VIIIth symposium

Stochastic Analysis onLarge Scale Interacting Systems

Oct. 7 (Wed) — Oct. 9 (Fri), 2009Graduate School of Mathematical Sciences,

University of Tokyo

VIII S A L S I S

DATE ： Oct. 7 (WED) — Oct. 9 (FRI), 2009PLACE ： Graduate School of Mathematical Sciences, University of TokyoORGANIZERS： Tadahisa F (Univ. of Tokyo)

Hirofumi O (Kyushu Univ.)Yoshiki O (Shinshu Univ.)

This symposium is supported by

• Global COE Program: The research and training center for new development in mathe-matics, University of Tokyo.

• Japan Society for the Promotion of Science, KAKENHI (18204007) Grant-in-Aid for Scien-tific Research (A), HI: T. Funaki.

• Japan Society for the Promotion of Science, KAKENHI (21340031) Grant-in-Aid for Scien-tific Research (B), HI: H. Osada.

Oct. 7 (WED)

10:20–11:05 (45) Hideki T (Chiba Univ.)Markov Property of Dyson’s Model with an Infinite Number of Particles

11:15–12:00 (45) Makoto K (Chuo Univ.)Zeros of entire functions and relaxation processes

13:15–14:15 (60) Jean-Dominique D (Technische Universitat Berlin)Scaling limits of (1 + 1)-dimensional pinning models with Laplacian interaction

14:30–15:15 (45) Hirofumi O (Kyushu Univ.)Tagged particles of interacting Brownian motions with the 2D Coulomb potentialand the stochastic domination of the Ginibre random point field

15:15–15:45 (30) Coffee Break

15:45–16:10 (25) Daisuke S (Kyoto Univ.)Exact value of the resistance exponent for four dimensional random walk trace

16:10–16:35 (25) Bin X (Shinshu Univ.)Impulsive noise driven fractional partial differential equations

16:45–17:10 (25) Yoshiko O (Univ. of Tokyo)Large deviations in quantum spin chains

17:10–17:35 (25) Yoshiki O (Shinshu Univ.)Recurrence theorem and ergodicity of quantum dynamics

Oct. 8 (THU)

9:30–9:55 (25) Kouji Y (Kobe Univ.)Cameron–Martin formula for σ-finite measure unifying Brownian penalisations

9:55–10:20 (25) Makoto N (Kyoto Univ.)On the behavior of the population density of branching random walks

10:30–11:15 (45) Akira S (Hokkaido Univ.)Asymptotic behavior of the gyration radius for long-range self-avoiding walk andlong-range oriented percolation

11:20–11:45 (25) Jun M (Univ. of Tokyo)Random walks on two dimensional continuum percolation clusters

13:15–14:00 (45) Tomohiro S (Chiba Univ.)On the maximum of Dyson Brownian motion

14:10–14:55 (45) Norio K (Yokohama National Univ.)Localization of inhomogeneous coined quantum walks on the line

14:55–15:30 (35) Coffee Break

15:30–15:55 (25) Masahiro K (Univ. of Tokyo)Exact partition function of the zero-range process and expectation values in thethermodynamic limit

15:55–16:20 (25) Hiroshi W (Univ. of Tokyo)Huge-scale molecular dynamics simulation on gas-liquid multi-phase flow

16:30–17:30 (60) Short Communications

Oct. 9 (FRI)

9:30–9:55 (25) Kenshi H (Ritsumeikan Univ.)An alternative condition for stochastic domination

9:55–10:20 (25) Masato T (Osaka Electro-Communication Univ.)2D Ising percolation near critical external fields

10:30–11:15 (45) Nobuo Y (Kyoto Univ.)Power Law Fluids with Random Forcing

11:20–11:45 (25) Hironobu S (Keio Univ.)Confinement of the two dimensional discrete Gaussian free field between twohard walls

13:15–13:40 (25) Yukio N (Osaka Univ.)Spectral gap for multi-species exclusion processes

13:45–14:30 (45) Takashi K (Kyoto Univ.)Convergence of discrete Markov chains to jump processes

14:30–15:00 (30) Coffee Break

15:00–15:25 (25) Tomoyuki S (Kyushu Univ.)Ginibre-type determinantal processes

15:25–15:50 (25) Makiko S (Univ. of Tokyo)Hydrodynamic limit and fluctuations for an evolutional model of 2D Young dia-grams

16:00–17:00 (60) Claudio L (Instituto Nacional de Mathematica Pura e Aplicada)Metastability of reversible condensed zero range processes on a finite set

Markov Property of Dyson’s Modelwith an Infinite Number of Particles

Hideki Tanemura, Chiba University(joint work with Makoto Katori, Chuo University)

We consider the process Ξ(t) =∑N

j=1 δXj(t) with the SDEs

dXj(t) = dBj(t) +∑

1≤k≤N,k =j

dt

Xj(t) − Xk(t), 1 ≤ j ≤ N, t ∈ [0,∞),

where Bj(t)’s are independent one-dimensional standard Brownian motions. The state spaceof the process is the space of nonnegative integer-valued Radon measures M on R, whichis a Polish space with the vague topology. In [2] we gave sufficient conditions for an initial

configuration ξ =∞∑

j=1

δxj∈ M such that the process (Pξ∩[−L,L], Ξ(t)) converges to an M-

valued process, say (Pξ, Ξ(t)), as L → ∞, in the sense of finite dimensional distributions,

where ξ ∩ [−L,L] is the restriction of ξ on [−L, L], i.e. ξ ∩ [−L, L] =∑

j:xj∈[−L,L]

δxj.

In this talk we discuss the following:

1. Thightness of the sequence of the processes (Pξ∩[−L,L], Ξ(t)), L ∈ N.

2. Markov property of the limit process (Pξ, Ξ(t)).

References

[1] Katori, M., Tanemura, H.: Noncolliding Brownian motion and determinantal processes.J. Stat. Phys. 129, 1233-1277 (2007)

[2] Katori, M., Tanemura, H.: Non-equilibrium dynamics of Dyson’s model with aninfinite number of particles. Commun. Math. Phys. DOI:10.1007/s00220-009-0912-3;arXiv:math.PR/0812.4108.

[3] Katori, M., Tanemura, H.: Zeros of Airy function and relaxation process. J. Stat. Phys.DOI:10.1007/s10955-009-9829-7; arXiv:math.PR/0906.3666.

ZEROS OF ENTIRE FUNCTIONS AND RELAXATION PROCESSES

MAKOTO KATORI (Chuo University)katori@phys.chuo-u.ac.jp

We show in talk three infinite particle systems on R exhibiting typical non-equilibrium dynamics, relaxation phenomena. The first one is Dyson’s model withβ = 2, starting from a configuration in which every point of Z (the zero of sin(πz)) isoccupied by one particle and converging to the determinantal point process (DPP)with the sine kernel. The second one is a kind of drift transformation of Dyson’smodel with β = 2, starting from a configuration in which every zero of the Airyfunction Ai(z) on the negative R is occupied by one particle and converging to theDPP with the Airy kernel. The last one is the noncolliding squared Bessel processeswith index ν > −1, starting from the configuration in which every point (jν,i)

2

is occupied by one particle, i ∈ N, and converging to the DPP with the Besselkernel. Here jν,i denotes the i-th positive zero of the Bessel function Jν(z). Inthe random matrix theory, these three DPPs are obtained in the scaling limits ofeigenvalue distributions of the Gaussian random matrices called the bulk, the soft-edge, and the hard-edge scaling limit, respectively. In our relaxation processes, theyare attractors of dynamics, and the equilibrium dynamics in them are realized inthe long time limits. This is a joint work with Hideki Tanemura (Chiba University).

