The 12th MSJ-SI
The Mathematical Society of Japan, Seasonal Institute

Stochastic Analysis,
Random Fields and Integrable Probability

Program and Abstracts

July 31 (Wed.) – August 9 (Fri.), 2019
Lecture Theater 5, Shiiki Hall, Ito Campus, Kyushu University

This symposium is sponsored by the Mathematical Society of Japan (MSJ)and supported by Faculty of Mathematics, Institute of Mathematics for Industry and Stochastic Analysis Research Center at Kyushu University, as
well as by Japan Society for the Promotion of Science, Grant-in-Aid for:
Scientific Research (S)Scientific Research (A)Scientific Research (A)16H06338
17H01093
19H00643

PI: H. Osada (Kyushu University);
PI: T. Kumagai (Kyoto University);
PI: M. Hino (Kyoto University).

Scientific Committee
Hirofumi Osada
Yuzuru Inahama
Takashi Kumagai
Tomohiro Sasamoto
Tomoyuki Shirai
Hideki Tanemura(Chair, Kyushu University)(Kyushu University)(Kyoto University)(Tokyo Institute of Technology)(Kyushu University)(Keio University)

Local Organizers
Syota Esaki
Yuu Hariya
Masato Hoshino
Yuzuru Inahama
Masato Shinoda
Tomoyuki Shirai(Fukuoka University)(Tohoku University)(Kyushu University)(Kyushu University)(Nara Women’s University)(Kyushu University)

Timetable & Program (1st Week)
July 31
10:00-11:00
den Hollander
11:20-12:20
Corwin13:40-14:20
Funaki
14:30-15:10
Zhu
15:20-15:40
Hoshino16:10-16:50
Ghosh
17:00-17:20
Tsunoda18:00-20:00
WelcomeParty

August 1
August 2
10:00-11:00
10:00-11:00
Corwinden Hollander
11:20-12:20
11:20-12:20
Corwinden Hollander
Lunch
13:40-14:20
13:40-14:20
ImamuraRodriguez
14:30-15:10
14:30-15:10
HoldenBerestycki
15:20-15:40
15:20-15:40
Abe
Okada
Coffee Break
16:10-16:50
16:10-16:50
KatoriTsai
17:00-17:20
17:00-17:20
MaleckiLi

July 31 (Wed.)
9:00-

Registration

9:45-

Opening Speech
Tomohide Terasoma (President of MSJ)

10:00-11:00 Frank den Hollander (Universiteit Leiden)
Complex Networks: Structure and Functionality. I. Spectra
11:20-12:20 Ivan Corwin (Columbia University)
Stochastic Vertex Models I
13:40-14:20 Tadahisa Funaki (Waseda University)
Large Deviation for Lozenge Tiling Dynamics
14:30-15:10 Rongchan Zhu (Beijing Institute of Technology)
Stochastic Cahn-Hilliard Equation
15:20-15:40 Masato Hoshino (Kyushu University)
Paracontrolled Calculus and Regularity Structures
16:10-16:50 Subhroshekhar Ghosh (National University of Singapore)
Two Manifestations of Rigidity in Point Sets: Forbidden Regions andMaximal Degeneracy
17:00-17:20 Kenkichi Tsunoda (Osaka University)
Derivation of Viscous Burgers Equations from Weakly AsymmetricExclusion Processes
17:20-

Group Photo1

18:00-20:00 Welcome Party (Big Orange)
August 1 (Thu.)
10:00-11:00 Ivan Corwin (Columbia University)
Stochastic Vertex Models II
11:20-12:20 Ivan Corwin (Columbia University)
Stochastic Vertex Models III
13:40-14:20 Takashi Imamura (Chiba University)
Determinantal Structures in the q-Whittaker Measure
14:30-15:10 Nina Holden (ETH-ITS Zurich)
Cardy Embedding of Random Planar Maps
15:20-15:40 Yoshihiro Abe (Chiba University)
Exceptional Points of Two-Dimensional Random Walks at Multiples of the
Cover Time
16:10-16:50 Makoto Katori (Chuo University)
Gaussian Free Fields with Boundary Points, Multiple SLEs, and Log-Gases
17:00-17:20 Jacek Malecki (Wroclaw University of Science and Technology)
Universality Classes for General Random Matrix Flows
August 2 (Fri.)
10:00-11:00 Frank den Hollander (Universiteit Leiden)
Complex Networks: Structure and Functionality. II. Equivalence
11:20-12:20 Frank den Hollander (Universiteit Leiden)
Complex Networks: Structure and Functionality. III. Exploration
13:40-14:20 Pierre-François Rodriguez (Institut des Hautes Études
Scientifiques)
Isomorphism Theorems and the Sign Cluster Geometry of the Gaussian
Free Field
14:30-15:10 Julien Berestycki (The University of Oxford)
Branching Particle Systems with Selection and Free Boundary Problems
15:20-15:40 Izumi Okada (Kyushu University)
Exponents for High Points of Simple Random Walks in Two Dimensions
16:10-16:50 Li-Cheng Tsai (Rutgers University)
Lower-Tail Large Deviations of the KPZ Equation
17:00-17:20 Xinyi Li (The University of Chicago)
Natural Parametrization for the Scaling Limit of Loop-Erased Random
Walk in Three Dimensions

