Probability and Analysis on Random Structures
and Related Topics
RIMS Symposium, 8-10 August 2022
Research Institute for Mathematical Sciences, Kyoto University
Organisers: David Croydon (Kyoto University), Ryoki Fukushima (University of Tsukuba),
Naotaka Kajino (Kyoto University)

Monday 8th August

Tuesday 9th August

Wednesday 10th August

10:00–10:40 . . . . . Seiichiro Kusuoka

09:30–10:10 . . . . . . . Shuta Nakajima

09:30–10:10 . . . . . . . . Yuki Tokushige

10:50–11:30 . . . . . Reika Fukuizumi*

10:20–11:00 . . . . . . . . . Naoki Kubota

10:20–11:00 . . . . . . Daisuke Shiraishi

11:40–12:20 . . . . . . . Masato Hoshino

11:20–12:00. . . . Makoto Nakashima

11:20–12:00. . . . . . Takashi Kumagai

14:00–14:50 . . . . . . Stefan Neukamm

13:30–14:20. . . . . . Sebastian Andres

13:30–14:10 . . . . . . . . Makiko Sasada

15:00–15:25 . . . . . . Yutaka Takeuchi

14:30–14:55 . . . . . . . . . . . Takumu Ooi

14:20–15:00 . . . . . . . Hirofumi Osada

15:40–16:20 . . . . . . . . . . . . Akira Sakai

15:05–15:30 . . . . . Satomi Watanabe

16:30–17:10 . . . Tomohiro Sasamoto

15:50–16:30 . . . . . . Ryokichi Tanaka
16:40–17:20 . . . . . . . . . . Stefan Junk*∗ = online presentation

Sebastian Andres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First passage percolation with long-range correlations
Reika Fukuizumi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pattern formation in stochastic Swift-Hohenberg equation
Masato Hoshino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A semigroup approach to the multi-level Schauder estimate
Stefan Junk . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuations for the partition function of directed polymers beyond the L2 phase
Naoki Kubota . . . . . . . . . . . . Strict comparison for the Lyapunov exponents of the simple random walk in random potentials
Takashi Kumagai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Periodic homogenization of jump-type processes with drifts
Seiichiro Kusuoka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic quantization associated with the exp(Φ)2 -quantum field model
Shuta Nakajima . . . . . . . . . . . . . . . A variational formula for large deviations in first-passage percolation under tail estimates
Makoto Nakashima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A remark of the free energy for some disordered systems
Stefan Neukamm . . . . . . . . Large-scale Schauder estimate and applications to error estimates in stochastic homogenization
Takumu Ooi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence of time-changed α-stable processes by GMC
Hirofumi Osada . . . . . . . . . . . . . . . . . . The sub-diffusivity and diffusivity of tagged particles for interacting Brownian particles
Akira Sakai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spread-out limit of the critical points for various statistical-mechanics models
Makiko Sasada . . . . . . . . . . . . . . . . . On the stationary solutions of random polymer models and their zero-temperature limits
Tomohiro Sasamoto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuations of the log gamma polymer in half space
Daisuke Shiraishi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exceptional sets for simple random walks
Yutaka Takeuchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogenization of reflecting diffusions in a continuum percolation cluster
Ryokichi Tanaka . . . . . . . . . . Quasi-conformal measures and harmonic measures for hyperbolic groups: multifractal analysis
Yuki Tokushige . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On scaling limits of SRWs on long-range percolation clusters
Satomi Watanabe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat kernel fluctuations for the three-dimensional uniform spanning tree
https://www.kurims.kyoto-u.ac.jp/∼croydon/ProbAnalRandStruRelaTopi.html