[1] Katori, M., Tanemura, H.: Non-equilibrium dynamics of Dyson’s model with aninfinite number of particles. Commun. Math. Phys. DOI:10.1007/s00220-009-0912-3; arXiv:math.PR/0812.4108.[2] Katori, M., Tanemura, H.: Zeros of Airy function and relaxation process. J.Stat. Phys. DOI:10.1007/s10955-009-9829-7; arXiv:math.PR/0906.3666.

SCALING LIMITS OF (1 + 1)-DIMENSIONAL PINNINGMODELS WITH LAPLACIAN INTERACTION

JEAN-DOMINIQUE DEUSCHEL

We consider a random field f : 1, ..., N → R with Laplacian interac-tion of the form

∑i V (∆f(i)), where ∆ is the discrete Laplacian and

the potential V (·) is symmetric and uniformly strictly convex. Thepinning model is defined by giving the field a reward ε ≥ 0 each time ittouches the x-axis, that plays the role of a defect line. It is known thatthis model exhibits a phase transition between a delocalized regime(ε < εc), and a localized one (ε > εc) where 0 < εc < ∞.

We give a a precise pathwise description of the transition, extractingthe full scaling limits of the model. At the critical regime (ε = εc) weshow that field suitably rescaled converges in distribution towards thederivative of a symmetric Levy process of index 2/5.

This is a joint work with Caravenna.

1

TAGGED PARTICLES OF INTERACTING BROWNIANMOTIONS WITH THE 2D COULOMB POTENTIAL

AND THE STOCHASTIC DOMINATION OF THE GINIBRERANDOM POINT FIELD

HIROFUMI OSADA

We prove the tagged particles of the interacting Brownian motions in R2 withthe 2D Coulomb potential Ψ(x) = −2 log |x| (x ∈ R2) is sub-diffusive. We alsogive an explicit formula of the density of the stochastic domination of the Ginibrerandom point field.

The Ginibre random point field µ is a probability measure on the configurationS over R2. It is known that µ is translation and rotation invariant. Moreover, µis so called a determinantal random point field whose n-correlation function ρn isgiven by

ρn(x1, . . . , xn) = det[K(xi, xj)]1≤i,j≤n, (1)

where K :R2 × R2→C is the kernel defined by

K(x, y) =1π

exp(−|x|2

2− |y|2

2) · exy. (2)

Here we identify R2 as C by the obvious correspondence: R2 3 x = (x1, x2) 7→x1 +

√−1x2 ∈ C, and y = y1 −√−1y2 means the complex conjugate under this

identification.Intuitively, µ is a measure interacting 2D Coulomb potentials Ψ defined by

Ψ(x) = −2 log |x| (x ∈ R2). (3)

We remark that the DLR equation for µ does not make sense. However, we willprove that µ is the reversible measure of the unlabeled diffusion Xt =

∑i∈N δXi

t,

where the associated labeled dynamics Xt = (Xit) ∈ (R2)N is the solution of the

infinitely dimensional SDE:

dXit = dBi

t + limr→∞

∑

|Xit−Xj

t |<r

Xit −Xj

t

|Xit −Xj

t |2dt (X0 = (xi)i∈N). (4)

We remark the unlabeled diffusion Xt is translation and rotation invariant. By (4)one can say µ is a measure with 2D Coulomb interaction potential Ψ.

Theorem 1. There exists a set S ⊂ (R2)N such that µ({∑i∈N δxi ;x = (xi) ∈ S}) =1 and that (4) has a solution for all initial points x = (xi)i∈N ∈ S. Moreover, forall initial points x ∈ S,

P (Xt ∈ S ∩ Ssingle for all t) = 1.

Here Ssingle = {s = (si) ∈ (R2)N ; si 6= sj if i 6= j}. More precisely, there exist(R2)N-valued process X = (Xi)i∈N and Brownian motion B = (Bi)i∈N such thatthe pair (X,B) satisfies the SDE (4).

The key point of Theorem 1 is to calculate the log derivative of the one momentmeasure µ1 of µ and to introduce a coupling of an infinite system of Dirichletspaces describing (4). Here µ1 is the measure on R2 × S such that µ1(A × B) =

1

2 HIROFUMI OSADA

∫A

ρ1(x)µx(B)dx = (1/π)∫

Aµ0(B), where µ0 is the Palm measure of µ conditioned

at x.

Theorem 2. Let α be the self-diffusion matrix of Ginibre interacting Brownianmotion (4). Then α = 0.

We remark (4) has only repulsive interaction; there are no center force. If theinteraction is of Ruelle class and d ≥ 2, then α is always non degenerate. This wasproved mathematically when the particle have convex hard cores [O. 98, PTRF].So the result above caused by the strength of the long range part of the interactionpotential Ψ.

A key point of the proof is a µ-almost sure equality on functions of the configu-ration space. We also use the invariant principle and, moreover, the necessary andsufficient condition for the non degeneracy of the limit coefficients under diffusivescaling obtained in [O. 98, IHP].

The second topic is the stochastic domination of the Ginibre random point fieldµ. By construction µ is a probability measure on S. We construct a measure ν onR2 × S such that

ν ◦ ι−1 = µ. (5)

Here ι :R2 × S→ S defined by ι((x,∑

i δyi)) = δx +∑

i δyi . The existence of ν isproved by Goldman. He also proved the marginal distribution of the first componentof ν is the Gaussian measure on R2, that is, a probability measure!. We remark theexistence of such a probability marginal never happens in the case of translationinvariant Ruelle’s class Gibbs measure. So, as well as the sub diffusivity of thetagged particles, it reflects the strength at infinity of the 2D Coulomb potentialpotential. We give here an explicit formula of the density dν/dx× µ0.

Exact value of the resistance exponent for four dimensional

random walk trace

Daisuke Shiraishi

Department of MathematicsFaculty of ScienceKyoto University

daisuke@math.kyoto-u.ac.jp

Abstract

Let S be a simple random walk starting at the origin in Z4. We consider G = S[0,∞) to bea random subgraph of the integer lattice and assume that a resistance of unit 1 is put on eachedge of the graph G. Let Rn be the effective resistance between the origin and Sn. We derive the

exact value of the resistance exponent; more precisely, we prove that n−1E(Rn) ≈ (log n)−12 .

Furthermore, we derive the precise exponent for the heat kernel of a random walk on G at thequenched level. These results give the answer to the problem raised by Burdzy and Lawler(1990) in four dimensions.

References

[1] Burdzy, K.; Lawler, G. F. : Rigorous exponent inequalities for random walks. Journal of PhysicsA: Mathematical and General, Volume 23, Issue 1, pp. L23-L28 (1990).