2

Timetable & Program (2nd Week)
August 5
August 6
10:00-11:00
10:00-11:00
LawlerAddario-Berry
11:20-12:20
11:20-12:20
Addario-Berry
LawlerLunch
13:40-14:40
13:40-14:20
LawlerSasada14:50-15:30
14:30-15:10
Shiraishi
Basu
15:4015:20-15:40
PosterNakajima
Presentations
Coffee Break
16:4016:10-16:50
Poster Session
Seo
18:30-20:30
BanquetAugust 7
10:00-11:00
Miermont
11:20-12:20
Addario-Berry
Excursion

August 8
August 9
10:00-11:00 10:00-11:00
Bufetov
Bufetov
11:20-12:20 11:20-12:20
Miermont
Bufetov
Lunch
13:40-14:40 13:40-14:20
Miermont
Kajino14:50-15:30 14:30-15:10
Sakai
Fukushima
Coffee Break
16:00-16:40
Kusuoka
16:50-17:30
Qiu

August 5 (Mon.)
10:00-11:00 Gregory F. Lawler (The University of Chicago)
Loop Measures and Loop-Erased Random Walk I
11:20-12:20 Louigi Addario-Berry (McGill University)
Algorithms and Random Trees I
12:20-

Group Photo

13:40-14:40 Gregory F. Lawler (The University of Chicago)
Loop Measures and Loop-Erased Random Walk II
14:50-15:30 Daisuke Shiraishi (Kyoto University)
Scaling Limit of Uniform Spanning Tree in Three Dimensions
15:40-

Poster (Short Presentations Followed by the Poster Session)

August 6 (Tue.)
10:00-11:00 Louigi Addario-Berry (McGill University)
Algorithms and Random Trees II
11:20-12:20 Gregory F. Lawler (The University of Chicago)
Loop Measures and Loop-Erased Random Walk III
13:40-14:20 Makiko Sasada (The University of Tokyo)
Generalized Pitman’s Transform and Discrete Integrable Systems
14:30-15:10 Riddhipratim Basu (Tata Institute of Fundamental Research)
Bigeodesics and Polymer Watermelons in Last Passage Percolation
15:20-15:40 Shuta Nakajima (Nagoya University)
Divergence of Non-Random Fluctuation in First-Passage Percolation3

16:10-16:50 Insuk Seo (Seoul National University)
The Cut-Off Phenomenon for the Random-Cluster Model
18:30-20:30 Banquet (Hyakunen-Gura)
August 7 (Wed.)
10:00-11:00 Grégory Miermont (École Normale Supérieure de Lyon)
Brownian Surfaces I
11:20-12:20 Louigi Addario-Berry (McGill University)
Algorithms and Random Trees III
Excursion (Karatsu)
August 8 (Thu.)
10:00-11:00 Alexander I. Bufetov (Aix-Marseille Université)
Determinantal Point Processes and Extrapolation I
11:20-12:20 Grégory Miermont (École Normale Supérieure de Lyon)
Brownian Surfaces II
13:40-14:40 Grégory Miermont (École Normale Supérieure de Lyon)
Brownian Surfaces III
14:50-15:30 Akira Sakai (Hokkaido University)
Critical Two-Point Function for Long-Range Models with Power-Law
Couplings: The Marginal Case for d ≥ dc
16:00-16:40 Seiichiro Kusuoka (Kyoto University)
Approach to the Quantum Field with Exponential Interactions by Singular
SPDEs
16:50-17:30 Yanqi Qiu (Chinese Academy of Sciences)
A Class of Positive Functions on the Circle and Their Applications on
Hankel Operators
August 9 (Fri.)
10:00-11:00 Alexander I. Bufetov (Aix-Marseille Université)
Determinantal Point Processes and Extrapolation II
11:20-12:20 Alexander I. Bufetov (Aix-Marseille Université)
Determinantal Point Processes and Extrapolation III
13:40-14:20 Naotaka Kajino (Kobe University)
The Laplacian on Some Self-Conformal Fractals and Weyl’s Asymptotics
for Its Eigenvalues
14:30-15:10 Ryoki Fukushima (Kyoto University)
Zero Temperature Limits for Directed Polymers in Random Environment
15:15-

Closing Speech
Hirofumi Osada (Chair of the Scientific Committee)
4Serial Lectures

Algorithms and Random Trees
Louigi Addario-Berry (McGill University)
I will present a variety of recent and ongoing work at the interface of
probabilistic analysis of algorithms and the study of random trees.
First, I will discuss an ongoing project on how certain recursive distributional equations can be solved by importing rigorous results on the convergence of approximation schemes for degenerate PDEs, from numericalanalysis.
Second, I will discuss joint work with Pascal Maillard on the algorithmic
hardness of finding low-energy states in the continuous random energy modelof Bovier and Kurkova.
Finally, I will also discuss one or both of (a), (b) and (c), where (a) isrecently completed work with Marie Albenque on symmetrization of random
trees and its application to convergence of random maps, (b) is an ongoingproject about eigenvalues of random trees, and (c) is joint work with Sanchayan Sen, on distributional convergence of the minimum spanning tree ofrandom 3-regular graphs.

