For each $n \ge 1$, let $\mathrm{d}^n = (d^{n}(i),1 \le i \le n)$ be a sequence of positive integers with even sum $\sum_{i=1}^n d^n(i) \ge 2n$. Let $(G_n, T_n, \Gamma_n)$ be uniformly distributed over the set of simple graphs $G_n$ with degree sequence $\mathrm{d}^n$, endowed with a spanning tree $T_n$ and rooted along an oriented edge $\Gamma_n$ of $G_n$ which is not an edge of $T_n$. Under a finite variance assumption on degrees in $G_n$, we show that, after rescaling, $T_n$ converges in distribution to the Brownian continuum random tree as $n \to \infty$. Our main tool is a new version of Pitman's additive coalescent [P], which can be used to build both random trees with a fixed degree sequence, and random tree-weighted graphs with a fixed degree sequence. As an input to the proof, we also derive a Poisson approximation theorem for the number of loops and multiple edges in the superposition of a fixed graph and a random graph with a given degree sequence sampled according to the configuration model; we find this to be of independent interest.

We present a first introduction to determinantal point processes, focussing on multiplicative functionals, Palm theory, and key applications such as random matrices, Young diagrams and random holomorphic functions.

We prove that the values of discrete directed polymer partition functions involving multiple non-intersecting paths remain invariant under replacing the background weights by their images under the geometric RSK correspondence. This result is inspired by a recent and remarkable identity proved by Dauvergne, Orthmann and Virág which is recovered as the zero-temperature, semi-discrete limit of our main result.

The goal of the present paper is to describe three topics in the area of complex networks that constituted a mini-course delivered at the 12th Mathematical Society of Japan – Seasonal Institute, held at Kyushu University in Fukuoka, Japan, 31/07–09/08, 2019. The topics are: (I) Spectra of adjacency matrices, (II) Equivalence of ensembles, (III) Exploration and mixing times. Topics (I) and (II) are part of the ongoing attempt to understand structure of networks, topic (III) to elucidate functionality of networks.

We consider several problems about random walks, or equivalently, discrete harmonic function on the planar integer lattice. The questions deal with the rate of convergence of quantities to their Brownian counterparts.

We study the scaling limit of uniform random quadrangulations of a connected, orientable and compact surface of a fixed topology, when the number of quadrangles tends to infinity, as well as the perimeters of the boundary faces. The limiting random metric spaces are called the (orientable) compact Brownian surfaces. This result generalises the known cases where the surface is either the sphere or the disk. We use an interpretation of a bijection by Chapuy-Marcus-Schaeffer (generalised by Bettinelli) as a way to decompose a quadrangulation along well-chosen geodesics into a finite number of elementary pieces whose scaling limits are obtained separately, by using in a crucial way the known results for the sphere and the disk, and their extensions to the non-compact settings of the Brownian plane and half-plane This paper presents the main objects and ideas of the proof, as well as some further properties of the law of the Brownian cylinder. This paper and its pictures are mostly based on joint work with Jérémie Bettinelli.

This is a review of research on the cover time and avoided points for the two-dimensional simple random walk.

In this note, we consider the directed first passage percolation introduced in [F. Comets, R. Fukushima, S. Nakajima and N. Yoshida: Journal of Statistical Physics, 161-(3), 577–597 (2015)]. It is proved that the shortest path from the origin to the $n$-th hyperplane makes a jump larger than a positive power of $\log n$. Some numerical results are also provided, which indicates that the maximal jump size is much larger in a certain parameter region.

In this notes, we study the large deviation principle for the dynamic model of three dimensional Young diagrams, having the jump rate originally given by [10]. We employ the level-set function approach discussed in [8]. Our argument is rather heuristic by relying on the assumption of local equilibrium and the local ergodicity for slope variables. We are planning to give rigorous proofs in [4] based on a different formulation due to height functions and dimer-cover picture as in [9].