1

Impulsive noise driven fractional partial differentialequations

Bin XIEInternational Young Researchers Empowerment Center, Shinshu University

We intend to consider a stochastic fractional differential equation driven by animpulsive noise, which is initially introduced by Z. Peszat and J. Zabczyk in 2005 andis singular not only in time but also in space. We will first study the existence anduniqueness of solutions and then investigate the regularities of solution, especiallyin its space variable which depends on the order of the fractional operator. Ourresults deeply rely on precise analysis of the kernel generated by our operator.

1

LARGE DEVIATIONS IN QUANTUM SPIN CHAINS

YOSHIKO OGATA

A quantum spin chain is a noncommutative extension of a classicalspin chain. It has some additional freedom about the choice of observ-ables. In this talk, I will explain about large deviations in quantumspin chains. The large deviation in classical spin chain can be seen asa special case.

1

RECURRENCE THEOREM AND ERGODICITYOF QUANTUM DYNAMICS

YOSHIKI OTOBE

It is quite well-known that Hamiltonian flows in classical mechanics confined toa finite volume in phase space will eventually return to its initial state, namely,denoting the flow by Tt, we have lim inft→∞ d(Tt(x), x) = 0 for almost every x.This is a typical consequence of the Poincare recurrence theorem, and the celebratedLiouville theorem which asserts that every Hamiltonian flow preserves the Lebesguemeasure. We note here that a linear motion of constant speed clearly never returnsto the initial state, which means that the confining property is essential in this case.

It seems natural to try to extend this classical result to dynamics in quantummechanics. To our knowledge, Bocchieri and Loinger [BL57] are the first peoples inthis direction: Let Ψ(t) be a wave function evolving in time t under a HamiltonianH which has only discrete spectra. Then for each ε > 0 there exists a T > 1 suchthat ‖Ψ(T ) − Ψ(0)‖ < ε. The assumption that the Hamiltonian has only discretespectra seems natural; it is a reflection of the confining property of the classicalcase. It is also easy to prove that a Schrodinger flow generated by a Hamiltonianwith an absolutely continuous spectrum never returns to its initial state by virtueof the Riemann–Lebesgue theorem (due to Sasaki[Sas09]).

We point out here that there is another approach to the quantum recurrence the-orem; see e.g. [Duv02]. Their aim is to formulate the quantum recurrence theoremsimilarly to the classical one. They formulated the theorem in terms of C∗-algebraand obtained a quantum analogue of the Liouville theorem.

We will here give yet another approach which is based on a standard measuretheory (elementary probability theory). It will be a simple and entirely direct ex-tension of classical settings and it also enables us to formulate an ergodic propertyof the quantum dynamics. To do so we in addition need a slight stronger assump-tion of the spectra of the Hamiltonian and a little infinite dimensional analysis(stochastic analysis in a narrow sense).

Finally we note that Hida has studied the ergodicity of unitary (especially shift)operators on L2-space after extending it to an operator on S ′[Hid02]. Our situationis different from his.

References

[BL57] P. Bocchieri and A. Loinger, Quantum recurrence theorem, Phys. Rev. 107 (1957), no. 2,337–338.

[Duv02] Rocco Duvenhage, Recurrence in quantum mechanics, Intern. J. Theor. Phys 41 (2002),no. 1, 45–61.

[Hid02] Takeyuki Hida, White noise and functional analysis, vol. 60, SEMINAR on PROBABIL-ITY, 2002, (in Japanese).

[Sas09] Itaru Sasaki, A remark on Poincare’s recurrence theorem in quantum systems, Privatememorandom, 2009.

1

Cameron–Martin formula for σ-finite measure

unifying Brownian penalisations

Kouji Yano (Kobe University)

Quasi-invariance under translation is established for the σ-finite measureunifying Brownian penalisations, which has been introduced by Najnudel,Roynette and Yor ([2] and [3]). For this purpose, the theory of Wienerintegrals for centered Bessel processes, due to Funaki, Hariya and Yor ([1]),plays a key role.

Let {(Xt), (Ft), W} denote the canonical representation of 1-dimensionalBrownian motion starting from the origin. The measure mentioned above isdefined as follows:

W =

∫

∞

0

du√2πu

(

Π(u) • R)

(1)

where Π(u) stands for the law of Brownian bridge from 0 to 0 of length u, andR for the law of symmetrized 3-dimensional Bessel process. Now the maintheorem is stated as follows.

Theorem 1 ([5],[4]). Suppose that ht =∫

t

0f(s)ds with f ∈ L2(ds)∩L1(ds).

Then it holds that

W [F (X + h)] = W [F (X)E(h; X)] (2)

for any non-negative F∞

-measurable functional F where

E(h; X) = exp

(∫

∞

0

f(s)dXs −1

2

∫

∞

0

f(s)2ds

)

. (3)

References

[1] T. Funaki, Y. Hariya, and M. Yor. Wiener integrals for centered Besseland related processes. II. ALEA Lat. Am. J. Probab. Math. Stat., 1:225–240 (electronic), 2006.

[2] J. Najnudel, B. Roynette, and M. Yor. A remarkable σ-finite measure onC(R+,R) related to many Brownian penalisations. C. R. Math. Acad.

Sci. Paris, 345(8):459–466, 2007.

[3] J. Najnudel, B. Roynette, and M. Yor. A global view of Brownian pe-

nalisations, volume 19 of MSJ Memoirs. Mathematical Society of Japan,Tokyo, 2009.

[4] K. Yano. Cameron–Martin formula for the σ-finite measure unifyingBrownian penalisations. preprint, arXiv:0909.5132, 2009.

[5] K. Yano. Wiener integral for the coordinate process under the σ-finitemeasure unifying brownian penalisations. preprint, arXiv:0909.5130,2009.

On the behavior of the population density of branching

random walks

Makoto NakashimaDivision of Mathematics, Graduate School of Science, Kyoto University

We consider the branching random walk in d dimensional integer lattice with time-spacei.i.d. offspring distributions. Then, the normalization of total population is a non-negativemartingale and it converges to a certain random variable almost surely. Moreover, thefollowing phase transition occurs. If the environment is not too random, then the growthrate of the total population is the same as the one of its expectation (the regular growthphase). On the other hand, if the environment is random enough, then the one of the totalpopulation is slower than the one of its expectation (the slow growth phase). We will lookat the behavior of the population density in each phase.