6Determinantal Point Processes and Extrapolation
Alexander I. Bufetov (Aix-Marseille Université)
Point processes whose correlation functions are given by determinants
arise in the study of very different objects such as, for example, randommatrices (Dyson), fermions (Macchi), random series (Peres-Virag, Krishnapur), spanning trees (Burton-Pemantle, Benjamini-Lyons-Peres-Schramm),
Young diagrams (Baik-Deift-Johansson, Borodin-Okounkov-Olshanski), nonintersecting diffusions (Johansson, Osada-Tanemura), representations of infinitedimensional groups (Borodin-Olshanski).
For general determinantal point processes a rich general theory has been
built, which includes existence theorems (Macchi, Soshnikov, Shirai-Takahashi),description of Palm measures (Shirai-Takahashi, Lyons), rigidity (GhoshPeres), the Central Limit Theorem (Soshnikov) and its functional analogue(joint work with Dymov).
The course will start with an elementary introduction to determinantal
point processes, proceeding to recent developments and open problems.
For example, consider a Gaussian Analytic Function on the disk, that is,a random series whose coefficients are independent complex Gaussians. In
joint work arXiv:1612.06751 with Yanqi Qiu and Alexander Shamov, we showthat the zero set of a Gaussian Analytic Function is a uniqueness set for the
Bergman space on the disk: in other words, almost surely, there does not exist
a nonzero square-integrable holomorphic function having these zeros. The
key rôle in our argument is played by the determinantal structure of the zeros
due to Peres and Viràg. In general, we prove that the family of reproducing
kernels along a realization of a determinantal point process generates thewhole ambient Hilbert space, thus settling a conjecture of Lyons and Peres.
The key lemma in our argument is that the determinantal property ispreserved under conditioning on a fixed restriction of our configuration ina part of the phase space. In full generality, the explicit description of thekernels of these conditional measures remains an open problem.
It is nevertheless possible to write the analogue of the Gibbs property for
one-dimensional determinantal point processes with integrable kernels: forthese processes, conditional measures with respect to fixing the configurationin the complement of an interval is an orthogonal polynomial ensemble with
explicitly found weight (arXiv:1605.01400). Note here that a reproducing
kernel must have integrable form as soon as the corresponding Hilbert spacesatisfies a weak form of division property (joint work with Roman Romanov);in this case, our space must be a Hilbert space of holomorphic functions.
The proof of the Lyons-Peres conjecture raises the problem of extrapolating a function from its restriction to a realization of a determinantal point
process. In joint work with Yanqi Qiu (arXiv:1806.02306), extrapolation for
Bergman functions from zero sets of Gaussian Analytic Functions is obtainedusing the Patterson-Sullivan construction.

7

Stochastic Vertex Models
Ivan Corwin (Columbia University)
We will describe some recent advances in the study of stochastic vertex
models. Topics may include the Yang Baxter equation and Bethe ansatz,
Markov dualities, and asymptotics to various stochastic PDEs and the KPZ
fixed point. No prior knowledge of these subjects will be assumed.
Complex Networks: Structure and Functionality.
I. Spectra; II. Equivalence; III. Exploration
Frank den Hollander (Universiteit Leiden)
Lecture I considers inhomogeneous Erdős-Rényi random graphs on n vertices. We study the empirical spectral distribution of the adjacency matrix
An in the limit as n → ∞ in a regime that interpolates between sparse anddense. In particular, we show that the empirical spectral distribution of An
when properly scaled converges to a deterministic limit weakly in probability. For the special case where the connectivity probability between two
vertices has the product property, we give an explicit characterisation of thelimit distribution. The result is applied to statistical inference of sociability
patterns in social networks. (Based on joint work with Arijit Chakrabarty,
Rajat Hazra and Matteo Sfragara.)
Lecture II considers random graphs subject to topological constraints.
We compare two probability distributions on the set of simple graphs onn vertices induced by a given constraint: (1) The microcanonical ensemble,
where the constraint is hard, i.e., has to be satisfied for every realisation of thegraph; (2) The canonical ensemble, where the constraint is soft, i.e., has to besatisfied on average. We say that breaking of ensemble equivalence occurs in
the limit as n → ∞ when the relative entropy of the two ensembles per vertex(in the sparse regime), respectively, per edge (in the dense regime) is strictly
positive. We present two examples of constraints where breaking of ensemble
equivalence occurs, namely, when the constraint is on the degree sequence
and when the constraint is on the total number of edges and triangles. The
result is applied to model selection for real-world networks. (Based on joint
work with Diego Garlaschelli, Michel Mandjes, Andrea Roccaverde, Tiziano
Squartini and Nicos Starreveld.)
Lecture III considers the mixing time of random walks on random graphs.
Many real-world networks, such as WWW, are dynamic in nature. It is therefore natural to study random walks on dynamic random graphs. We consider
random walk on a random graph with prescribed degrees. We investigate
what happens when at each unit of time a fraction αn of the edges is randomly rewired, where n is the number of vertices. We identify three regimes
for the mixing time in the limit as n → ∞, depending on the choice of
αn . These regimes exhibit surprising behaviour. The results are relevant for
Google PageRank. (Based on joint work with Luca Avena, Hakan Guldas
and Remco van der Hofstad.)8