1Sebastian Andres (University of Manchester)First passage percolation with long-range correlations
In this talk we consider first passage percolation (FPP) with passage times generated by ageneral class of models with long-range correlations, including discrete Gaussian free fields as
a prominent example. We will discuss conditions under which the associated time constant is
positive and the FPP distance is comparable to the Euclidean distance. We will also present
two applications to random conductance models (RCM) with possibly unbounded and strongly
correlated conductances, namely a Gaussian heat kernel upper bound for RCMs with a generalclass of speed measures, and an exponential decay estimate for the Green function of RCMs withrandom killing measures. This talk is based on a joined work with Alexis Prévost (Cambridge).
Reika Fukuizumi (Tohoku University)
Pattern formation in stochastic Swift-Hohenberg equation
We study a phenomenological model for pattern formations in the electroconvection, and theeffect of noise on the pattern. As such model we consider an anisotropic Swift-Hohenberg
equation adding an additive noise. We prove the existence of a global solution of that equationon the two dimensional torus. In addition, inserting a scaling parameter, we consider theequation on a large domain near its change of stability, we observe numerically that, under theappropriate scaling, its solutions can be approximated by a periodic wave, which is modulatedby the solutions to a stochastic Ginzburg-Landau equation. This is a joint work with Yueyuan
Gao, Guido Schneider, and Motomitsu Takahashi.
Masato Hoshino (Osaka University)
A semigroup approach to the multi-level Schauder estimate
We introduce a semigroup approach to prove the reconstruction theorem and the multi-level
Schauder estimate. These two theorems play important roles in the analytic part of the theoryof regularity structures. The first one is an application of the method by Otto and Weber, but
the second one is new. This approach allows us to treat semilinear/quasilinear parabolic SPDEs
with variable coefficients. This talk is based on a joint work with Ismaël Bailleul (Université
Rennes 1) and an ongoing work with Bailleul and Seiichiro Kusuoka (Kyoto University).
Stefan Junk (Tohoku University)
Fluctuations for the partition function of directed polymers beyond the L2 phase
The directed polymer model describes random paths under the influence of a space-time random environment. We consider the high-temperature, weak disorder regime in spatial dimensiond ≥ 3, where it is known that the polymer obeys a central limit theorem. Many author focus
on a (strict) sub-regime of the weak disorder phase, which is characterized by L2 -boundednessof an associated martingale (Wn )n . In contrast, we study the situation with unbounded second moments, specifically the spatial correlations of the field (Wnx )x , where the superscript x
indicates the starting location of the polymer. In the case of a bounded environment, we showthat a suitably re-centered spatial average over a set of diameter n1/2 converges to zero at rate
n−a+o(1) , with an explicit exponent that is different from the corresponding exponent in theL2 -bounded case.
2

Naoki Kubota (Nihon University)
Strict comparison for the Lyapunov exponents of the simple random walk in random potentials
We consider the simple random walk in i.i.d. nonnegative random potentials on the lattice.
In this model, the so-called Lyapunov exponent describes the “cost” paid by the walker for
traveling from the origin to a remote location in a landscape of potential. The Lyapunov
exponent is related to not only the traveling cost but also the large deviation principle. In
fact, the rate function is represented by the Lyapunov exponent. Hence, it is important tostudy the properties of the Lyapunov exponent, but there are still many unknown aspects. In
particular, there are a few results on the comparison between Lyapunov exponents for different
laws of potential. We focus on this topic, and prove the “strict” comparison between Lyapunov
exponents with different laws in some sense.
Takashi Kumagai (Waseda University)
Periodic homogenization of jump-type processes with drifts
We study homogenization problem for strong Markov processes on Rd having infinitesimal
generatorsZ
Lf (x) =f (x + z) − f (x) − h∇f (x), zi1{|z|≤1} k(x, z) Π(dz) + hb(x), ∇f (x)i, f ∈ Cb2 (Rd )RdRin periodic media, where Rd (1 ∧ |z|2 ) Π(dz) < ∞ and Π can be singular with respect to theLebesgue measure. Under a proper scaling, we show the scaled processes converge weakly toLévy processes on Rd . In particular, we completely characterize the limiting processes whenb(x) is bounded continuous, k(x, z) is non-negative bounded continuous and 1-periodic both forx and z, and in spherical coordinate
1{|z|>1} Π(dz) = 1{r>1} %0 (dθ)

dr
r1+α

with α ∈ (0, ∞) and %0 being any finite measure on the unit sphere. Different scaling limits
appear depending on the values of α. This talk is based on joint work with Xin Chen, Zhen-Qing
Chen and Jian Wang (Ann. Probab. 2021).
Seiichiro Kusuoka (Kyoto University)
Stochastic quantization associated with the exp(Φ)2 -quantum field model
We consider the stochastic quantization equations of the quantum field model with the exponential interactions on the two-dimensional torus. Under the full L1 -regime, we construct theunique time-global solution in the sense of singular stochastic partial differential equations, thestationary solution, and the Dirichlet form. Moreover, we obtain the identification of the processes obtained by the methods. This talk is based on jointworks with Masato Hoshino andHiroshi Kawabi.