In the previous study [6], the author provided an algebraic proof of the multicomponent commutator estimate in Besov spaces $C^{\alpha}=B_{\infty,\infty}^{\alpha}$ with $0 < \alpha < 1$. In this paper, we extend that result to general Besov spaces $B_{p,q}^{\alpha}$ with $p, q \in [1,\infty]$ and $0 < \alpha < 1$.

The $q$-Whittaker measure is a probability measure on the set of partitions. We show a Fredholm determinant formula for an expectation value with respect to this measure, which is the $q$-Laplace transform of the marginal distribution on the last element of the partition. Contrary to the typical approaches in integrable probability, our method does not focus on the $q$-moment generating function but explains the origin of its determinantal structure using Ramanujan's summation formula and the Frobenius determinant. This method can be applied to analyze on the fluctuations in the stationary situations in $q$-TASEP, higher spin exclusion process, and the directed polymer in the beta-distributed random environment.

This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a “geometrically canonical” Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of Möbius transformations on the Riemann sphere $\widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$), including the classical Apollonian gasket and some round Sierpiński carpets. The main result on Weyl's asymptotics for its eigenvalues is of the same form as that by Oh and Shah [Invent. Math. 187 (2012), 1–35, Theorem 1.4] on the asymptotic distribution of the circles in a very large class of such fractals.

Miller and Sheffield introduced the notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $\mathbb{C}$ and Gaussian free fields (GFFs) on them under the conformal equivalence. They considered the situation in which the conformal maps are given by a chordal Schramm–Loewner evolution (SLE). In the present paper, we construct GFF-valued processes on $\mathbb{H}$ (the upper half-plane) and $\mathbb{O}$ (the first orthant of $\mathbb{C}$) by coupling a GFF with a multiple SLE evolving in time on each domain. We prove that a GFF on $\mathbb{H}$ and $\mathbb{O}$ is locally coupled with a multiple SLE if the multiple SLE is driven by the stochastic log-gas called the Dyson model defined on $\mathbb{R}$ and the Bru–Wishart process defined on $\mathbb{R}_{+}$, respectively. We obtain pairs of time-evolutionary domains and GFF-valued processes.

We consider stochastic quantization associated with the weighted exponential quantum field model on the two-dimensional torus via a method of singular stochastic partial differential equations and show the tightness of the solutions to approximating equations of the stochastic quantization equation. If the model is not weighted, then the drift term of the stochastic quantization equation, which includes a renormalization term, is nonpositive or nonnegative. However, in the weighted case, generally the drift term is neither nonpositive nor nonnegative. We modify the argument in the case without weights and discuss the weighted model.

The non-random fluctuation is one of the central objects in first passage percolation. It was proved in [11] that for a particular asymptotic direction, it diverges in a lattice first passage percolation with an explicit lower bound. In this paper, we discuss the non-random fluctuation in Euclidean first passage percolations and show that it diverges in dimension $d \geq 2$ in this model also. Compared with the result in [11], the present result is proved for any direction and improves the lower bound.

We survey recent work that relates Pitman's transformation to a variety of classical integrable systems, including the box-ball system, the ultra-discrete and discrete KdV equations, and the ultra-discrete and discrete Toda lattice equations. It is explained how this connection enables the dynamics of the integrable systems to be initiated from infinite configurations, which is important in the study of invariant measures. In the special case of spatially independent and identically distributed configurations, progress on the latter topic is also reported.

Let ${\mathcal U}_{\delta}$ be the uniform spanning tree on $\delta \mathbb{Z}^{3}$. A spanning cluster of $\mathcal{U}_{\delta}$ is a connected component of the restriction of $\mathcal{U}_{\delta}$ to the unit cube $[0,1]^{3}$ that connects the left face $\{ 0 \} \times [0,1]^{2}$ to the right face $\{ 1 \} \times [0,1]^{2}$. In this note, we will prove that the number of the spanning clusters is tight as $\delta \to 0$, which resolves an open question raised by Benjamini in [4].

In this article we review the ideas in [Tsa18] toward proving the one-point, lower-tail large deviation principle for the Kardar–Parisi–Zhang equation.