Loop Measures and Loop-Erased Random Walk
Gregory F. Lawler (The University of Chicago)
I will give an introduction to (discrete time, discrete space) loop measuresand the loop-erased random walk (LERW), along with the closely related
objects, uniform spanning trees and the Gaussian free field. The first lecture
will emphasize the discrete loop measures and will be combinatorial in nature;
the second will discuss the LERW in dimensions 2 through 4; and the third
will discuss conformal invariance and the scaling limits in two dimensions(Schramm-Loewner evolution, Brownian loop measure).
Brownian SurfacesGrégory Miermont (École Normale Supérieure de Lyon)
I will review the topic of random maps and its connection to classical objects of probability theory, namely random walks and peeling processes, andbranching processes arising in slice decompositions and layer decompositions.
The first lecture will focus on the “local” model of Boltzmann maps andwill comment on its relation to “non-local” models of maps, and will introduce peeling processes and some applications.
The second and third lectures will focus on slice decompositions, whichis, from the geometric point of view, a convenient way to understand the
Bouttier-di Francesco-Guitter bijection between maps and trees. We willshow how this decomposition can be used to show convergence of random
maps to Brownian surfaces in various topologies.
Time permitting, I will comment on the use by Curien-Le Gall of layer
decompositions to show robustness of these results when modifying locallythe distances, and how it is used by Carrance to show convergence of Eulerian
triangulations to the Brownian map.

9

Invited Talks (40 min.)Bigeodesics and Polymer Watermelons in Last Passage
Percolation
Riddhipratim Basu
(Tata Institute of Fundamental Research)
I shall describe two results for the exactly solvable model of exponential
last passage percolation that are connected by the common theme of understanding rarity of multiple disjoint geodesics across on-scale (i.e., n × n2/3 )rectangles. The first result shows that almost surely there does not exist
any non-trivial bi-infinite geodesic. In the second, we consider the “polymer
watermelon” given by the maximal weight collection of k disjoint geodesics
between origin and the point (n, n) and identify its fluctuation exponents in
k. I shall also discuss the role of integrable probability estimates in theseresults and some possible extensions if time permits.
Branching Particle Systems with Selection and Free Boundary
Problems
Julien Berestycki (The University of Oxford)
The N-Branching Brownian motion is a particle system in which independent particles move on the real-line as Brownian motions, branch at rate one
and the total population size is kept constant equal to N by removing theleftmost particle at each branching event. This model, which was introduced
by Brunet and Derrida to study certain noisy reaction-diffusion equations,can also be seen as a model the evolution of a population under selection. Its
hydrodynamic limit was shown recently to exist and satisfies a free boundary problem. I will present an analogous result in higher dimension (theso-called Brownian bees model) for which we can prove the existence of thehydrodynamic result, the global existence of the solution to the associatedfree boundary problem and describe the large time behaviour of the particle
system and of its deterministic limit.
This is based on joint works with E. Brunet, J. Nolen and S. Penington.
Zero Temperature Limits for Directed Polymers in Random
Environment
Ryoki Fukushima (Kyoto University)
In this talk, I discuss the zero-temperature limit for directed polymer
models. In many cases, the limit is simply the first/last passage percolation.
However, there are several natural settings where the entropy remains nondegenerate at zero-temperature. I explain a motivation to consider such asituation and present a few results on the continuity of the free energy in thezero-temperature limit.

