ X_n(t) = \frac 1 {\sqrt n} \sum_{k=1}^n \left( \xi_k \sin (kt) +
\eta_k \cos (kt)\right), $$ where $(\xi_1,\eta_1),(\xi_2,\eta_2),\ldots$ are
i.i.d. bivariate real random vectors with zero mean and unit covariance matrix.
Let $(s_n)_{n\in\mathbb N}$ be any sequence of real numbers. We prove that as
$n\to\infty$, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+
b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$
of a stationary, zero-mean Gaussian process with correlation function $(\sin
t)/t$.
... For several models of spin glasses it is known that the logpartition function has asymptotically Gaussian fluctuations in the high temperature regime. This was shown for the Sherrington-Kirkpatrick model in [1], for the Random Energy Model and the p-spin model in [8], and for the Generalized Random Energy Model in [18], to give just an incomplete list of examples. We are interested in the Biggins martingale W n (θ) associated with a supercritical branching random walk (BRW), to be defined below. ...
Let $(W_n(\theta))_{n\in\mathbb N_0}$ be the Biggins martingale associated
with a supercritical branching random walk and denote by $W_\infty(\theta)$ its
limit. Assuming essentially that the martingale $(W_n(2\theta))_{n\in\mathbb
N_0}$ is uniformly integrable and that $\text{Var} W_1(\theta)$ is finite, we
prove a functional central limit theorem for the tail process
$(W_\infty(\theta) - W_{n+r}(\theta))_{r\in\mathbb N_0}$ and a law of the
iterated logarithm for $W_\infty(\theta)-W_n(\theta)$, as $n\to\infty$.
... In particular, the case of arbitrary correlations between the imaginary and real parts of the energies was considered in [11]. The same authors answered in [12] similar questions about the Generalized Random Energy model (GREM) -a model with hierarchical correlations -and obtained the full phase diagram. In the complex GREM, the phase diagram turned out to have a much richer structure than that of the complex REM. ...
We identify the fluctuations of the partition function for a class of random
energy models, where the energies are given by the positions of the particles
of the complex-valued branching Brownian motion (BBM). Specifically, we provide
the weak limit theorems for the partition function in the so-called "glassy
phase" -- the regime of parameters, where the behaviour of the partition
function is governed by the extrema of BBM. We allow for arbitrary correlations
between the real and imaginary parts of the energies. This extends the recent
result of Madaule, Rhodes and Vargas, where the uncorrelated case was treated.
In particular, our result covers the case of the real-valued BBM energy model
at complex temperatures.
... The law of the zero set of ξ as defined in the present paper is invariant with respect to real translations only. The function ξ and its complex analogue appeared as limits of certain random partition functions; see [19], [20]. ...
... be a sequence of random variables defined on a probability space (Ω, F , P) and taking values in a Polish space E. We say that X n converges stably to a kernel Q : Ω → M 1 (E) if the sequence of kernels Q n : ω → δ Xn(ω) converges stably to Q. That is to say, for every set A ∈ F and every bounded continuous function f : E → R, we have (20) lim ...
... By taking g = ½ A in (25) we obtain the required relation (20). ...
Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_n(\beta)$
associated to a supercritical branching random walk with mean number of
offspring $m$. We prove a functional central limit theorem stating that as
$n\to\infty$ the process $$ D_n(u):= m^{\frac 12 n}
\left(W_{\infty}\left(\frac{u}{\sqrt n}\right) - W_{n}\left(\frac{u}{\sqrt
n}\right) \right) $$ converges weakly, on a suitable space of analytic
functions, to a Gaussian random analytic function with random variance. Using
this result we prove central limit theorems for the total path length of random
trees. In the setting of binary search trees, we recover a recent result of R.
Neininger [Refined Quicksort Asymptotics, Rand. Struct. and Alg., to appear],
but we also prove similar results for uniform random recursive trees and, more
generally, linear recursive trees. Moreover, we replace weak convergence in
Neininger's theorem by the almost sure weak (a.s.w.) convergence of probability
transition kernels. In the case of binary search trees, our result states that
$$ \mathcal{L}\left\{\sqrt{\frac{n}{2\log n}} \left(EPL_{\infty} -
\frac{EPL_n-2n\log n}{n}\right)\Bigg | \mathcal{G}_{n}\right\} \to
\{\omega\mapsto N_{0,1}\} \quad a.s.w.,$$ where $EPL_n$ is the external path
length of a binary search tree $X_n$ with $n$ vertices, $EPL_{\infty}$ is the
limit of the R\'egnier martingale, and $\mathcal{L}(\cdot |\mathcal{G}_n)$
denotes the conditional distribution w.r.t. the $\sigma$-algebra
$\mathcal{G}_n$ generated by $X_1,\ldots,X_n$. A.s.w. convergence is stronger
than weak and even stable convergence, and contains a.s. and weak convergence
as special cases. We prove several basic properties of the a.s.w. convergence.
We study a number of further examples in which the a.s.w. convergence appears
naturally. These include the classical central limit theorem for Galton--Watson
processes and the P\'olya urn.
Final version of my thesis (accepted).
425 pages.
From the Korteweg de Vries equation to the D-brane theory