3

Shuta Nakajima (Meiji University)
A variational formula for large deviations in first-passage percolation under tail estimates
We consider the first-passage percolation with identical and independent weight distributions. In
this talk, we discuss the upper tail large deviations under a tail assumption on the distributions.
We prove that the corresponding rate function is described by the so-called discrete p-capacity,
and we study its asymptotic. The talk is based on joint work with Clement Cosco and Florian
Schweiger.
Makoto Nakashima (Nagoya University)
A remark of the free energy for some disordered systems
In this talk, we focus on disordered pinning models and directed polymers in random environment(DPRE). It is known that disordered pinning models have the scaling limits in anintermediate regime when the distribution of the underlying renewal process has a polynomial
tail with exponent α ∈ ( 12 , 1). The same phenomena happens for 1 + 1 dimensional directed
polymers in random environment. Moreover, it has been proved that under a certain technicalcondition, their free energies have a universal asymptotic behavior in the weak coupling regime
which is explicitly given in terms of the scaling limits. Recently, the same asymptotic behavior
is obtained without the technical condition.
Stefan Neukamm (TU Dresden)
Large-scale Schauder estimate and applications to error estimates in stochastic homogenization
Since the seminal work by Avellaneda and Lin it is known that elliptic operators with periodic
coefficients feature – thanks to homogenization – the same regularity theory on large scales as
the Laplacian. In the talk we extend this observation to elliptic systems with stationary andergodic, random coefficients. In particular, we shall discuss large scale C 1,α estimates for such
systems. Next, we consider elliptic equations with correlated stationary coefficients. Using thelarge scale regularity theory, we establish error estimates with optimal scaling for the two-scale
expansion of stochastic homogenization.
Takumu Ooi (Kyoto University)
Convergence of time-changed α-stable processes by GMC
We consider Gaussian fields whose covariance kernels are 1-order green functions of α-stableprocesses on Rd and their Gaussian multiplicative chaos (GMC). It is known that α-stableprocesses converge weakly with Skorokhod’s topology as α → d and, by using the result of[Shamov, 2016], GMC converges weakly with the weak topology. In this talk, we will consider
the convergence of the time-changed processes of α-stable processes by GMC, and I will talkabout results to date (mainly results of the tightness).

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Hirofumi Osada (Chubu University)
The sub-diffusivity and diffusivity of tagged particles for interacting Brownian particles
The celebrated Nash theory for symmetric diffusions clarifies the relation of the time decay of
the transition probability density of diffusion with translation-invariant measure and the Nashinequality. If the Nash inequality holds, then the diffusion is diffusive. However, there exists no
analogy for infinite-dimensional space.
In this talk, we present a criterion for sub-diffusivity of tagged particles of the stochastic
dynamics associated with the translation-invariant random point fields. For this we introduce
the concept of decomposability for random point fields. We prove that the tagged particles are
sub-diffusive if the random point field is decomposable. The decomposability follows from thenumber rigidity of random point fields. This result implies the tagged particles of the Ginibre
interacting Brownian motion and the diffusions related to the planner Gaussian analytic function
are sub-diffusive. We also show a sufficient condition of diffusivity in terms of reduced Palm
measures.

Akira Sakai (Hokkaido University)
Spread-out limit of the critical points for various statistical-mechanics models
Self-avoiding walk and lattice trees/animals are statistical-mechanics models for (linear or
branched) polymers in a good solvent. We define these models in Zd with the spread-out
edge set {{x, y} ⊂ Zd : 0 < kx − yk ≤ L}. The statistical properties of each model are encodedin the generating function with fugacity p/∆, where ∆ = O(Ld ) is the degree of a vertex. We
are interested in the spread-out limit (L ↑ ∞) of the model-dependent critical point pc .
With Remco van der Hofstad, I was able to show in [2] that pc = 1 + CL−d + O(L−d−1 )for self-avoiding walk, percolation, oriented percolation (in d + 1 space-time dimensions) andthe contact process above their respective upper-critical dimension dc (e.g., = 4 for self-avoidingwalk), where the model-dependent constant C can be represented in terms of continuous-space
random walks:
∞
X



U ∗n (o)[self-avoiding walk, the contact process],




n=2


∞
 X
1
C=
U ∗2n (o)[oriented percolation],

2


n=2

∞

X

n + 1 ∗n

∗2

U
(o)
+
U (o) [percolation],


2
n=3

where U ∗n is the n-fold convolution in Rd of the uniform probability distribution U on {x ∈Rd : kxk ≤ 1}. (A similar result was proven in [1] for long-range oriented percolation in
spatial dimensions d > 2(α ∧ 2), where the bond-occupation probability for an oriented edge((x, t), (y, t + 1)) is proportional to (|x − y| ∨ L)−d−α with α > 0.) Recently, Noe Kawamoto andI were able to show in [3] that pc = 1/e + CL−d + O(L−d−1 ) for lattice trees (LT) and latticeanimals (LA) above the common upper-critical dimension dc = 8, where
CLT =

∞
X
n+1
n=2

2e

∞

U

∗n

1 X ∗n
CLA = CLT − 2
U (o).
2e

(o),

n=3

At the workshop, I will explain the key ideas for the proof of the above results andpotential future work on those models on random conductors.
5[1] L.-C. Chen and A. Sakai. Critical behavior and the limit distribution for long-range oriented
percolation. I. Probab. Theory Relat. Fields, 142 (2008): 151–188.
[2] R.v.d. Hofstad and A. Sakai. Critical points for spread-out self-avoiding walk, percolationand the contact process above the upper critical dimensions. Probab. Theory Relat. Fields,
132 (2005): 438–470.
[3] N. Kawamoto and A. Sakai. Spread-out limit of the critical points for lattice trees andlattice animals in dimensions d > 8. Preprint: arXiv:2205.09451.