11

Large Deviation for Lozenge Tiling Dynamics
Tadahisa Funaki (Waseda University)
We consider dynamics of lozenge tilings, or equivalently that of dimer
covers of honeycomb lattice, or three-dimensional Young diagrams. One canchoose the flip rate to satisfy the so-called gradient condition, which was
originally proposed by Luby, Randall and Sinclair. The hydrodynamic limitof this model was studied by Laslier and Toninelli (CMP, 2018) under theperiodic boundary condition. We discuss the corresponding large deviation
principle. This is joint work with Fabio Toninelli.
Two Manifestations of Rigidity in Point Sets: Forbidden Regions
and Maximal Degeneracy
Subhroshekhar Ghosh (National University of Singapore)
A point process is said to be “rigid” if its local observables are completely
determined (as deterministic functions of) the point configuration outside a
local neighbourhood. For example, it has been shown in earlier work that,in the Ginibre ensemble (a.k.a. the 2D Coulomb gas at inverse temperature
beta=2), the point configuration outside any bounded domain determines,
almost surely, the number of points in the domain.
In this talk, we will explore two recent manifestations of such rigidity
phenomena. For the zeros of the planar Gaussian analytic function, we showthat outside every large “hole”, there is a “forbidden region” which contains
a vanishing density of points. This should be seen in contrast with the corresponding situation for classically understood models (e.g. random matrix
ensembles), where no such effects are known to occur.
In the second part of the talk, we will consider “stealthy” hyperuniform
systems, which are systems whose structure function (i.e., the Fourier transform of the two-point correlation) vanishes near the origin. We show thatsuch systems exhibit “maximal degeneracy”, namely the points outside abounded domain determine, almost surely, the entire point configuration inside the domain. En route, we establish a conjecture of Zhang, Stillinger and
Torquato that such systems have (deterministically) bounded holes.
Based on joint works with Joel Lebowitz and Alon Nishry.
Cardy Embedding of Random Planar Maps
Nina Holden (ETH-ITS Zurich)
A random planar map is a canonical model for a discrete random surface
which is studied in probability theory, combinatorics, mathematical physics,
and geometry. Liouville quantum gravity is a canonical model for a random
2D Riemannian manifold with roots in the physics literature. In a joint workwith Xin Sun, we prove a strong relationship between these two naturalmodels for random surfaces. Namely, we prove that the random planar map
converges in the scaling limit to Liouville quantum gravity under a discrete
conformal embedding which we call the Cardy embedding. We also provethat the percolation interface on the triangulation converges to the SchrammLoewner evolution (SLE) with parameter 6 in a quenched sense.
12

Determinantal Structures in the q-Whittaker Measure
Takashi Imamura (Chiba University)
The q-Whittaker measures are probability measures on partition and are
written in terms of a product of two q-Whittaker functions. Since the introduction by Borodin and Corwin in 2011, it has been playing an importantrole in recent progresses in the integrable probability. Their interesting feature is that the q-Laplace transform of some marginal distributions can be
represented as a single Fredholm determinant, from which we can analyze
asymptotic behavior of interacting particle processes belonging to the KPZclass. The Fredholm determinant formula was first obtained by BorodinCorwin by using Macdonald difference operator. Since then, various techniques such as Markov duality, the Yang-Baxter equation etc, have beendeveloped to obtain the determinantal formulas for more generalized setting.
All these techniques focus on some mathematical structures of q-moments ofsome observables.
In this talk, I will talk on another approach to analyze the q-Whittakermeasures without relying on the q-moments. We utilize the Ramanujan summation formula and an elliptic version of the Cauchy determinant identity.
We also discuss some relations of the q-Whittaker measures to determinantal
point processes. This is the joint work with Matteo Mucciconi and Tomohiro
Sasamoto.

13

The Laplacian on Some Self-Conformal Fractals and Weyl’s
Asymptotics for Its Eigenvalues
Naotaka Kajino (Kobe University)
This talk will present the speaker’s recent results on the construction ofa “geometrically canonical” Laplacian on round Sierpiński carpets invariant
with respect to certain Kleinian groups (i.e., discrete groups of Möbius transb := C ∪ {∞}) and on Weyl’s asymptotics for its eigenvalues.
formations on Cb homeomorphic to theHere a round Sierpiński carpet refers to a subset of Cb consists of disjoint
standard Sierpiński carpet, such that its complement in Cb
open disks in C.
The construction of the Laplacian is based on the speaker’s preceding
study of the simplest case of the Apollonian gasket, the compact fractal subset
of C obtained from an ideal triangle (a triangle formed by mutually tangent
three circles) by repeating indefinitely the process of removing the interiorof the inner tangent circles of the ideal triangles. On this fractal, Teplyaev(2004) had constructed a canonical Dirichlet form as one with respect towhich the coordinate functions on the gasket are harmonic, and the author
later proved its uniqueness and discovered an explicit expression of it in termsof the circle packing structure of the gasket.
The expression of the Dirichlet form obtained for the Apollonian gasket infact makes sense on general circle packing fractals, including round Sierpiński
carpets, and defines (a candidate of) a “geometrically canonical” Laplacianon such fractals. When the circle packing fractal is the limit set (i.e., theminimum invariant non-empty compact set) of a certain class of Kleiniangroups, some explicit combinatorial structure of the fractal is known andmakes it possible to prove Weyl’s asymptotic formula for the eigenvalues ofthis Laplacian, which is of the same form as the circle-counting asymptotic
formula by Oh and Shah [Invent. Math. 187 (2012), 1–35].
The overall structure of the proof of Weyl’s asymptotic formula is thesame as in the case of the Apollonian gasket and is based on a serious application of Kesten’s renewal theorem [Ann. Probab. 2 (1974), 355–386] toa certain Markov chain on the “space of all possible Euclidean shapes” ofthe fractal. There is, however, a crucial difficulty in the case of a round
Sierpiński carpet; since it is infinitely ramified, i.e., the cells in its cellular decomposition intersect on infinite sets, it is highly non-trivial to show that the
principal order term of the eigenvalue asymptotics is not affected by the cellular decomposition, namely by assigning the vanishing Dirichlet boundary
condition on the boundary of the cells.
If time permits, a possible approach toward extensions to the case of selfconformal fractals in C consisting of nowhere rectifiable curves will also bementioned.