1

Asymptotic behavior of the gyration radius for long-rangeself-avoiding walk and long-range oriented percolation

Akira Sakai1

Let α > 0 and suppose that the 1-step distribution D for random walk on Zd

decays as D(x) ≈ |x|−d−α such that its Fourier transform D(k) ≡ ∑x∈Zd eik·xD(x)

satisfies

1− D(k) = vα|k|α∧2 ×{

1 + O(|k|ε) (α 6= 2),log 1

|k| + O(1) (α = 2),(1)

for some vα ∈ (0,∞) and ε > 0. The following long-range Kac potential, for anyL ∈ [1,∞), satisfies the above property with vα = O(Lα∧2) [3]:

D(x) =h(y/L)∑

y∈Zd h(y/L)(x ∈ Zd), (2)

where h(x) ≡ |x|−d−α(1 + O(|x|ε)) is a rotation-invariant function on Rd.Let ϕt(x) denote the two-point functions for random walk and self-avoiding walk

whose 1-step distribution is given by the above D and for oriented percolation onZd×Z+ whose bond-occupation probability for each bond ((u, s), (v, s+1)) is givenby pD(v − u), independently of s ∈ Z+, where p ≥ 0 is the percolation parameter.More precisely,

ϕt(x) =

∑ω:o→x|ω|=t

t∏

s=1

D(ωs − ωs−1) (RW),

∑ω:o→x|ω|=t

t∏

s=1

D(ωs − ωs−1)∏

0≤i<j≤t

(1− δωi,ωj ) (SAW),

Pp

((o, 0) −→ (x, t)

)(OP),

(3)

where∏

0≤i<j≤t(1− δωi,ωj ) is the self-avoiding constraint on ω, and {(o, 0) → (x, t)}is the event that either (x, t) = (o, 0) or there is a consecutive sequence of occupiedbonds from (o, 0) to (x, t) in the time-increasing direction. The order-r gyrationradius ξ(r)

t , defined as

ξ(r)

t =(∑

x∈Zd |x|rϕt(x)∑x∈Zd ϕt(x)

)1/r

, (4)

represents a typical end-to-end distance of a linear structure of length t or a typicalspatial size of a cluster at time t. It has been expected (and is certainly true forrandom walk in any dimension) that, above the common upper-critical dimensiondc = 2(α ∧ 2) for self-avoiding walk and oriented percolation, for every r ∈ (0, α),

ξ(r)

t =

{O(t

1α∧2 ) (α 6= 2),

O(√

t log t) (α = 2).(5)

The conjecture was proved to be affirmative for self-avoiding walk, but only for smallr < α ∧ 2 [4].

In my recent joint work with L.-C. Chen [3], we have proved the following sharpasymptotics:

1Creative Research Institution “Sousei”, Hokkaido University. sakai@cris.hokudai.ac.jp

1

Theorem 1 ([3]). Consider the aforementioned three long-range models. For ran-dom walk in any dimension with any L, and for self-avoiding walk and critical/sub-critical oriented percolation for d > 2(α ∧ 2) with L À 1, the following holds forevery r ∈ (0, α): there are constants C1, C2 = 1 + O(L−d) (C1 = C2 = 1 for randomwalk) and ε > 0 such that, as m ↗ mc,

∞∑

t=0

∑

x∈Zd

|x1|rϕt(x) mn =2 sin rπ

α∨2

(α ∧ 2) sin rπα

Γ(r + 1)C1(C2vα)

rα∧2

(1− mmc

)1+ rα∧2

×

1 + O((1− m

mc)ε

)(α 6= 2),

(log 1√

1− mmc

)r/2+ O(1) (α = 2).

(6)

where mc is the radius of convergence for the sequence∑

x∈Zd ϕt(x).

In fact, the above C1, C2 are the following model-dependent constants [1, 2, 4]:

∑

x∈Zd

ϕt(x) ∼t↑∞

C1m−tc ,

∑x∈Zd eikt·xϕt(x)∑

x∈Zd ϕt(x)∼

t↑∞e−C2|k|α∧2

. (7)

Theorem 2 ([3]). Under the same condition as in Theorem 1,∑

x∈Zd |x1|rϕt(x)∑x∈Zd ϕt(x)

∼t↑∞

2 sin rπα∨2

(α ∧ 2) sin rπα

Γ(r + 1)Γ( r

α∧2 + 1)(C2vα)

rα∧2

×{

tr

α∧2 (α 6= 2),(t log

√t)r/2 (α = 2).

(8)

As far as we notice, even for random walk, the sharp asymptotics in the abovetwo theorems are new. By |x1|r ≤ |x|r ≤ dr/2

∑dj=1 |xj |r and the Zd-symmetry of

the models, Theorem 2 immediately proves the conjecture (5) for all r ∈ (0, α).The proof is based on the derivatives of the lace expansion and the new fractional-

moment analysis for the derivatives of the expansion coefficients, initiated in [2]. Itis worth emphasizing that the same proof applies to finite-range models, for whichα is considered to be infinity.

In the talk, I explain the general framework to treat all three models simultane-ously and show some complex analysis for the derivation of the right constants inthe asymptotics.

References

[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] L.-C. Chen and A. Sakai. Critical behavior and the limit distribution forlong-range oriented percolation. II: Spatial correlation. Probab. Theory Relat.Fields 145 (2009): 435–458.

[3] L.-C. Chen and A. Sakai. Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation. In preparation.

[4] M. Heydenreich. Long-range self-avoiding walk converges to alpha-stable pro-cesses. Preprint: arXiv:0809.4333v1 (2008).

2

Random walks on two dimensionalcontinuum percolation clusters

JUN MISUMI

Graduate School of Mathematical Sciences, The University of Tokyo

We consider random graphs defined by a continuum percolation. The vertex set is given by thePoisson points in Rd, and the edge set is determined by the random radii of the spheres centered ateach points. Here, the radii of the spheres may not be bounded. When a connected subgraph withinfinite size exists, we consider the simple random walk on it. For d = 2, under some conditionson the moment of the radii, we have the recurrence of the random walk.

1

On the maximum of Dyson Brownian motion

T. Sasamoto

Dyson’s Brownian motion is a stochastric process described by the SDE,

dXi = dBi +β

2

∑1≤j≤m

j 6=i

dt

Xi −Xj

, 1 ≤ i ≤ m

where Bi, 1 ≤ i ≤ m are independent 1D Brownian motions. In this talk, we only considerthe case β = 2. It is known that the initial conditions can be taken to be Xi(0) = 0, 1 ≤i ≤ m and that the process satisfies X1(t) < X2(t) < · · · < Xm(t), ∀t > 0. This canbe interpreted as a system of m Brownian particles conditioned to never collide with eachother.

When m = 1, this is simply a single Brownian motion. It is well known that max0≤s≤t B(s)has the same distribution as the reflected BM.

One can also define Dyson’s Brownian motion of type C, X(C), as a system of mBrownian particles conditioned to never collide with each other or the wall and that oftype D, X(D), as as a system of m reflected Brownian particles conditioned to never collidewith each other. Then we show

Theorem. Let X and X(C), X(D) start from the origin. Then for each fixed t ≥ 0, onehas

sup0≤s≤t

Xn(s)d=

{X

(C)m (t), for n = 2m,

X(D)m (t), for n = 2m− 1.

This is a multi-dimensional generalization of the above mentioned relation between maxi-mum of BM and BM with a reflecting boundary.

Originally this relation was anticipated from a consideration of the diffucion scalinglimit in the TASEP with two speeds but can be shown without refering to TASEP. Theidea of the proof is to introduce two new new processesZ, Y and show

max X = max Z = Y = X(C,D).

.The work is based on a collaboration with A. Borodin, P. L. Ferrari, M. Prahofer, J.

Warren.

References[1] A. Borodin, P. L. Ferrari, M. Prahofer, T. Sasamoto, J. Warren, arXiv:0905.3989.[2] A. Borodin, P. L. Ferrari, T. Sasamoto, arXiv:0904.4655, to appear in J. Stat. Phys.