Makiko Sasada (University of Tokyo)
On the stationary solutions of random polymer models and their zero-temperature limits
In this talk, I will discuss stationary measures for positive-temperature random polymer models and their zero-temperature limits. In particular, we derive new stationary measures forthe zero-temperature limit of the beta random polymer, that has been called the river delta
model. To do this, we apply techniques developed for understanding the stationary measuresof the corresponding positive-temperature random polymer models, highlighting how the maps
underlying these systems can each be reduced to one of two basic bijections, and that throughan ‘independence preservation’ property, these bijections characterize the associated stationary
measures. We then derive similar descriptions for the corresponding zero-temperature maps,
whereby each is written in terms of one of two bijections. I will also show that our systematic
approach helps clarify why particular stationary measures arise for the different random polymer models considered, and the connections between them. This talk is based on the joint workwith David Croydon.
Tomohiro Sasamoto (Tokyo Institute of Technology)
Fluctuations of the log gamma polymer in half space
Log gamma polymer, introduced by Seppalainen in 2009, is one of the most well studied modelof a directed polymer in random media. It has a natural statistical mechanical interpretation
at positive temperature and admits exact analysis. While previous studies mostly focus on thewhole space version, there have been much less results for its half-space version.
Recently we discovered a direct connection between solvable models in the KPZ classand free fermionic models at positive temperature (or determinant point processes). The key
ingredient in our theory is a new identity between marginals of the q-Whittaker measure andthe periodic Schur measure, which is proved in bijective fashion by substantially generalizing
the RSK algorithm. Once the connection is established, subsequent analysis becomes rather
standard and it allows to study various half-space models by taking certain limits. In this talkwe present a Fredholm Pfaffian formula for log gamma polymer in half-space and establish thelimit theorems.
The talk is based on collaborations with Takashi Imamura and Matteo Mucciconi [1].
[1] T. Imamura, M. Mucciconi, T. Sasamoto, Solvable models in the KPZ class: approachthrough periodic and free boundary Schur measure, arXiv: 2204.0842.

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Daisuke Shiraishi (Kyoto University)
Exceptional sets for simple random walks
I will discuss several exceptional sets for simple random walk (SRW) paths: cut points, looperased random walk and outer boundary of 2D SRW. I will show that the renormalized counting
measure for these sets converges as the lattice spacing goes to zero. This is joint work with
Yifan Gao, Xinyi Li and Petr Panov.
Yutaka Takeuchi (Keio University)
Homogenization of reflecting diffusions in a continuum percolation cluster
Homogenization is one of the important problems of random media. After the 2000s, many
researchers showed quenched results. For instance, quenched invariance principle for random
walks in random conductance models and diffusions in random environments were shown. In
these settings, quenched local central limit theorems are also considered. In this talk, we
consider reflecting diffusions in a continuum percolation cluster built over stationary ergodic
point processes. Under some geometrical conditions, we obtain a quenched invariance principle
and a local central limit theorem.
Ryokichi Tanaka (Kyoto University)
Quasi-conformal measures and harmonic measures for hyperbolic groups: multifractal analysis
We describe the multifractal structure of measures at infinity for hyperbolic groups. Joint workwith Steve Cantrell (Chicago).
Yuki Tokushige (Kyushu University)
On scaling limits of SRWs on long-range percolation clusters
In this talk, we will discuss a Long-Range Percolation (LRP) on Zd , which is a variant of theclassical nearest-neighbor Bernoulli percolation. In particular, we consider a problem concerninga SRW on the infinite cluster of a LRP. In the paper published in 2013, Crawford and Sly provedthat a SRW on the infinite cluster scales to α-stable process. However, they completed the proof
only when α ∈ (0, 1). It is because the remaining case involves a technical problem to do witha control of short jumps of a SRW, which is related to the fact that sample paths of α-stableprocess have infinite variations for α ∈ [1, 2).
In this talk, we will give a brief overview of this subject and explain our idea to overcome
the technical issue depicted above and complete the proof of the case α ∈ [1, 2). This talk isbased on a joint work with Noam Berger (Technical University of Munich).
Satomi Watanabe (Kyoto University)
Heat kernel fluctuations for the three-dimensional uniform spanning tree
Heat kernel fluctuations have been observed in some random walks in random environments.
In this talk, I will consider the three-dimensional uniform spanning tree (UST) as a random
environment. I will first show the occurrence of log-logarithmic fluctuations around the leading
order for the volume of intrinsic balls in the UST. This is then used to obtain similar fluctuations
for the quenched heat kernel.
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