14

Gaussian Free Fields with Boundary Points, Multiple SLEs, and
Log-Gases
Makoto Katori (Chuo University)
A quantum surface (QS) (resp. an imaginary surface (IS)) is an equivalence class of pairs of simply commenced domains D ( C and the Gaussian
free fields (GFFs) on D with the free (resp. Dirichlet) boundary condition
induced by the conformal equivalence for random metric spaces. We define a
QS with N +1 marked boundary points (MBPs) and an IS with N +1 boundary condition changing points (BCCPs) on ∂D with N ∈ Z≥1 , in which thereal (resp. imaginary) part of the sum of C-valued logarithmic (2D Coulomb)
potentials arising from the MBPs (resp. BCCPs) is added to GFF in D. We
consider the situation such that the boundary points evolve in time as astochastic log-gas on ∂D and multiple random slits are generated in D by
the multiple Schramm–Loewner evolution (SLE) driven by that stochastic
log-gas. Then we cut the domain D along the SLE slits, restrict the GFF
on the resulting domain, and pull it back to D following the reverse flow of
the multiple SLE. We prove that if the log-gases on ∂D follow the stochastic
differential equations well-studied in random matrix theory (e.g., Dyson’s
Brownian motion model, the Bru–Wishart processes), and parameters areproperly chosen, then the coupled systems of GFFs and multiple SLE slits
provide stationary processes. The obtained random systems are used to solveinteresting geometric problems called the conformal welding problem and theflow line problem. The present study extends the previous results for a QS
with two MBPs reported by Sheffield and for an IS with two BCCPs by Miller
and Sheffield. This is a joint work with Shinji Koshida (Chuo University);see https://arxiv.org/abs/1903.09925.
Approach to the Quantum Field with Exponential Interactions by
Singular SPDEs
Seiichiro Kusuoka (Kyoto University)
We consider the stochastic quantization of the quantum field model withexponential interactions on the two-dimensional torus, which is called HoeghKrohn model. The model has been studied by quasi-regular Dirichlet forms.
In this talk, we study the model by singular stochastic differential equations,which is recently developed. By the method, we construct the time-global
solution and the invariant probability measure of the stochastic quantization,and see the relation to the process obtained by quasi-regular Dirichlet forms.
This is a joint work with Masato Hoshino and Hiroshi Kawabi.
A Class of Positive Functions on the Circle and Their
Applications on Hankel Operators
Yanqi Qiu (Chinese Academy of Sciences)
We propose a definition of hyper-positive functions on the circle related tostochastic processes on homogeneous trees. After giving a complete criterionof such functions, we are going to talk about their applications on inequalities
of Hankel operators. The talk is based on a joint work with Zipeng WANG.
15

Isomorphism Theorems and the Sign Cluster Geometry of theGaussian Free Field
Pierre-François Rodriguez(Institut des Hautes Études Scientifiques)
We consider the Gaussian free field (GFF) on a large class of transient
weighted graphs G, and prove that its sign clusters contain an infinite connected component. In fact, we show that the sign clusters fall into a regimeof strong supercriticality, in which two infinite sign clusters dominate (onefor each sign), and finite sign clusters are necessarily tiny, with overwhelming
probability. Examples of graphs G belonging to this class include cases inwhich the random walk on G exhibits anomalous diffusive behavior. Amongother things, our proof exploits a certain relation (isomorphism theorem)
relating the GFF to Sznitman’s random interlacements. Our findings also
imply the existence of a nontrivial percolating regime for the vacant set of
random interlacements on G and relate to certain disconnection problems
for random walks. Based on joint work with A. Prévost and A. Drewitz (U.
Köln).
Critical Two-Point Function for Long-Range Models with
Power-Law Couplings: The Marginal Case for d ≥ dc
Akira Sakai (Hokkaido University)
Consider the long-range models on Z d of random walk, self-avoiding walk,
percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as |x|−d−a for some a > 0. In the previous
work (Ann. Probab., 43, 639–681, 2015), we have shown in a unified fashion
for all a other than 2 that, assuming a bound on the “derivative” of then-step distribution (the compound-zeta distribution satisfies this assumed
bound), the critical two-point function Gpc (x) decays as |x|a∧2−d above theupper-critical dimension dc = (a ∧ 2)m, where m = 2 for self-avoiding walkand the Ising model and m = 3 for percolation.
In this talk, I will show in a much simpler way, without assuming a boundon the derivative of the n-step distribution, that Gpc (x) for the marginalcase a = 2 decays as |x|2−d / log |x| whenever d ≥ dc (with a large spreadout parameter L). This solves the conjecture in the previous work, extended
all the way down to d = dc , and confirms a part of predictions in physics(Brezin, Parisi, Ricci-Tersenghi, J. Stat. Phys., 157, 855–868, 2014). The
proof is based on the lace expansion and new convolution bounds on powerfunctions with log corrections.