Localization of inhomogeneous coined quantum walks on the line

Norio Konno (Yokohama National University)

As a quantum counterpart of the classical random walk, the quantum walk (QW) has recently attractedmuch attention for various fields. There are two types of QWs. One is the discrete-time (or coined) walk andthe other is the continuous-time one. The discrete-time QW in one dimension (1D) was intensively studiedby Ambainis et al. [1]. One of the most striking properties of the 1D QW is the spreading property of thewalker. The standard deviation of the position grows linearly in time, quadratically faster than classicalrandom walk. The review and book on QWs are Kempe [2], Kendon [3], Venegas-Andraca [4], Konno [5],for examples. In this talk we focus on discrete-time case. The model mainly considered here [6] is a space-inhomogeneous two-state 1D QW. The two-state corresponds to left and right chiralities. Let pn(0) denotethe probability that the walker returns to the origin at time n. The model is said to exhibit localization iflim supn→∞ pn(0) > 0. The homogeneous two-state 1D QW except a trivial case (see [1, 7, 8], for examples)and a class of inhomogeneous two-state 1D QWs [9] do not exhibit localization. The decay order of pn(0)is closely related to the recurrence. As for the recurrence property of QWs, see Stefanak et al. [10, 11, 12].Localization of the homogeneous model was shown for a three-sate 1D QW in [13], a four-state 1D QW in[14], and a multi-state QW on tree in [15]. Mackay et al. [16] and Tregenna et al. [17] found numerically thata homogeneous 2D QW exhibits localization. Inui et al. [18] and Watabe et al. [19] showed the phenomenon.In higher dimensions, a d-dimensional homogeneous tensor-product coin model does not exhibit localization[11]. Oka et al. [20] analyzed localization of a two-state QW on a semi-infinite 1D lattice, which is closelyrelated to the Landau-Zener transition dynamics. Through numerical simulations, Buerschaper and Burnett[21] and Wojcik et al. [22] reported that the dynamics of the two-state 1D QWs exhibits from dynamicallocalization, spreading more slowly than in the classical case, to linear diffusion like the homogeneous two-state 1D QW as the period of the perturbation is varied. Linden and Sharam [23] investigated a similarinhomogeneous two-state 1D QW where the inhomogeneity is periodic in position.

An interesting question is whether localization emerges even for a simpler inhomogeneous two-state 1DQW compared with the previous models. We give an affirmative answer to the question (Theorem 1 in thisabstract). Our very non-trivial result could be useful for quantum information processing by controllingthe spreading of the walker. The model has a simple inhomogeneity at the origin depending on a singleparameter. Despite of this rather small deviation from the homogeneous Hadamard walk, the presentedmodel differs significantly from the previous ones. In particular, it is shown that it leads to localization. Wedescribe the above mentioned result more precisely. For a given sequence {ωx : x ∈ Z} with ωx ∈ [0, 2π), weconsider the inhomogeneous 1D QW given by

Ux = Ux(ωx) =1√2

[1 eiωx

e−iωx −1

],

where the subscript x indicates the location. In particular, we concentrate on a simple inhomogeneous modeldepending only on a one-parameter ω ∈ [0, 2π) as follows: U0 = U0(ω), and Ux = Ux(0) if x �= 0. So whenω �= 0, our model is homogeneous except the origin. If ω = 0, then this model becomes homogeneous and isequivalent to the Hadamard walk. For our inhomogeneous two-state 1D QW with the parameter ω ∈ [0, 2π),we have

Theorem 1 ([6])

limn→∞ p2n(0) =

(2(1 − cosω)3 − 2 cosω

)2

=: c(ω).

As for the above limit c(ω), see Fig. 1. If the model is inhomogeneous, i.e., ω ∈ (0, 2π), then it exhibitslocalization, i.e., c(ω) > 0. In addition to Theorem 1, we also discuss some results on the related models([9, 24], for instance).

1

1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

ω

c(ω)

Figure 1: The plot of c(ω)

References

[1] Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceed-ings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 37–49 (2001)

[2] Kempe, J.: Quantum random walks - an introductory overview. Contemporary Physics 44, 307–327 (2003)

[3] Kendon, V.: Decoherence in quantum walks - a review. Math. Struct. in Comp. Sci. 17, 1169–1220 (2007)

[4] Venegas-Andraca, S. E.: Quantum Walks for Computer Scientists. Morgan and Claypool (2008)

[5] Konno, N.: Quantum Walks. In: Quantum Potential Theory, Franz, U., and Schurmann, M., Eds., LectureNotes in Mathematics: Vol. 1954, pp. 309–452, Springer-Verlag, Heidelberg (2008)

[6] Konno, N.: Localization of an inhomogeneous discrete-time quantum walk on the line. Quantum Inf. Proc. (inpress), arXiv:0908.2213

[7] Bressler, A., Pemantle, R.: Quantum random walks in one dimension via generating functions. Discrete Math-ematics and Theoretical Computer Science (DMTCS) Proceedings of the 2007 Conference on Analysis of Algo-rithms, 403–414 (2007)

[8] Cantero, M. J., Grunbaum, F. A., Moral, L., Velazquez, L.: Matrix valued Szego polynomials and quantumrandom walks. arXiv:0901.2244

[9] Konno, N.: One-dimensional discrete-time quantum walks on random environments. Quantum Inf. Proc. 8,387–399 (2009), arXiv:0904.0392

[10] Stefanak, M., Jex, I., Kiss, T.: Recurrence and Polya number of quantum walks. Phys. Rev. Lett. 100, 020501(2008)

[11] Stefanak, M., Kiss, T., Jex, I.: Recurrence properties of unbiased coined quantum walks on infinite d-dimensionallattices. Phys. Rev. A 78, 032306 (2008)

[12] Stefanak, M., Kiss, T., Jex, I.: Recurrence of biased quantum walks on a line. New J. Phys. 11, 043027 (2009)

[13] Inui, N., Konno, N., Segawa, E.: One-dimensional three-state quantum walk. Phys. Rev. E 72, 056112 (2005)

[14] Inui, N., Konno, N.: Localization of multi-state quantum walk in one dimension. Physica A 353, 133–144 (2005)

[15] Chisaki, K., Hamada, M., Konno, N., Segawa, E.: Limit theorems for discrete-time quantum walks on trees.Interdisciplinary Information Sciences (in press), arXiv:0903.4508

[16] Mackay, T. D., Bartlett, S. D., Stephenson, L. T., Sanders, B. C.: Quantum walks in higher dimensions. J. Phys.A: Math. Gen. 35, 2745–2753 (2002)

[17] Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states.New J. Phys. 5, 83 (2003)

[18] Inui, N., Konishi, Y., Konno, N.: Localization of two-dimensional quantum walks. Phys. Rev. A 69, 052323(2004)

[19] Watabe, K., Kobayashi, N., Katori, M., Konno, N.: Limit distributions of two-dimensional quantum walks.Phys. Rev. A 77, 062331 (2008)

[20] Oka, T., Konno, N., Arita, R., Aoki, H.: Breakdown of an electric-field driven system: a mapping to a quantumwalk. Phys. Rev. Lett. 94, 100602 (2005)

[21] Buerschaper, O., Burnett, K.: Stroboscopic quantum walks. quant-ph/0406039

[22] Wojcik, A., �Luczak, T., Kurzynski, P., Grudka, A., Bednarska, M.: Quasiperiodic dynamics of a quantum walkon the line. Phys. Rev. Lett. 93, 180601 (2004)

[23] Linden, N., Sharam, J.: Inhomogeneous quantum walks. arXiv:0906.3692

[24] Bourget, O., Howland, J. S., Joye, A.: Spectral analysis of unitary band matrices. Commun. Math. Phys. 234,191–227 (2003)

2

Exa t partition fun tion of the zero-range pro essand expe tation values in the thermodynami limitKANAI, MasahiroGraduate S hool of Mathemati al S ien es, The University of Tokyokanai�ms.u-tokyo.a .jpIn this talk, we give the partition fun tion for the zero-range pro ess inthe nonequilibrium steady state using hypergeometri fun tions, and then al ulate some expe tation values. All the al ulations are done for an ar-bitrary �nite system, and thereafter we onsider the thermodynami limit.From the viewpoint of appli ation (e.g. traÆ ow, and pedestrian dynam-i s), we also onsider parallel updating, i.e., all sites are updated in on ert.The parallel updating seems to give rise to a global intera tion, but never-theless the partition fun tion an be obtained in the same manner above.

HUGE-SCALE MOLECULAR DYNAMICS SIMULATIONON GAS-LIQUID MULTI-PHASE FLOW

HIROSHI WATANABE

Gas-liquid multi-phase simulations are achieved by the moleculardynamics method. The full-particle simulation allows us to investigatetransportation behavior through phase interfaces from the microscopicpoint of view. In this talk, we report the energy-transportation andnuclei in the boiling flow.

1

An alternative condition for stochastic domination

KENSHI HOSAKA

Research Organization of Social Science, Ritsumeikan University

In this talk, we will propose an alternative condition for stochastic dom-ination. This condition differs in an essential way from the strong likelihoodratio property. We also show an example, which satisfies the new condition,however does not satisfy the strong likelihood ratio property.

KEY WORDS : Stochastic domination, Strong likelihood ratio property

1

2D Ising percolation near critical external fields

Masato Takei

(Osaka Electro-Communication University)

This talk is based on the joint work with Yasunari Higuchi (Kobe University)

and Yu Zhang (University of Colorado).

We consider the percolation problem for Ising model on the two-dimensional

square lattice Z2. For T > Tc and h ∈ R, there exists a unique Gibbs measure

µT,h. The (+)-cluster containing the origin is denoted by C+0 . For each T > 0,

the critical external field is defined by

hc(T ) := inf{h : µT,h(#C+0 = ∞) > 0}.

It is known (see e.g. [1]) that hc(T ) > 0 whenever T > Tc. Hereafter we fix a

T > Tc, and abbreviate µT,h to µh and hc(T ) to hc, respectively. The expectation

under µh is denoted by Eh.

The following power laws are widely believed to hold:

I Percolation probability:

θ(h) := µh(#C+0 = ∞) ≈ (h − hc)

β as h ↘ hc.

I Mean cluster size:

χ(h) := Eh[#C+0 : #C+

0 < ∞] ≈ |h − hc|−γ as h → hc.

I Correlation length:

ξ(h) :=

[1

χ(h)

∑v∈Z2

|v|2µh

(O

+↔ v, #C+0 < ∞

)]1/2

≈ |h − hc|−ν as h → hc.

＊ For S(n) = [−n, n]2, we define

L(h, ε0) :=

{min

{n : µh

(LS of S(n)

+↔ RS of S(n))≥ 1 − ε0

}(h > hc),

min{n : µh

(LS of S(n)

+↔ RS of S(n))≤ ε0

}(h < hc).

Then ξ(h) ³ L(h, ε0).

I One-arm probability: πhc(n) := µhc

(O

+↔ ∂S(n))≈ n−1/δr .

I Connectivity function: τhc(n) := µhc{O+↔ (n, 0)} ≈ n−η.

We extend some of Kesten’s scaling relations [2] for 2D Bernoulli percolation

to our case. Our main result is as follows.

Theorem. 1) If

πhc(n) ≈ n−1/δr or τhc(n) ≈ n−η (1)

holds, then both statements as well as

µhc{#C+0 ≥ n} ≈ n−1/δ

hold. Moreover,

θ(h) ³ L(h, ε0)−1/δr = L(h, ε0)

−2/(δ+1)

and

η =2

δr

, δ = 2δr − 1 =4

η− 1.

In addition, if

ξ(h) ≈ |h − hc|−ν , (2)

then β =2ν

δ + 1.

2) Assume that (1) and (2) hold.

• For t ≥ 2,Eh[(#C+

0 )t : #C+0 < ∞]

Eh[(#C+0 )t−1 : #C+

0 < ∞]≈ ξ(h)2πhc(ξ(h)),

• For t > 0,

[1

χ(h)

∑v∈Z2

|v|tµh

(O

+↔ v, #C+0 < ∞

)]1/t

³ ξ(h).

Moreover,

• For k ≥ 2,Eh[(#C+

0 )k : #C+0 < ∞]

Eh[(#C+0 )k−1 : #C+

0 < ∞]≈ |h − hc|−∆k ,

• For k ≥ 1,

[1

χ(h)

∑v∈Z2

|v|kµh

(O

+↔ v, #C+0 < ∞

)]1/k

≈ |h − hc|−νk ,

and

γ = 2νδ − 1

δ + 1, ∆k = 2ν

δ

δ + 1(k ≥ 2), νk = ν (k ≥ 1).

References

[1] Higuchi, Y. : Sugaku Expositions 10, 143–158, (1997).

[2] Kesten, H. : Probab. Theory Related Fields 73, 369–394, (1986); IMA Vol.

Math. Appl. 8, 203–212, (1987); Comm. Math. Phys. 109, 109–156, (1987).

Power Law Fluids with Random Forcing

Nobuo Yoshida1

This is a joint work with Yutaka Terasawa (Tohoku Univ).We consider viscous, imcompressible fluid subject to a random perturbation. As a part ofidealization, the container of the fluid is supposed to be the torus Td = (R/Z)d ∼= [0, 1]d. Weassume that the extra stress tensor

τ(v) : Td → Rd ⊗ Rd

depends on the velocity field v : Td → Rd via the following power law: for ν > 0 (the kinematicviscosity), p > 1 (the power index) and a constant κ > 0,

τ(v) = 2ν(κ+ |e(v)|2)p−22 e(v) with e(v) =

(∂ivj + ∂jvi

2

). (0.1)

The linearly dependent case p = 2 is the Newtonian fluid, which is described by the Navier-Stokes equation, the special case of (0.2)–(0.3) below. On the other hand, both the shearthinning (p < 2) and the shear thickenning (p > 2) cases are considered in many fields inscience and engneering (e.g., suspension of cement in water, paper pulp in water, latex paint,blood flow, earth’s mantle convection, cf. [MNRR96, Wi09]).