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Generalized Pitman’s Transform and Discrete Integrable Systems
Makiko Sasada (The University of Tokyo)
The Korteweg-de Vries equation (KdV equation) and the Toda lattice are
typical and well-known classical integrable systems. For the KdV equation,the (almost-sure) well-posedness of a solution starting from a general ergodic
random field on R, such as the white noise, is still an open problem, though
the invariance of the white noise is known on T. On the other hand, for theinfinite Toda lattice, the invariance under the generalized Gibbs ensembles(GGE) are standard. Recently, Spohn shows that the generalized Gibbs freeenergy of the Toda chain is related to the β-ensembles of random matrix
theory in the mean-field regime and obtained an exact variational formulafor the density of states of the Lax matrix, when its matrix elements are
distributed according to some GGE.
In this talk, I will present our recent results on discrete time versionsof the KdV equation and the Toda lattice. First, I will introduce some
generalization of Pitman’s transform and show that the dynamics of several
discrete integrable systems, such as the discrete KdV equation, the ultradiscrete KdV equation, the discrete Toda lattice and the ultra-discrete Toda
lattice are given by them. We apply this observation to define the dynamics
uniquely on the infinite product space and study their invariant measures.
If time allows, I will also talk about the generalized Gibbs free energy ofthe discrete Toda chain and its connection to the Wishart β-ensembles ofrandom matrix theory.
This talk is mostly based on a joint work with David Croydon, Tsuyoshi
Kato and Satoshi Tsujimoto.
The Cut-Off Phenomenon for the Random-Cluster Model
Insuk Seo (Seoul National University)
In this presentation, we consider the random-cluster model which is a
generalization of the standard edge percolation model. For the randomcluster model on lattice, we prove that the Glauber dynamics exhibits aphenomenon known as the cut-off, especially for the very subcritical regime
for all dimensions. This is a joint work with Shirshendu Ganguly.
Scaling Limit of Uniform Spanning Tree in Three Dimensions
Daisuke Shiraishi (Kyoto University)
We will show that the properly rescaled three-dimensional uniform spanning tree converges weakly with respect to a Gromov-Hausdorff-Prohorovtype topology in a space whose elements are measured, rooted real trees
continuously embedded into Euclidean space. We will describe various properties of the intrinsic metrics, measures and embeddings of the limit in thisspace. This is based on a joint work with Omer Angel (UBC), David Croydon(Kyoto University) and Sarai Hernandez Torres (UBC).

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Lower-Tail Large Deviations of the KPZ Equation
Li-Cheng Tsai (Rutgers University)
Consider the solution of the KPZ equation with the narrow wedge initial condition. We prove the one-point, lower-tail Large Deviation Principle(LDP) of the solution, with time t → ∞ being the scaling parameter, andwith an explicit rate function. This result confirms existing physics predictions. We utilize a formula from Borodin and Gorin (2016) to convert the
LDP of the KPZ equation to calculating an exponential moment of the Airy
point process, and analyze the latter via stochastic Airy operator and Riccati
transform.
Stochastic Cahn-Hilliard Equation
Rongchan Zhu (Beijing Institute of Technology)
We study the asymptotic limit, as ε & 0, of solutions to the stochastic
Cahn-Hilliard equation:


1
ε
ε
ε
∂t u = ∆ −ε∆u + f (u ) + Ẇtε ,εwhere W ε = εσ W or W ε = εσ W ε , W is a Q-Wiener process and W ε issmooth in time and converges to W as ε & 0. In the case that W ε = εσ W ,we prove that for all σ > 12 , the solution uε converges to a weak solution toan appropriately defined limit of the deterministic Cahn-Hilliard equation.
In radial symmetric case we prove that for all σ ≥ 12 , uε converges to thedeterministic Hele-Shaw model. In the case that W ε = εσ W ε , we prove thatfor all σ > 0, uε converges to the weak solution to the deterministic limit
Cahn-Hilliard equation. In radial symmetric case we prove that uε converges
to deterministic Hele-Shaw model when σ > 0 and converges to a stochastic
model related to stochastic Hele-Shaw model when σ = 0.