Given an initial velocity u0 : Td → Rd, the dynamics of the fluid is described by the followingSPDE:

div u = 0, (0.2)

∂tui +d∑

j=1

uj∂jui = −∂iΠ +d∑

j=1

∂jτij(u) + ∂tWi, i = 1, ..., d. (0.3)

The unknown process in the SPDE are the velocity field u = u(t, x) = (ui(t, x))di=1 and the

pressure Π = Π(t, x). The Brownian motion W = W (t, x) = (Wi(t, x))di=1 with values in

L2(Td → Rd) is added as the random perturbation. Physical interpretation of (0.2) and (0.3)are the conservation laws of the mass and the momentum, repectively. Note that the SPDE(0.2)–(0.3) for the case p = 2 is the stochastic Navier-Stokes equation.

For certain ranges of the power p (e.g.,p ∈ (3/2,∞) for d = 2, p ∈ (9/5, 6) for d = 3),we construct a weak solution to the SPDE (0.2)–(0.3) globally in time. For d = 2, 3, thisgeneralizes the known result for the stochastic Navier-Stokes equation ([Fl08] and referencestherein). Also, by considering the degenerate noise, our result recovers the PDE result for p 6= 2[MNRR96].

References

[Fl08] Flandoli, Franco : An introduction to 3D stochastic fluid dynamics. SPDE in hydrodynamic:recent progress and prospects, 51–150, Lecture Notes in Math., 1942, Springer, Berlin, 2008.

[MNRR96] Malek, J.; Necas, J.; Rokyta, M.; R◦uzicka, M.: Weak and measure-valued solutions to

evolutionary PDEs. Applied Mathematics and Mathematical Computation, 13. Chapman & Hall,London, 1996. xii+317 pp. ISBN: 0-412-57750-X

[Wi09] Non-Newtonian fluid–Wikipedia.

1Division of Mathematics Graduate School of Science Kyoto University, Kyoto 606-8502, Japan. email:nobuo@math.kyoto-u.ac.jp URL: http://www.math.kyoto-u.ac.jp/enobuo/

1

CONFINEMENT OF THE TWO DIMENSIONALDISCRETE GAUSSIAN FREE FIELD BETWEEN TWO

HARD WALLS

HIRONOBU SAKAGAWA

We consider the two dimensional discrete Gaussian free field con-fined between two hard walls. We show that the field becomes massiveand identify the precise asymptotic behavior of the mass and the vari-ance of the field as the height of the wall goes to infinity. By largefluctuation of the field, asymptotic behaviors of these quantities in thetwo dimensional case differ greatly from those of the higher dimensionalcase.

Spectral gap for multi-species exclusion processesYukio Nagahata1

Department of Mathematical ScienceGraduate School of Engineering Science

Osaka University, Toyonaka, 560–8531, JAPANMakiko Sasada2

Graduate School of Mathematical Science,The University of Tokyo, Komaba, Tokyo, 153–8914, JAPAN

Let Λn be the d-dimensional cube with width 2n+1, centered at the origin.Fix r ∈ N such that r ≥ 2, we define Σn := {0, 1, 2, . . . , r}Λn and Σn,k :={η ∈ Σn;

∑x∈Λn

1(ηx = i) = ki for all 0 ≤ i ≤ r} for k = (k0, k1, . . . , kr)with

∑ri=0 ki = |Λn|. For η ∈ Σn and x, y ∈ Λn, we define the configuration

ηx,y ∈ Σn by

(ηx,y)z =

ηy if z = xηx if z = yηz otherwise,

and the operator πx,y by

πx,yf(η) = f(ηx,y) − f(η).

Let {pi}0≤i≤r be finite range, translation invariant, irreducible symmetrictransition probabilities on Zd.

Given a function g : {0, 1, . . . , r} → R such that g(i) > 0, and pi, wedefine the infinitesimal generator Ln by

(Lnf)(η) :=∑

x,y∈Λn

pηx(x, y)g(ηx)1(ηy = 0)πx,yf(η)

for f : Σn → R. The process is regarded as a gas of multi-species particles.The site x is occupied by an i-th particle if ηx = i for 1 ≤ i ≤ r and vacantif ηx = 0. An i-th particle at site x jumps to site y at rate g(i)pi(x, y) if it isvacant.

Assumption 0.1 There exists n0 such that for each n ≥ n0 and k =(k0, k1, . . . , kr) with

∑ri=1 ki = |Λn| and k0 = 0, Σn,k is an ergodic component

of Ln.

1E-mail address: nagahata@sigmath.es.osaka-u.ac.jp2E-mail address: sasada@ms.u-tokyo.ac.jp

1

Assumption 0.1 should be rewritten by the condition on the range of {pi}1≤i≤r.It seems difficult to give an equivalent condition, hence we give a reasonablesufficient condition:

Lemma 0.2 Suppose that the dimension d ≥ 2 and pi(x) > 0 for all ∥x∥1 =1 and for all 1 ≤ i ≤ r, or pi(x) > 0 for all ∥x∥1 = 1 and for all 1 ≤ i ≤ r andthere exists l = (l1, l2, . . . , lr) ∈ (Zd)r such that ∥li∥1 ≥ 2, pi(0, li) > 0 for alli, where ∥x∥1 =

∑di=1 |xi| for x = (x1, x2, . . . , xd) ∈ Zd. Then Assumption

0.1 holds true.

Let νn,k be the uniform measure on Σn,k. Due to Assumption 0.1, therestriction of Ln on Σn,k which is denoted by Ln,k is symmetric with respectto νn,k. In view of this symmetry the second smallest eigenvalue of −Ln,k isgiven by

λ = λ(n, k) := inf

{Eνn,k

[f(−Ln)f ]

Eνn,k[f 2]

∣∣∣∣∣ Eνn,k[f ] = 0

}.

We call λ the spectral gap of Ln,k.

Theorem 0.3 Suppose that the number of species of particles is at leasttwo, i.e., there are 1 ≤ i < j ≤ r such that ki, kj > 0. Then we have

λ(n, k) ≃ ρ0

n2,

where ρ0 = k0

|Λn| and f ≃ g means that there exist positive constants C,C ′,

which do not depend on n nor k, such that Cf ≤ g ≤ C ′f .

Remark 0.4 Suppose that the number of species of particles is one. Thenthis gives a symmetric simple exclusion process. Hence we have

λ(n, k) ≃ 1

n2,

i.e., it does not depend on the density of vacant site.

2

CONVERGENCE OF DISCRETE MARKOV CHAINSTO JUMP PROCESSES

TAKASHI KUMAGAI

In this talk, we will discuss general criteria on tightness and weakconvergence of discrete Markov chains to symmetric jump processes onmetric measure spaces under mild conditions. As an application, weinvestigate convergence of Markov chains with random conductancesto symmetric jump processes. This is a on-going join work with Z.Q.Chen (Seattle) and P. Kim (Seoul).

1

Ginibre-type determinantal processes

Tomoyuki Shirai (Kyushu University) ∗

We consider the Landau Hamiltonian on L2(R2), which is the Schrodingeroperator with a constant magnetic field in R2. It is known that its spectrumconsists of only eigenvalues with infinite multiplicity, which is so-called theLandau levels. The eigenspace corresponding to the lowest eigenvalues (thefirst Landau level) can be identified with the space of L2-entire functionswith respect to the complex gaussian measure, and the other eigenspacescorresponding to higher Landau levels are obtained from the first one byapplying a creation operator repeatedly. Determinantal point processes canbe attached to closed subspaces of the space of square integrable functions.For instance, the determinantal process associated with the first Landau levelis known as the Ginibre point process on C, which is the eigenvalue processof certain complex Gaussian matrix ensemble. In this talk, I will discuss thedeterminantal processes attached to higher Landau levels.