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Invited Talks (20 min.)Exceptional Points of Two-Dimensional Random Walks at
Multiples of the Cover Time
Yoshihiro Abe (Chiba University)
I will talk about the local time of the continuous-time simple random
walk on a large finite subset of the two-dimensional lattice with the wired
boundary condition. The running time of the walk is parametrized by the
local time at the “boundary vertex” and approximated by a multiple of thecover time. The talk will focus on statistics of avoided points (those notvisited at all). This is based on joint work with Marek Biskup (UCLA).
Paracontrolled Calculus and Regularity Structures
Masato Hoshino (Kyushu University)
First we introduce the Bailleul-Hoshino’s result, which links the theory of
regularity structures with the paracontrolled calculus. As an application of
their result, we give another algebraic proof of the multicomponent commutator estimate, which is a general version of the Gubinelli-Imkeller-Perkowski’s
commutator estimate.
Natural Parametrization for the Scaling Limit of Loop-Erased
Random Walk in Three Dimensions
Xinyi Li (The University of Chicago)
In this talk, I will talk about some recent progress on loop-erased random
walk (LERW), a model that plays a central role in statistic physics. I will
focus on the case of three dimensions and talk about its convergence innatural parametrization to its scaling limit. This is joint work with Daisuke
Shiraishi (Kyoto).
Universality Classes for General Random Matrix Flows
Jacek Malecki(Wroclaw University of Science and Technology)
We consider matrix-valued processes described as solutions to stochastic differential equations of very general form. We study the family of theempirical measure-valued processes constructed for the corresponding eigenvalues. We show that the family indexed by the size of the matrix is tightunder very mild assumptions on the coefficients of the initial SDE. We characterize the limiting distributions of its sub-sequences as solutions to theintegro-differential equation. We use the result to study universality classesof random matrix flows. These generalize the classical results related toDyson Brownian motion and squared Bessel particle systems, but even inthese cases we show new phenomenons as the lack of uniqueness of solution
and existence of the generalized Marchenko-Pastur distributions supported
on the real line. Assuming the uniqueness of the solution to the integrodifferential equation we characterize the limiting process as a free diffusion
solving free SDE. This is based on a joint work with José Luis Pérez.
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Divergence of Non-Random Fluctuation in First-Passage
Percolation
Shuta Nakajima (Nagoya University)
We consider the First-passage percolation on Z d . There are two typesof fluctuations in First-passage percolation: the random fluctuation and thenon-random fluctuation. They are both important and well studied. Although there has been a significant progress of the upper bound due to thedevelopment of concentration inequalities, it is hard to get a non-trivial lower
bound because of the lack of techniques. In this talk, we discuss the divergence of the non-random fluctuation.
Exponents for High Points of Simple Random Walks in Two
Dimensions
Izumi Okada (Kyushu University)
We consider the favorite points of random walk, i.e. points where the local
time is a fraction of the maximum. These are conjectured to coincide withthe exponents for the numbers of pairs of late points and the one for highpoints of the Gaussian free field for which their exact values are known. We
estimate the exponents for the numbers of a multipoint set of late points,
favorite points and high points in average and in probability and verified thatthe conjecture is correct. In addition, we extend this result to the generaldomain which a simple random walk frequently visits.
Derivation of Viscous Burgers Equations from Weakly
Asymmetric Exclusion Processes
Kenkichi Tsunoda (Osaka University)
We consider in this talk the so-called incompressible limit for the weakly
asymmetric exclusion processes. A fundamental question in mathematicalphysics is the derivation of the master equation of fluids such as the Burgers equation or the Navier-Stokes equation. By the celebrated Varadhan’snongradient method, Esposito, Marra and Yau (1996) and Quastel and Yau(1998) derived the Navier-Stokes equation as the incompressible limits in dimensions strictly larger than 2. On the other hand, the derivation of theNavier-Stokes equation in low dimensions is achieved only from a stochastic
lattice gas dynamics with long jumps. In this talk, we discuss the incompressible limit for the weakly asymmetric exclusion processes via recent notableentropy estimates developed by Menezes and Jara. This talk is based on
joint work with Milton Jara (IMPA) and Claudio Landim (IMPA).

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List of Contributed Posters(as of July 25)

Eleanor Archer (The University of Warwick)“Brownian Motion on Stable Looptrees”
Sung-Soo Byun (Seoul National University)“Annulus SLE Partition Functions and Martingale-Observables”
Taiki Endo (Chuo University)“Three-Parametric Marcenko–Pastur Density”
Xiang Fang (National Central University)“Random Weighted Shifts”
Yueyuan Gao (AIST, Tohoku University)“Uniqueness of the Entropy Solution of a Stochastic Conservation Law witha Q-Brownian Motion”
Masayuki Kageyama (Nagoya City University)“A New Criteria of Risk in Markov Decision Processes”
Yoshinori Kamijima (Hokkaido University)“Mean-Field Behavior for the Quantum Ising Model”
Shinji Koshida (Chuo University)“Free Field Approach to the Macdonald Process”
Atef Lechiheb (The University of Tunis El Manar)
TBA
Matteo Mucciconi (Tokyo Institute of Technology)“Yang Baxter Field and Stochastic Vertex Models”
Takuya Murayama (Kyoto University)“Loewner Chains and Evolution Families on Parallel Slit Half-Planes”
Takuya Nakagawa (Ritsumeikan University)“Lα−1 Distance between Two One-Dimensional Stochastic Differential Equations Driven by a Symmetric α-Stable Process”
Shota Osada (Kyushu University)“Isomorphisms between DPPs and PPPs”
Yuki Tokushige (Kyoto University)“The Differentiability of the Speed of Biased RWs on Galton-Watson Trees”
Linjie Zhao (Peking University)“Scaling Limits for the Exclusion Process with a Slow Site”

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