∗VIIIth workshop on Stochastic Analysis on Large Scale Interacting Systems,2009/10/7～10/9 at Komaba, the University of Tokyo

1

Hydrodynamic limit and fluctuations for

an evolutional model of two-dimensional Young diagrams

Makiko Sasada (University of Tokyo)

We construct dynamics of two-dimensional Young diagrams, which are naturally as-sociated with their grandcanonical ensembles and show that, as the averaged size of thediagrams diverges, the corresponding height variable converges to a solution of a certainnon-linear partial differential equation under a proper hydrodynamic scaling. We discussboth uniform and restricted uniform statistics for the Young diagrams. Also, we studythe corresponding dynamic fluctuation problem in a non-equilibrium situation.

To each partition p = {p1 ≥ p2 ≥ · · · ≥ pj ≥ 1} of a positive integer n by positiveintegers {pi}j

i=1 (i.e., n =∑j

i=1 pi), the height function of the Young diagram is definedby ψp(u) =

∑ji=1 1{u<pi} for u ≥ 0. For each fixed n, the uniform statistics (U-statistics

in short) µnU assigns an equal probability to each of possible partitions p of n, i.e., to the

Young diagrams of area n. The restricted uniform statistics (RU-statistics in short) µnR

also assigns an equal probability, but restricting to the partitions satisfying q = {q1 >q2 > · · · > qj ≥ 1}. Grandcanonical ensembles µε

U and µεR with parameter 0 < ε < 1

are defined by superposing the canonical ensembles. Vershik [1] proved that, under thecanonical U- and RU- statistics µN2

U and µN2

R (with n = N2), the law of large numbersholds as N →∞ for the scaled height variable

(1) ψNp (u) :=

1N

ψp(Nu), u ≥ 0,

and for ψNq (u) defined similarly, and the limit shapes ψU and ψR are given by

(2) ψU (u) = − 1α

log(1− e−αu

)and ψR(u) =

1β

log(1 + e−βu

), u ≥ 0,

with α = π/√

6 and β = π/√

12, respectively. These results can be extended to thecorresponding grandcanonical ensembles µε

U and µεR, if the averaged size of the diagrams

is N2 under these measures.The purpose of our talk is to study and extend these results from a dynamical point

of view. First, we construct dynamics of two-dimensional Young diagrams, which have µεU

and µεR as their invariant measures, respectively. Let pt ≡ pε

t = (pi(t))i∈N be the Markovprocess defined by means of the infinitesimal generator Lε,U :

(3) Lε,Uf(p) =∑

i∈N

[ε1{pi−1>pi}{f(pi,+)− f(p)}+ 1{pi>pi+1}{f(pi,−)− f(p)}],

where

(4) pi,±j =

{pj if j 6= i,

pi ± 1 if j = i.

jointly with T. Funaki

1

In (3), we regard p0 = ∞. Let qt ≡ qεt = (qi(t))i∈N be the Markov process with the

infinitesimal generator Lε,R:

(5) Lε,Rf(q) =∑

i∈N

[ε1{qi−1>qi+1}{f(qi,+)− f(q)}+ 1{qi>qi+1+1 or qi=1}{f(qi,−)− f(q)}],

where qi,± are defined by the formula (4) and we regard q0 = ∞.Under the diffusive scaling in space and time and choosing the parameter ε = ε(N)

of the grandcanonical ensembles such that the averaged size of the Young diagrams isN2, we derive the hydrodynamic equations in the limit and show that the Vershik curvesdefined by (2) are actually stationary solutions to the limiting non-linear partial differentialequations in both cases. Denote by ψN

p (t, u) := 1N ψptN2 (Nu) the scaled height variable

and define ψNq (t, u) similarly.

Theorem 1. (U-case) If ψNp (0, u) converges to a function ψ0(u) in probability as N →∞,

then ψNp (t, u) converges to ψU (t, u) in probability where the limit ψU (t, u) is the solution

of the non-linear partial differential equation (PDE):

(6)

∂tψ = {ψ′/(1− ψ′)}′ + αψ′/(1− ψ′), u > 0,

ψ(0, ·) = ψ0(·),ψ(t, 0+) = ∞, ψ(t,∞) = 0,

where ∂tψ = ∂ψ/∂t, ψ′ = ∂ψ/∂u(< 0).

Theorem 2. (RU-case) If ψNq (0, u) converges to a function ψ0(u) in probability as N →

∞, then ψNq (t, u) converges to ψR(t, u) in probability where the limit ψR(t, u) is the solution

of the non-linear partial differential equation (PDE):

(7)

∂tψ = ψ′′ + βψ′(1 + ψ′), u > 0,

ψ(0, ·) = ψ0(·),ψ′(t, 0+) = −1

2, ψ(t,∞) = 0.

Next, we consider the fluctuations of ψNp (t, u) and ψN

q (t, u) around their limits, re-spectively:

ΨNp (t, u) :=

√N

(ψN

p (t, u)− ψU (t, u)), ΨN

q (t, u) :=√

N(ψN

q (t, u)− ψR(t, u)).

By formal computations, we conjecture that ΨNp (t, u) and ΨN

q (t, u) weakly converge to thesolutions of the following stochastic partial differential equations respectively:

dΨ(t, u) =(( Ψ′(t, u)

(1 + ρU (t, u))2)′ + α

Ψ′(t, u)(1 + ρU (t, u))2

)dt +

√2ρU (t, u)

1 + ρU (t, u)dW (t, u)

for ΨNp (t, u) and

dΨ(t, u) =(Ψ′′(t, u) + β(1− 2ρR(t, u))Ψ′(t, u)

)dt +

√2ρR(t, u)(1− ρR(t, u))dW (t, u),

for ΨNq (t, u) where ρU (t, u) = −∂uψU (t, u), ρR(t, u) = −∂uψR(t, u) and W (t, u) is the

space-time white noise on [0, T ]× R+.

References

[1] A. Vershik, Statistical mechanics of combinatorial partitions and their limit shapes,Func. Anal. Appl., 30 (1996), 90–105.

2

METASTABILITY OF REVERSIBLE CONDENSEDZERO RANGE PROCESSES ON A FINITE SET

CLAUDIO LANDIM

Let r : S × S → R+ be the jump rates of a irreducible random walkon a finite set S, reversible with respect to some probability measurem. For α > 2, let g : N → R+ be given by g(0) = 0, g(1) = 1,g(k) = (k/k− 1)α, k ≥ 2. Consider a zero-range process on S in whicha particle jumps from a site x, occupied by k particles, to a site y atrate g(k)r(x, y). Let N stand for the total number of particles. In thestationary state, as N ↑ ∞, all particles but a finite number accumulateon one single site. We show in talk that in the time scale N1+α thesite which concentrates almost all particles evolves as a random walkon S whose transition rates R(x, y) are a multiple of the capacities ofthe underlying random walk: R(x, y) = C0CapS(x, y).

1

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15 Lucy

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16 Nonkoi

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17ＥＬ

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|~~~~~~~~~

18 Sonso「o

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